Topic 15 - Probabilities
Test ID: 8450917
Question #1 of 21
Question ID: 438710
A company says that whether its earnings increase depends on whether it increased its dividends. From this we know: ✗ A) P(dividend increase or earnings increase) = P(both dividend and earnings increase). ✗ B) P(both dividend increase and earnings increase) = P(dividend increase). ✗ C) P(dividend increase | earnings increase) is not equal to P(earnings increase). ✓ D) P(earnings increase | dividend increase) is not equal to P(earnings increase). Explanation If two events A and B are dependent, then the conditional probabilities of P(A|B) and P(B|A) will not equal their respective unconditional probabilities (of P(A) and P(B), respectively). The other choices may or may not occur, e.g., P(A | B) = P(B) is possible but not necessary.
Question #2 of 21
Question ID: 438722
A conditional expectation involves: ✗ A) determining the expected joint probability. ✗ B) estimating the skewness. ✗ C) calculating the conditional variance. ✓ D) refining a forecast because of the occurrence of some other event. Explanation Conditional expected values are contingent upon the occurrence of some other event. The expectation changes as new information is revealed.
Question #3 of 21
Question ID: 438709
An investor is choosing one of twenty securities. Ten of the securities are stocks and ten are bonds. Four of the ten stocks were issued by utilities, the other six were issued by industrial firms. Two of the ten bonds were issued by utilities, the other eight were issued by industrial firms. If the investor chooses a security at random, the probability that it is a bond or a security issued by an industrial firm is: ✓ A) 0.80. ✗ B) 0.50. ✗ C) 0.60. ✗ D) 0.70. Explanation Let B represent the set of bonds and I the set of industrial firms. The desired probability is the probability of the union of sets B and I, P(B I). According to the theorems of probability, P(B I)=P(B)+P(I)-P(B∩I), where P(B) is the probability that a security is a
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bond=P(B)=10/20, P(I) is the probability that a security was issued by an industrial firm=P(I)=14/20, and P(B∩I) is the probability that a security is both a bond and issued by an industrial firm=P(B∩I)=8/20. 10/20+14/20-8/20=16/20=0.80.
Question #4 of 21
Question ID: 438725
A joint probability of A and B must always be: ✓ A) less than or equal to the conditional probability of A given B. ✗ B) greater than or equal to than the probability of A or B. ✗ C) greater than or equal to the conditional probability of A given B. ✗ D) less than the probability of A and the probability of B. Explanation By the formula for joint probability: P(AB)=P(A|B) × P(B), since P(B) ≤ 1, then P(AB) ≤P(A|B). None of the other choices must hold.
Question #5 of 21
Question ID: 438723
Dependent random variables are defined as variables where their joint probability is: ✓ A) not equal to the product of their individual probabilities. ✗ B) equal to zero. ✗ C) equal to the product of their individual probabilities. ✗ D) greater than the product of their individual probabilities. Explanation Dependence results between random variables when their joint probabilities are not equal products of individual probabilities. If they are equal, then an independent relationship exists.
Question #6 of 21
Question ID: 438711
Thomas Baynes has applied to both Harvard and Yale. Baynes has determined that the probability of getting into Harvard is 25% and the probability of getting into Yale (his father's alma mater) is 42%. Baynes has also determined that the probability of being accepted at both schools is 2.8%. What is the probability of Baynes being accepted at either Harvard or Yale? ✗ A) 7.7%. ✗ B) 67.0%. ✗ C) 10.5%. ✓ D) 64.2%. Explanation Using the addition rule, the probability of being accepted at Harvard or Yale, is equal to: P(Harvard) + P(Yale) − P(Harvard and Yale) = 0.25 + 0.42 − 0.028 = 0.642 or 64.2%.
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Question #7 of 21
Question ID: 438726
The probability of a new Wal-Mart being built in town is 64%. If Wal-Mart comes to town, the probability of a new Wendy's restaurant being built is 90%. What is the probability of a new Wal-Mart and a new Wendy's restaurant being built? ✗ A) 0.306. ✗ B) 0.675. ✗ C) 0.240. ✓ D) 0.576. Explanation P(AB) = P(A|B) × P(B) The probability of a new Wal-Mart and a new Wendy's is equal to the probability of a new Wendy's "if Wal-Mart" (0.90) times the probability of a new Wal-Mart (0.64). (0.90)(0.64) = 0.576.
Question #8 of 21
Question ID: 438721
A dealer in a casino has rolled a five on a single die three times in a row. What is the probability of her rolling another five on the next roll, assuming it is a fair die? ✗ A) 0.200. ✗ B) 0.500. ✓ C) 0.167. ✗ D) 0.001. Explanation The probability of a value being rolled is 1/6 regardless of the previous value rolled.
Question #9 of 21
Question ID: 438720
X and Y are discrete random variables. The probability that X = 3 is 0.20 and the probability that Y = 4 is 0.30. The probability of observing that X = 3 and Y = 4 concurrently is closest to: ✗ A) 0.06. ✗ B) 0. ✗ C) 0.50. ✓ D) Cannot answer with the information provided. Explanation If you knew that X and Y were independent, you could calculate the probability as 0.20(0.30) = 0.06. Without this knowledge, you would need the joint probability distribution.
Question #10 of 21
Question ID: 438707
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Which of the following statements about probability is most accurate? ✗ A) An outcome is the calculated probability of an event. ✗ B) Out of a sample of 100 widgets 10 were found to be defective, 20 were perfect, and 70 were OK. The probability of picking a perfect widget at random is 29%. ✓ C) An event is a set of one or more possible values of a random variable. ✗ D) A conditional probability is the probability that two or more events will happen concurrently. Explanation Conditional probability is the probability of one event happening given that another event has happened. An outcome is the numerical result associated with a random variable. The probability of picking a perfect widget is 20 / 100 = 0.20 or 20%.
Question #11 of 21
Question ID: 438717
The probabilities that three students will earn an A on an exam are 0.20, 0.25, and 0.30, respectively. If each student's performance is independent of that of the other two students, the probability that all three students will earn an A is closest to: ✓ A) 0.0150. ✗ B) 0.0075. ✗ C) 0.0010. ✗ D) 0.7500. Explanation Since the events are independent, the probability is obtained by multiplying the individual probabilities. Probability of three As = (0.20) × (0.25) × (0.30) = 0.0150.
Question #12 of 21
Question ID: 438718
An investor has an A-rated bond, a BB-rated bond, and a CCC-rated bond where the probabilities of default over the next three years are 4 percent, 12 percent, and 30 percent, respectively. What is the probability that all of these bonds will default in the next three years if the individual default probabilities are independent? ✗ A) 1.44%. ✓ B) 0.14%. ✗ C) 46.00%. ✗ D) 23.00%. Explanation Since the probability of default for each bond is independent, P(A BB CCC) = P(A) × P(BB) × P(CCC) = 0.04 × 0.12 × 0.30 = 0.00144 = 0.14%.
Question #13 of 21
Question ID: 438713
If the outcome of event A is not affected by event B, then events A and B are said to be:
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✗ A) mutually exclusive. ✓ B) statistically independent. ✗ C) conditionally dependent. ✗ D) collectively exhaustive. Explanation If the outcome of one event does not influence the outcome of another, then the events are independent.
Question #14 of 21
Question ID: 438724
If the probability of both a new Wal-Mart and a new Wendy's being built next month is 68% and the probability of a new Wal-Mart being built is 85%, what is the probability of a new Wendy's being built if a new Wal-Mart is built? ✗ A) 0.85. ✓ B) 0.80. ✗ C) 0.70. ✗ D) 0.60. Explanation P(AB) = P(A|B) × P(B) 0.68 / 0.85 = 0.80
Question #15 of 21
Question ID: 438708
If X and Y are independent events, which of the following is most accurate? ✗ A) P(X or Y) = (P(X)) × (P(Y)). ✓ B) P(X | Y) = P(X). ✗ C) X and Y cannot occur together. ✗ D) P(X or Y) = P(X) + P(Y). Explanation Note that events being independent means that they have no influence on each other. It does not necessarily mean that they are mutually exclusive. Accordingly, P(X or Y) = P(X) + P(Y) − P(X and Y). By the definition of independent events, P(X|Y) = P(X).
Question #16 of 21
Question ID: 438706
Let A and B be two mutually exclusive events with P(A) = 0.40 and P(B) = 0.20. Therefore:
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✗ A) P(A and B) = 0.08. ✗ B) P(A or B) = 0.52. ✓ C) P(A and B) = 0. ✗ D) P(B|A) = 0.20. Explanation If the two evens are mutually exclusive, the probability of both ocurring is zero.
Question #17 of 21
Question ID: 438712
The following table summarizes the availability of trucks with air bags and bucket seats at a dealership.
Bucket
No Bucket
seats
Seats
Air Bags
75
50
125
No Air Bags
35
60
95
Total
110
110
220
Total
What is the probability of randomly selecting a truck with air bags and bucket seats? ✓ A) 0.34. ✗ B) 0.57. ✗ C) 0.28. ✗ D) 0.16. Explanation 75 / 220 = 0.34.
Question #18 of 21
Question ID: 438714
The following table summarizes the availability of trucks with air bags and bucket seats at a dealership. Bucket Seats No Bucket Seats
Total
Air Bags
75
50
125
No Air Bags
35
60
95
Total
110
110
220
What is the probability of selecting a truck at random that has either air bags or bucket seats? ✓ A) 73%. ✗ B) 50%. ✗ C) 34%. ✗ D) 107%. Explanation
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The addition rule for probabilities is used to determine the probability of at least one event among two or more events occurring. The probability of each event is added and the joint probability (if the events are not mutually exclusive) is subtracted to arrive at the solution. P(air bags or bucket seats) = P(air bags) + P(bucket seats) − P(air bags and bucket seats) = (125 / 220) + (110 / 220) − (75 / 220) = 0.57 + 0.50 − 0.34 = 0.73 or 73%. Alternative: 1 − P(no airbag and no bucket seats) = 1 − (60 / 220) = 72.7%
Question #19 of 21
Question ID: 438719
If two fair coins are flipped and two fair six-sided dice are rolled, all at the same time, what is the probability of ending up with two heads (on the coins) and two sixes (on the dice)? ✗ A) 0.8333. ✗ B) 0.4167. ✗ C) 0.0039. ✓ D) 0.0069. Explanation For the four independent events defined here, the probability of the specified outcome is 0.5000 × 0.5000 × 0.1667 × 0.1667 = 0.0069.
Question #20 of 21
Question ID: 438715
Given the following table about employees of a company based on whether they are smokers or nonsmokers and whether or not they suffer from any allergies, what is the probability of both suffering from allergies and not suffering from allergies? Suffer from Allergies Don't Suffer from Allergies
Total
Smoker
35
25
60
Nonsmoker
55
185
240
Total
90
210
300
✗ A) 0.50. ✗ B) 0.24. ✗ C) 1.00. ✓ D) 0.00. Explanation These are mutually exclusive, so the joint probability is zero.
Question #21 of 21
Question ID: 438716
If a fair coin is tossed twice, what is the probability of obtaining heads both times? ✗ A) 1.
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✗ B) 3/4. ✗ C) 1/2. ✓ D) 1/4. Explanation The probability of tossing a head, H, is P(H) = 1/2. Since these are independent events, the probability of two heads in a row is P(H H) = P(H) × P(H) = 1/2 × 1/2 = 1/4.
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Topic 16 - Basic Statistics
Test ID: 8450922
Question #1 of 54
Question ID: 438780
In a negatively skewed distribution, what is the order (from lowest value to highest) for the distribution's mode, mean, and median values? ✓ A) Mean, median, mode. ✗ B) Median, mode, mean. ✗ C) Mode, mean, median. ✗ D) Median, mean, mode. Explanation In a negatively skewed distribution, the mean is less than the median, which is less than the mode.
Question #2 of 54
Question ID: 438734
An investor is considering purchasing ACQ. There is a 30% probability that ACQ will be acquired in the next two months. If ACQ is acquired, there is a 40% probability of earning a 30% return on the investment and a 60% probability of earning 25%. If ACQ is not acquired, the expected return is 12%. What is the expected return on this investment? ✗ A) 17.4%. ✓ B) 16.5%. ✗ C) 12.3%. ✗ D) 18.3%. Explanation E(r) = (0.70 × 0.12) + (0.30 × 0.40 × 0.30) + (0.30 × 0.60 × 0.25) = 0.165.
Question #3 of 54
Question ID: 438758
Determine and interpret the correlation coefficient for the two variables X and Y. The standard deviation of X is 0.05, the standard deviation of Y is 0.08, and their covariance is −0.003. ✗ A) −1.33 and the two variables are negatively associated. ✓ B) −0.75 and the two variables are negatively associated. ✗ C) +0.75 and the two variables are positively associated. ✗ D) +1.33 and the two variables are positively associated. Explanation The correlation coefficient is the covariance divided by the product of the two standard deviations, i.e. −0.003 / (0.08 × 0.05).
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Question #4 of 54
Question ID: 438751
Which of the following statements about the correlation coefficient is TRUE? The correlation coefficient is: ✗ A) bounded between 0 and +1. ✗ B) boundless. ✗ C) bounded between -1 and 0. ✓ D) bounded between -1 and +1. Explanation The correlation coefficient is bounded between -1 and +1.
Question #5 of 54
Question ID: 438756
If X and Y are independent random variables: ✗ A) the variables are perfectly correlated. ✓ B) The covariance between the two variables is equal to zero and the correlation between the two variables is equal to zero. ✗ C) the covariance between the two variables is equal to zero. ✗ D) the correlation between the two variables is equal to zero. Explanation Variables have a zero correlation and covariance when they are independent.
Question #6 of 54
Question ID: 438772
Andrew Dawns holds a large position in the common stock of Savory Doughnuts, Inc (Savory). After an extensive executive search, Savory is about to announce a new CEO. There are three candidates for the CEO position, and each is viewed differently by the market. Dawns estimates the following probabilities for the rate of return on Savory's stock in the year following the announcement:
Candidate Probability of Being Chosen Rate of Return if Chosen One
.50
10%
Two
.15
-40%
Three
.35
20%
Based on Dawn's estimates, the expected rate of return on Savory's stock following the announcement of the new CEO is closest to: ✓ A) 6%. ✗ B) -2%. ✗ C) 9%. ✗ D) 10%.
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Explanation E(R) = (0.50 × 10%) + (0.15 × -40%) + (0.35 × 20%) = 6%.
Question #7 of 54
Question ID: 438785
Shawn Choate is thinking about his graduate thesis. Still in the preliminary stage, he wants to choose a variable of study that has the most desirable statistical properties. The statistic he is presently considering has the following characteristics: The expected value of the sample mean is equal to the population mean. The variance of the sampling distribution is smaller than that for other estimators of the parameter. As the sample size increases, the standard error of the sample mean rises and the sampling distribution is centered more closely on the mean. Select the best choice. Choate's estimator is:
✗ A) efficient and consistent. ✓ B) unbiased and efficient. ✗ C) unbiased, efficient, and consistent. ✗ D) unbiased and consistent. Explanation The estimator is unbiased because the expected value of the sample mean is equal to the population mean. The estimator is efficient because the variance of the sampling distribution is smaller than that for other estimators of the parameter. The estimator is not consistent. To be consistent, as the sample size increases, the standard error of the sample mean should fall and the sampling distribution will be centered more closely on the mean. A consistent estimator provides a more accurate estimate of the parameter as the sample size increases.
Question #8 of 54
Question ID: 438771
A security has the following expected returns and probabilities of occurrence: Return Probability 6.1%
10%
7.5%
40%
9.2%
50%
What is the expected return of the security?
✗ A) 7.6%. ✓ B) 8.2%. ✗ C) 7.8%. ✗ D) 8.4%. Explanation
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The expected return is calculated as the sum of the return times the probability that the return occurs. In this case, the expected return is: (0.061)(0.10) + (0.075)(0.40) + (0.092)(0.50) = 0.0821, or 8.2%.
Question #9 of 54
Question ID: 438753
Which of the following statements is least accurate regarding covariance? ✗ A) Covariance can exceed one. ✓ B) A covariance of zero rules out any relationship. ✗ C) Covariance can only apply to two variables at a time. ✗ D) Covariance can be negative. Explanation A covariance only measures the linear relationship. The covariance can be zero while a non-linear relationship exists. All the other statements are true.
Question #10 of 54
Question ID: 438759
For the case of simple linear regression with one independent variable, which of the following statements about the correlation coefficient is least accurate? ✓ A) If the regression line is flat and the observations are dispersed uniformly about the line, the correlation coefficient will be +1. ✗ B) The correlation coefficient describes the strength of the relationship between the X variable and the Y variable. ✗ C) The correlation coefficient can vary between −1 and +1. ✗ D) If the correlation coefficient is negative, it indicates that the regression line has a negative slope coefficient. Explanation Correlation analysis is a statistical technique used to measure the strength of the relationship between two variables. The measure of this relationship is called the coefficient of correlation. If the regression line is flat and the observations are dispersed uniformly about the line,there is no linear relationship between the two variables and the correlation coefficient will be zero. All other choices are TRUE.
Question #11 of 54
Question ID: 438762
SCU and QXA are two stocks in the same industry. The variance of returns for each stock is 0.3025 and the returns are perfectly positively correlated. The covariance between the returns is closest to: ✓ A) 0.3025. ✗ B) 0.2525.
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✗ C) 0.5500. ✗ D) 0.1000. Explanation Perfectly positively correlated means the correlation coefficient between the two stocks is 1. The standard deviation for both stocks is the square root of 0.3025, or
. Therefore:
Question #12 of 54
Question ID: 438784
If Estimator A is a more efficient estimator than Estimator B, it will have: ✓ A) the same mean and a smaller variance. ✗ B) the same mean and a larger variance. ✗ C) a smaller mean and the same variance. ✗ D) a smaller mean and a larger variance. Explanation The more efficient estimator is the one that has the smaller variance, given that they both have the same mean.
Question #13 of 54
Question ID: 438754
The correlation coefficient for two dependent random variables is equal to: ✗ A) the absolute value of the difference between the means of the two variables divided by the product of the variances. ✗ B) the product of the standard deviations for the two random variables divided by the covariance. ✓ C) the covariance between the random variables divided by the product of the standard deviations. ✗ D) the covariance between the random variables divided by the product of the variances. Explanation Correlation is equal to covariance divided by the product of the standard deviations.
Question #14 of 54
Question ID: 438761
For two (possibly dependent) random variables, X and Y, an upper bound on the covariance of X and Y is: ✗ A) zero. ✗ B) 1. ✓ C) σ(X) • σ(Y). ✗ D) there is no upper bound unless the variables are independent. Explanation
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Cov(X,Y) ≤ σ(X) • σ(Y).
Question #15 of 54
Question ID: 438777
When the tails of a distribution are fatter than that implied by a normal distribution, we say that the distribution is: ✓ A) leptokurtic. ✗ B) platykurtic. ✗ C) symmetrical. ✗ D) skewed. Explanation When the tails of a distribution are thicker than that of a normal distribution, we say that the distribution is leptokurtic. Platykurtic means that the tails are thinner than normal. The size of the tails is not indicative of a distribution's symmetry. Rather, the presence of skewness indicates that the distribution is not symmetrical.
Question #16 of 54
Question ID: 438757
In order to have a negative correlation between two variables, which of the following is most accurate? ✓ A) The covariance must be negative. ✗ B) The covariance can never be negative. ✗ C) Both the covariance and at least one of the standard deviations must be negative. ✗ D) Either the covariance or one of the standard deviations must be negative. Explanation In order for the correlation between two variables to be negative, the covariance must be negative. (Standard deviations are always positive.)
Questions #17-19 of 54 Use the following joint probability distribution to answer the questions below. Y=1
Y=2
Y=3
X=1
0.05
0.05
0.10
X=2
0.05
0.10
0.15
X=3
0.15
0.15
0.20
Question #17 of 54
Question ID: 438739
The expected value of Y is closest to: ✗ A) 0.2. ✓ B) 2.2. ✗ C) 1.0.
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✗ D) 2.3. Explanation p(Y=1)=0.05+0.05+0.15=0.25, p(Y=2)=0.05+0.10+0.15=0.30, and p(Y=3)=0.10+0.15+0.20=0.45, so E(Y)=0.25(1)+0.30(2)+0.45(3)=2.20.
Question #18 of 54
Question ID: 438740
If you know that Y is equal to 2, the probability that X is equal to 1 is closest to: ✗ A) 0.20. ✓ B) 0.17. ✗ C) 0.05. ✗ D) 0.25. Explanation p(X=1|Y=2)=0.05/(0.05+0.10+0.15)=0.17.
Question #19 of 54
Question ID: 438741
The variance of X is closest to: ✗ A) 0.74. ✓ B) 0.61. ✗ C) 0.67. ✗ D) 0.54. Explanation p(X=1)=0.05+0.05+0.10=0.20, p(X=2)=0.05+0.10+0.15=0.30, and p(X=3)=0.15+0.15+0.20=.50. Thus, the mean of X is equal to μ=0.20(1)+0.30(2)+0.50(3)=2.3, and the variance is calculated as 0.20(1-2.3)2+0.30(2-2.3)2+0.50(3-2.3)2=0.61.
Question #20 of 54
Question ID: 438767
A security has the following expected returns and probabilities of occurrence:
Return Probability 10%
40%
12%
40%
15%
20%
What is the expected return of the security?
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✓ A) 11.8%. ✗ B) 12.4%. ✗ C) 11.4%. ✗ D) 12.2%. Explanation The expected return is calculated as the sum of the return times the probability that the return occurs. In this case, the expected return is: (0.10)(0.40) + (0.12)(0.40) + (0.15)(0.20) = 0.118, or 11.8%.
Questions #21-23 of 54 Y=1
Y=2
Y=3
X=1
0.05
0.15
0.20
X=2
0.15
0.15
0.30
Question #21 of 54
Question ID: 438744
The expected value of X is closest to: ✓ A) 1.6 ✗ B) 1.2 ✗ C) 1.8 ✗ D) 1.5 Explanation p(X = 1) = 0.05 + 0.15 + 0.20 = 0.40, and p(X = 2) = 0.15 + 0.15 + 0.30 = 0.60, so E(X) = 0.40(1) + 0.60(2) = 1.6.
Question #22 of 54
Question ID: 438745
If you know that X is equal to 1, the probability that Y is equal to 2 is closest to: ✗ A) 0.30 ✓ B) 0.38 ✗ C) 0.50 ✗ D) 0.15 Explanation p(Y = 2|X = 1) = 0.15/(0.05 + 0.15 + 0.20) = 0.375.
Question #23 of 54
Question ID: 438746
The variance of Y is closest to: ✗ A) 1.51 ✓ B) 0.61 ✗ C) 0.76
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✗ D) 2.27 Explanation p(Y = 1) = 0.05 + 0.15 = 0.20, p(Y = 2) = 0.15 + 0.15 = 0.30, and p(Y = 3) = 0.20 + 0.30 = 0.50. Thus, the mean of Y is equal to μ = 0.20(1) + 0.30(2) + 0.50(3) = 2.3, and the variance is calculated as 0.20(1 - 2.3)2 + 0.30(2 - 2.3)2 + 0.50(3 - 2.3)2 = 0.61.
Question #24 of 54
Question ID: 438748
The Y variable is regressed against the X variable resulting in a regression line that is flat with the plot of the paired observations widely dispersed about the regression line. Based on this information, which statement is most accurate? ✗ A) The R2 of this regression is close to 100%. ✗ B) X is perfectly negatively correlated to Y. ✗ C) X is perfectly positively correlated to Y. ✓ D) The correlation between X and Y is close to zero. Explanation Perfect correlation means that the observations fall on the regression line. + means the line is going up and − means it is pointed down. An R2 of 100%, means perfect correlation. When there is no correlation, the regression line is flat and the residual standard error equals the standard deviation of Y.
Question #25 of 54
Question ID: 438781
If the variance of the sampling distribution of an estimator is smaller than all other unbiased estimators of the parameter of interest, the estimator is: ✗ A) consistent. ✓ B) efficient. ✗ C) unbiased. ✗ D) reliable. Explanation By definition.
Question #26 of 54
Question ID: 438783
The sample mean is an unbiased estimator of the population mean because the:
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✗ A) sampling distribution of the sample mean has the smallest variance of any other unbiased estimators of the population mean. ✓ B) expected value of the sample mean is equal to the population mean. ✗ C) sampling distribution of the sample mean is normal. ✗ D) sample mean provides a more accurate estimate of the population mean as the sample size increases. Explanation An unbiased estimator is one for which the expected value of the estimator is equal to the parameter you are trying to estimate.
Question #27 of 54
Question ID: 438776
A distribution that has positive excess kurtosis:
✗ A) is less peaked than a normal distribution. ✓ B) is more peaked than a normal distribution. ✗ C) is more skewed than a normal distribution. ✗ D) has thinner tails than a normal distribution. Explanation A distribution with positive excess kurtosis is one that is more peaked than a normal distribution.
Question #28 of 54
Question ID: 438774
Mike Palm, CFA, is an analyst with a large money management firm. Currently, Palm is considering the risk and return parameters associated with Alux, a small technology firm. After in depth analysis of the firm and the economic outlook, Palm estimates the following return probabilities: Probability (Pi) Return (Ri) 0.3
-5%
0.5
15%
0.2
25%
Palm's objective is to quantify the risk/return relationship for Alux. Given the returns and probability estimates above, what is the expected return for Alux? ✗ A) 9%. ✓ B) 11%. ✗ C) 15%. ✗ D) 45%. Explanation ERportfolio = S(ERstock)(W % of funds invested in each of the stocks)
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ER = w1ER1 + w2ER2 + w3ER3, where ER = Expected Return and w = % invested in each stock. ER = (0.3 * -5.0) + (0.50 * 15) + (0.20 * 25) = -1.5 + 7.5 + 5.0 = 11.0%
Question #29 of 54
Question ID: 438735
A two-sided but very thick coin is expected to land on its edge twice out of every 100 flips. And the probability of face up (heads) and the probability of face down (tails) are equal. When the coin is flipped, the prize is $1 for heads, $2 for tails, and $50 when the coin lands on its edge. What is the expected value of the prize on a single coin toss? ✗ A) $1.50. ✗ B) $26.50. ✓ C) $2.47. ✗ D) $17.67. Explanation Since the probability of the coin landing on its edge is 0.02, the probability of each of the other two events is 0.49. The expected payoff is: (0.02 × $50) + (0.49 × $1) + (0.49 × $2) = $2.47.
Question #30 of 54
Question ID: 438732
The characteristic function of the product of independent random variables is equal to the: ✗ A) square root of the product of the individual characteristic functions. ✗ B) exponential root of the product of the individual characteristic functions. ✗ C) sum of the individual characteristic functions. ✓ D) product of the individual characteristic functions. Explanation The characteristic function of the sum of independent random variables is equal to the product of the individual characteristic functions. E(XY) = E(X) × E(Y).
Question #31 of 54
Question ID: 438736
Use the following probability distribution to calculate the standard deviation for the portfolio.
State of the Economy Probability Return on Portfolio Boom
0.30
15%
Bust
0.70
3%
✗ A) 6.5%. ✓ B) 5.5%. ✗ C) 6.0%.
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✗ D) 7.0%. Explanation [0.30 × (0.15 − 0.066)2 + 0.70 × (0.03 − 0.066)2]1/2 = 5.5%.
Question #32 of 54
Question ID: 438737
There is a 30% chance that the economy will be good and a 70% chance that it will be bad. If the economy is good, your returns will be 20% and if the economy is bad, your returns will be 10%. What is your expected return? ✗ A) 18%. ✗ B) 15%. ✓ C) 13%. ✗ D) 17%. Explanation Expected value is the probability weighted average of the possible outcomes of the random variable. The expected return is: ((0.3) × (0.2)) + ((0.7) × (0.1)) = (0.06) + (0.07) = 0.13.
Question #33 of 54
Question ID: 438733
The variance of the sum of two independent random variables is equal to the sum of their variances: ✓ A) plus zero. ✗ B) minus a positive covariance term. ✗ C) plus a positive covariance term. ✗ D) plus a non-zero covariance term. Explanation Independent random variables have a covariance of zero. Hence, the variance of the sum of two variables will be the sum of the variables' variances.
Question #34 of 54
Question ID: 438769
As a fund manager, Bryan Cole, CFA, is responsible for assessing the risk and return parameters of the portfolios he oversees. Cole is currently considering a portfolio consisting of only two stocks. The first stock, Remba Co., has an expected return of 12 percent and a standard deviation of 16 percent. The second stock, Labs, Inc., has an expected return of 18 percent and a standard deviation of 25 percent. The correlation of returns between the two securities is 0.25. Cole has the option of including a third stock in the portfolio. The third stock, Wimset, Inc., has an expected return of 8% and a standard deviation of 10 percent. If Cole constructed an equally weighted portfolio consisting of all three stocks, the portfolio's expected return would be closest to:
✗ A) 17.1%. ✓ B) 12.7%.
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✗ C) 13.9%. ✗ D) 15.9%. Explanation ERportfolio = S(ERstock)(W% of funds invested in each of the stocks) ER = w1ER1 + w2ER2 + w3ER3, where ER = Expected Return and w = % invested in each stock. Here, use 1/3 for each of the weightings. (Note: If you use 0.33, you will calculate a slightly different result.) ER =( 1/3 * 12) + (1/3 * 18) + (1/3 * 8) = 4 + 6 + 2.7 = 12.7%
Question #35 of 54
Question ID: 438760
A scatter plot is a collection of points on a graph where each point represents the values of two variables (i.e., an X/Y pair). The pattern of data points will illustrate a correlation between these two variables that is between: ✗ A) -0.7 and +0.7. ✓ B) -1 to +1. ✗ C) slope of -1 and slope of +1. ✗ D) 0 and +1. Explanation Correlations range from - 1 to + 1. For r = 1 and r = -1 the data points lie exactly on a line, but the slope of that line is not necessarily +1 or -1.
Question #36 of 54
Question ID: 438747
A bond analyst is looking at historical returns for two bonds, Bond 1 and Bond 2. Bond 2's returns are much more volatile than Bond 1. The variance of returns for Bond 1 is 0.012 and the variance of returns of Bond 2 is 0.308. The correlation between the returns of the two bonds is 0.79, and the covariance is 0.048. If the variance of Bond 1 increased to 0.026 while the variance of Bond B decreased to 0.188 and the covariance remains the same, the correlation between the two bonds will: ✗ A) increase. ✓ B) decrease. ✗ C) the values given are not plausible. ✗ D) remain the same. Explanation P1,2 = 0.048/(0.0260.5 × 0.1880.5) = 0.69 which is lower than the original 0.79.
Question #37 of 54
Question ID: 438750
Suppose the covariance between Y and X is 12, the variance of Y is 25, and the variance of X is 36. What is the correlation coefficient (r), between Y and X?
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✗ A) 0.013. ✓ B) 0.400. ✗ C) 0.160. ✗ D) 0.000. Explanation The correlation coefficient is:
Question #38 of 54
Question ID: 438770
An analyst is currently considering a portfolio consisting of two stocks. The first stock, Remba Co., has an expected return of 12% and a standard deviation of 16%. The second stock, Labs, Inc., has an expected return of 18% and a standard deviation of 25%. The correlation of returns between the two securities is 0.25. If the analyst forms a portfolio with 30% in Remba and 70% in Labs, what is the portfolio's expected return? ✓ A) 16.2%. ✗ B) 15.0%. ✗ C) 17.3%. ✗ D) 21.5%. Explanation ERportfolio = Σ(ERstock)(W % of funds invested in each of the stocks) ER = w1ER1 + w2ER2, where ER = Expected Return and w = % invested in each stock. ER = (0.3 × 12) + (0.70 × 18) = 3.6 + 12.6 = 16.2%
Question #39 of 54
Question ID: 438768
An investor owns the following three-stock portfolio. Stock
Market Value
Expected Return
A
$5,000
12%
B
$3,000
8%
C
$4,000
9%
The expected return is closest to:
✗ A) 9.67%. ✗ B) 29.00%.
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✗ C) 10.50%. ✓ D) 10.00%. Explanation To calculate this result, we first need to calculate the portfolio value, then determine the weights for each stock, and then calculate the expected return. Portfolio Value: = sum of market values = 5,000 + 3,000 + 4,000 = 12,000 Portfolio Weights: WA = 5,000 / 12,000 = 0.4167 WB = 3,000 / 12,000 = 0.2500 WC = 4,000 / 12,000 = 0.333
Expected Return ERportfolio = S(ERstock)(W% of funds invested in each of the stocks) ER = wAERA + wBERB + wCERC, where ER = Expected Return and w = % invested in each stock. ER = (0.4167 * 12.0) + (0.2500 * 8.0) + (0.333 * 9.0) = 10.0%
Question #40 of 54
Question ID: 438742
Tully Advisers, Inc., has determined four possible economic scenarios and has projected the portfolio returns for two portfolios for their client under each scenario. Tully's economist has estimated the probability of each scenario as shown in the table below. Given this information, what is the expected return on portfolio A?
Scenario Probability Return on Portfolio A Return on Portfolio B A
15%
17%
19%
B
20%
14%
18%
C
25%
12%
10%
D
40%
8%
9%
✗ A) 12.95%. ✓ B) 11.55%. ✗ C) 9.25%. ✗ D) 10.75%. Explanation The expected return is equal to the sum of the products of the probabilities of the scenarios and their respective returns: = (0.15) (0.17) + (0.20)(0.14) + (0.25)(0.12) + (0.40)(0.08) = 0.1155 or 11.55%.
Question #41 of 54
Question ID: 438773
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Suppose you have chosen two stocks with an equity investment screener. The following data has been collected on these two securities: Stock A
Stock B
Expected Return
5%
8%
Variance
10%
16%
Covariance (A,B)
0.2
What is the mean and variance of the sum of the two stock's expected returns and variances assuming the returns are independent of one another?
✗ A) Mean = 6.5%; Variance = 13.4%. ✗ B) Mean = 13%; Variance = 26.4%. ✗ C) Mean = 6.5%; Variance = 13.0%. ✓ D) Mean = 13%; Variance = 26.0%. Explanation The mean of the sum of random variables is computed as: E(A +B) = E(A) + E(B) = 5% + 8% = 13%. The variance of the sum of random variables is computed as: Var(A + B) = Var(A) + Var(B) = 10% + 16% = 26%. Since the returns were assumed to be independent, the covariance was not a factor in the variance calculation.
Question #42 of 54
Question ID: 438731
An analyst has knowledge of the beginning-of-period expected returns, standard deviations of return, and market value weights for the assets that comprise a portfolio. The analyst does not require the covariances of returns between asset pairs to calculate the: ✗ A) variance of the return on the portfolio. ✗ B) reduction in risk due to diversification. ✗ C) correlations between asset pairs. ✓ D) expected return on the portfolio. Explanation All that is required to calculate the expected return for the portfolio is the portfolio weights and individual asset expected returns. All other items are functions of the covariance.
Question #43 of 54
Question ID: 438765
Rafael Garza, CFA, is considering the purchase of ABC stock for a client's portfolio. His analysis includes calculating the covariance between the returns of ABC stock and the equity market index. Which of the following statements regarding Garza's analysis is most accurate? ✗ A) The covariance of two variables is an easier measure to interpret than the correlation coefficient. ✗ B) A covariance of +1 indicates a perfect positive covariance between the two variables.
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✓ C) The actual value of the covariance is not very meaningful because the measurement is very sensitive to the scale of the two variables. ✗ D) The covariance measures the strength of the linear relationship between two variables. Explanation Covariance is a statistical measure of the linear relationship of two random variables, but the actual value is not meaningful because the measure is extremely sensitive to the scale of the two variables. Covariance can range from negative to positive infinity.
Question #44 of 54
Question ID: 438775
Left-skewed distributions exhibit: ✓ A) greater mass to the right of the expected value. ✗ B) greater mass close to the expected value. ✗ C) greater mass to the left of the expected value. ✗ D) a longer tail to the right of the distribution. Explanation Left-skewed distributions are those with a longer tail to the left side of the distribution, and whose mass is to the right of the expected value.
Question #45 of 54
Question ID: 438730
Tully Advisers, Inc., has determined four possible economic scenarios and has projected the portfolio returns for two portfolios for their client under each scenario. Tully's economist has estimated the probability of each scenario, as shown in the table below. Given this information, what is the standard deviation of returns on portfolio A?
Scenario Probability Return on Portfolio A Return on Portfolio B A
15%
18%
19%
B
20%
17%
18%
C
25%
11%
10%
D
40%
7%
9%
✗ A) 8.76%. ✗ B) 5.992%. ✗ C) 1.140%. ✓ D) 4.53%. Explanation E(RA) = 11.65% σ2 = 0.0020506 = 0.15(0.18 − 0.1165)2 + 0.2(0.17 − 0.1165)2 + 0.25(0.11 − 0.1165)2 + 0.4(0.07 − 0.1165)2
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σ = 0.0452836
Question #46 of 54
Question ID: 438752
Consider the case when the Y variable is in U.S. dollars and the X variable is in U.S. dollars. The 'units' of the covariance between Y and X are: ✗ A) U.S. dollars. ✓ B) squared U.S. dollars. ✗ C) a range of values from −1 to +1. ✗ D) the square root of U.S. dollars. Explanation The covariance is in terms of the product of the units of Y and X. It is defined as the average value of the product of the deviations of observations of two variables from their means. The correlation coefficient is a standardized version of the covariance, ranges from −1 to +1, and is much easier to interpret than the covariance.
Question #47 of 54
Question ID: 438764
Thomas Manx is attempting to determine the correlation between the number of times a stock quote is requested on his firm's website and the number of trades his firm actually processes. He has examined samples from several days trading and quotes and has determined that the covariance between these two variables is 88.6, the standard deviation of the number of quotes is 18, and the standard deviation of the number of trades processed is 14. Based on Manx's sample, what is the correlation between the number of quotes requested and the number of trades processed?
✗ A) 0.78. ✓ B) 0.35. ✗ C) 0.18. ✗ D) 0.98. Explanation Correlation = Cov (X,Y) / (Std. Dev. X)(Std. Dev. Y) Correlation = 88.6 / (18)(14) = 0.35
Question #48 of 54
Question ID: 438763
The covariance of returns on two investments over a 10-year period is 0.009. If the variance of returns for investment A is 0.020 and the variance of returns for investment B is 0.033, what is the correlation coefficient for the returns? ✗ A) 0.687. ✓ B) 0.350. ✗ C) 0.444. ✗ D) 0.785. Explanation
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The correlation coefficient is: Cov(A,B) / [(Std Dev A)(Std Dev B)] = 0.009 / [(√0.02)(√0.033)] = 0.350.
Question #49 of 54
Question ID: 438782
The sample mean is a consistent estimator of the population mean because the: ✓ A) sample mean provides a more accurate estimate of the population mean as the sample size increases. ✗ B) sampling distribution of the sample mean has the smallest variance of any other unbiased estimators of the population mean. ✗ C) sampling distribution of the sample mean is normal. ✗ D) expected value of the sample mean is equal to the population mean. Explanation A consistent estimator provides a more accurate estimate of the parameter as the sample size increases.
Question #50 of 54
Question ID: 438779
In a positively skewed distribution, what is the order (from lowest value to highest) for the distribution's mode, mean, and median values? ✗ A) Mode, mean, median. ✓ B) Mode, median, mean. ✗ C) Mean, median, mode. ✗ D) Median, mean, mode. Explanation In a positively skewed distribution, the mode is less than the median, which is less than the mean.
Question #51 of 54
Question ID: 438766
An investor owns the following portfolio today. Stock
Market Value
Expected Annual Return
R
$2,000
17%
S
$3,200
8%
T
$2,800
13%
The investor's expected total rate of return (increase in market value) after three years is closest to: ✓ A) 40.5%. ✗ B) 136.0%. ✗ C) 36.0%. ✗ D) 12.0%.
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Explanation To calculate this result, we first need to calculate the portfolio value, then determine the weights for each stock, and then calculate the expected return. Finally, we determine the compounded rate after three years. Portfolio Value: = sum of market values = 2,000 + 3,200 + 2,800 = 8,000 Portfolio Weights: W A = 2,000 / 8,000 = 0.25 W B = 3,200 / 8,000 = 0.40 W C = 2,800 / 8,000 = 0.35 Expected Return ERportfolio = Σ[(ERstock)(W % of funds invested in each of the stocks)] ER = wRERR + wSERS + wTERT, where ER = Expected Return and w = % invested in each stock. ER = (0.25 × 17.0) + (0.40 × 8.0) + (0.35 × 13.0) = 12.0% Expected Return after three years = (1 + return)3 = (1.12)3 − 1 = 1.405 − 1 = 0.405, or 40.5%.
Question #52 of 54
Question ID: 438778
It is often said that stock returns are leptokurtic. If this is true, relative to a normal distribution of the same mean and variance, the distribution of stock returns is: ✗ A) thin-tailed. ✗ B) negatively skewed. ✓ C) fat-tailed. ✗ D) positively skewed. Explanation A leptokurtic distribution is more peaked and has fatter tails than a normal distribution.
Question #53 of 54
Question ID: 438749
Which model does not lend itself to correlation coefficient analysis?
✓ A) Y = X3. ✗ B) Y - X = 2. ✗ C) X = Y × 2. ✗ D) Y = X + 2. Explanation The correlation coefficient is a measure of linear association. All of the functions except for Y = X3 are linear functions. Notice that Y - X = 2 is the same as Y = X + 2.
Question #54 of 54
Question ID: 438755
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The covariance:
✗ A) must be positive. ✗ B) must be between -1 and +1. ✗ C) must be less than +1. ✓ D) can be positive or negative. Explanation Cov(a,b) = σaσbρa,b. Since ρa,b can be positive or negative, Cov(a,b) can be positive or negative.
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Topic 17 - Distributions
Test ID: 8450933
Question #1 of 32
Question ID: 438802
Standardizing a normally distributed random variable requires the: ✓ A) mean and the standard deviation. ✗ B) variance and kurtosis. ✗ C) natural logarithm of X. ✗ D) mean, variance and skewness. Explanation All that is necessary is to know the mean and the variance. Subtracting the mean from the random variable and dividing the difference by the standard deviation standardizes the variable.
Question #2 of 32
Question ID: 438818
Which of the following statements is true regarding the mixture of distributions for risk modeling? I. Distributions should never be combined. II. It may be helpful to create a new distribution if the underlying data you are working with does not currently fit a pre-determined distribution. ✗ A) I only. ✓ B) II only. ✗ C) Neither I nor II. ✗ D) Both I and II. Explanation Distributions can be combined to create unique probability density functions. It may be helpful to create a new distribution if the underlying data you are working with does not currently fit a pre-determined distribution. In this case, a newly created distribution may assist with explaining the relevant data set.
Question #3 of 32
Question ID: 438789
Which of the following statements about a normal distribution is least accurate? ✓ A) A normal distribution has excess kurtosis of three. ✗ B) The mean, median, and mode are equal. ✗ C) Approximately 68% of the observations lie within +/- 1 standard deviation of the mean. ✗ D) The mean and variance completely define a normal distribution. Explanation Even though normal curves have different sizes, they all have identical shape characteristics. The kurtosis for all normal
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distributions is three; an excess kurtosis of three would indicate a leptokurtic distribution. The other choices are true.
Question #4 of 32
Question ID: 438816
The central limit theorem states that, for any distribution, as n gets larger, the sampling distribution: ✗ A) becomes smaller. ✗ B) approaches the mean. ✗ C) becomes larger. ✓ D) approaches a normal distribution. Explanation As n gets larger, the variance of the distribution of sample means is reduced, and the distribution of sample means approximates a normal distribution.
Question #5 of 32
Question ID: 438804
Given the probabilities N(-0.5) = 0.3085, N(0.75) = 0.7734, and N(1.50) = 0.9332 from a z-table, the probability of 0.2266 corresponds to: ✗ A) N(0.50). ✓ B) N(-0.75). ✗ C) N(0.25). ✗ D) N(-0.25). Explanation This problem is checking your knowledge of a normal distribution and gives you more information than you need to answer the question. The area of a normal distribution is 1, with two symmetric halves that equal 0.5 each. N(0.75) means that the area to the left of 0.75 on the positive portion of the curve is 0.7734. This means that the area to the right of 0.75 is (1.0 - 0.7734) = 0.2266. Since the halves of a normal distribution are symmetrical, that means the area to the left of (-0.75) is also 0.2266.
Question #6 of 32
Question ID: 438788
An analyst has determined that the probability that the S&P 500 index will increase on any given day is 0.60 and the probability that it will decrease is 0.40. The expected value and variance of the number of up days in a 5-day period are closest to: ✗ A) 2.0 and 2.1. ✓ B) 3.0 and 1.2. ✗ C) 3.0 and 1.1. ✗ D) 2.0 and 0.5. Explanation E(X)=np=5(0.60)=3.0. Var(X)=np(1-p)=5(0.60)(0.40)=1.2.
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Question #7 of 32
Question ID: 438790
A normal distribution can be completely described by its:
✗ A) standard deviation. ✗ B) skewness and kurtosis. ✗ C) mean and mode. ✓ D) mean and variance. Explanation The normal distribution can be completely described by its mean and variance.
Question #8 of 32
Question ID: 438800
Which of the following statements about probability distributions is least accurate? ✓ A) A probability distribution is, by definition, normally distributed. ✗ B) A probability distribution includes a listing of all the possible outcomes of an experiment. ✗ C) In a binomial distribution each observation has only two possible outcomes that are mutually exclusive. ✗ D) One of the key properties of a probability function is 0 ≤ p ≤ 1. Explanation Probabilities must be zero or positive, but a probability distribution is not necessarily normally distributed. Binomial distributions are either successes or failures.
Question #9 of 32
Question ID: 438805
The distribution of annual returns for a bond portfolio is approximately normal with an expected value of $120 million and a standard deviation of $20 million. Which of the following is closest to the probability that the value of the portfolio one year from today will be between $110 million and $170 million? ✓ A) 66%. ✗ B) 58%. ✗ C) 74%. ✗ D) 42%. Explanation Calculating z-values, z1 = (110 − 120) / 20 = −0.5. z2 = (170 − 120) / 20 = 2.5. Using the z-table, P(−0.5) = (1 − 0.6915) = 0.3085. P(2.5) = 0.9938. P(−0.5 < X < 2.5) = 0.9938 − 0.3085 = 0.6853. Note that on the exam, you will not have access to z-tables, so you would have to reason this one out using the normal distribution approximations. You know that the probability within +/− 1 standard deviation of the mean is approximately 68%, meaning that the area within −1 standard deviation of the mean is 34%. Since −0.5 is half of −1, the area under −0.5 to 0 standard deviations under the mean is approximately 34% / 2 = 17%. The probability under +/− 2.5 standard deviations of the mean is approximately 99%. The value $170 is mid way between +2 and +3 standard deviations, so the probability between these values must be (99% / 2) = 2%. The value from 0 to 2.5
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standard deviations must therefore be (99% / 2) − (2% / 2) = 48.5%. Adding these values gives us an approximate probability of (48.5% + 17%) = 65.5%.
Question #10 of 32
Question ID: 438815
The central limit theorem concerns the sampling distribution of the: ✗ A) sample standard deviation. ✗ B) population mean. ✗ C) population standard deviation. ✓ D) sample mean. Explanation The central limit theorem tells us that for a population with a mean m and a finite variance σ2, the sampling distribution of the sample means of all possible samples of size n will approach a normal distribution with a mean equal to m and a variance equal to σ2 / n as n gets large.
Question #11 of 32
Question ID: 438808
Which one of the following statements about the t-distribution is most accurate? ✓ A) The t-distribution approaches the standard normal distribution as the number of degrees of freedom becomes large. ✗ B) Compared to the normal distribution, the t-distribution is flatter with less area under the tails. ✗ C) Compared to the normal distribution, the t-distribution is more peaked with more area under the tails. ✗ D) The t-distribution is the appropriate distribution to use when constructing confidence intervals based on large samples. Explanation As the number of degrees of freedom grows, the t-distribution approaches the shape of the standard normal distribution. Compared to the normal distribution, the t-distribution is less peaked with more area under the tails. When choosing a distribution, three factors must be considered: sample size, whether population variance is known, and if the distribution is normal.
Question #12 of 32
Question ID: 438787
For a normal distribution, what approximate percentage of the observations falls within ± 2 standard deviations of the mean? ✗ A) 92%. ✗ B) 90%. ✓ C) 95%. ✗ D) 99%. Explanation
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For normal distributions, approximately 95 percent of the observations fall within ± 2 standard deviations of the mean.
Question #13 of 32
Question ID: 438799
Regarding a binomially distributed random variable, the variance of the number of successes, X, where n equals the number of trials and p equals the probability of success, is most likely equal to: ✗ A) np. ✗ B) p. ✗ C) p(1-p). ✓ D) np(1 - p). Explanation For a given series of n trials, the variance of X for a binomial random variable is equal to: np(1 - p) = npq, where q is the probability of failure.
Question #14 of 32
Question ID: 438796
An investment has a mean return of 15% and a standard deviation of returns equal to 10%. Which of the following statements is least accurate? The probability of obtaining a return:
✗ A) greater than 35% is 0.025. ✓ B) greater than 25% is 0.32. ✗ C) between 5% and 25% is 0.68. ✗ D) less than 5% is 0.16. Explanation Sixty-eight percent of all observations fall within +/- one standard deviation of the mean of a normal distribution. Given a mean of 15 and a standard deviation of 10, the probability of having an actual observation fall within one standard deviation, between 5 and 25, is 68%. The probability of an observation greater than 25 is half of the remaining 32%, or 16%. This is the same probability as an observation less than 5. Because 95% of all observations will fall within 20 of the mean, the probability of an actual observation being greater than 35 is half of the remaining 5%, or 2.5%.
Question #15 of 32
Question ID: 438793
The return on a portfolio is normally distributed with a mean return of 8 percent and a standard deviation of 18 percent. Which of the following is closest to the probability that the return on the portfolio will be between -27.3 percent and 37.7 percent? ✓ A) 92.5%. ✗ B) 68.0%. ✗ C) 81.5%. ✗ D) 96.5%. Explanation
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Note that if you memorize the basic intervals for a normal distribution, you do not need a normal distribution table to answer this question. -27.3% represents a loss of -35.3% from the mean return (-27.3 - 8.0), which is (-35.3/18) = -1.96 standard deviations to the left of the mean. For a normal distribution, we know that approximately 95 percent of all observations lie with +/- 1.96 standard deviations of the mean, so the probability that the return is between -27.3% and 8.0% must be (95%/2) = 47.5%. A return of 37.7 percent represents a gain of (37.7 - 8.0) = 29.7% from the mean return, which is (29.7/18) = 1.65 standard deviations to the right of the mean. For a normal distribution, we know that approximately 90 percent of all observations lie with +/- 1.65 standard deviations of the mean, so the probability that the return is between 8.0% and 37.7% must be (90%/2) = 45%. Therefore the probability that the return is between -27.3 percent and 37.7 percent = (47.5% + 45%) = 92.5%.
Question #16 of 32
Question ID: 438798
The lower limit of a normal distribution is: ✗ A) negative one. ✗ B) zero. ✗ C) one. ✓ D) negative infinity. Explanation By definition, a true normal distribution has a positive probability density function from negative to positive infinity.
Question #17 of 32
Question ID: 438810
With 60 observations, what is the appropriate number of degrees of freedom to use when carrying out a statistical test on the mean of a population? ✓ A) 59. ✗ B) 60. ✗ C) 58. ✗ D) 61. Explanation When performing a statistical test on the mean of a population based on a sample of size n, the number of degrees of freedom is n - 1 since once the mean is estimated from a sample there are only n - 1 observations that are free to vary. In this case the appropriate number of degrees of freedom to use is 60 - 1 = 59.
Question #18 of 32
Question ID: 438797
Approximately 50 percent of all observations for a normally distributed random variable fall in the interval: ✗ A) µ ± 2σ ✓ B) µ ± 0.67σ ✗ C) µ ± σ ✗ D) µ ± 3σ
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Explanation µ ± 0.67σ
Question #19 of 32
Question ID: 438806
The annual returns for a portfolio are normally distributed with an expected value of £50 million and a standard deviation of £25 million. What is the probability that the value of the portfolio one year from today will be between £91.13 million and £108.25 million? ✗ A) 0.075. ✗ B) 0.090. ✓ C) 0.040. ✗ D) 0.025. Explanation Calculate the standardized variable corresponding to the outcomes. Z1 = (91.13 − 50) / 25 = 1.645, and Z2 = (108.25 − 50) / 25 = 2.33. The cumulative normal distribution gives cumulative probabilities of F(1.645) = 0.95 and F(2.33) = 0.99. The probability that the outcome will lie between Z1 and Z2 is the difference: 0.99 − 0.95 = 0.04. Note that even though you will not have a z-table on the exam, the probability values for 1.645 and 2.33 are commonly used values that you should have memorized.
Question #20 of 32
Question ID: 438809
When is the t-distribution the appropriate distribution to use? The t-distribution is the appropriate distribution to use when constructing confidence intervals based on: ✗ A) small samples from populations with known variance that are at least approximately normal. ✗ B) large samples from populations with known variance that are nonnormal. ✓ C) small samples from populations with unknown variance that are at least approximately normal. ✗ D) large samples from populations with known variance that are at least approximately normal. Explanation The t-distribution is the appropriate distribution to use when constructing confidence intervals based on small samples from populations with unknown variance that are either normal or approximately normal.
Question #21 of 32
Question ID: 438817
To apply the central limit theorem to the sampling distribution of the sample mean, the sample is usually considered to be large if n is greater than: ✓ A) 30. ✗ B) 25. ✗ C) 15. ✗ D) 20.
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Explanation A sample size is considered sufficiently large if it is larger than or equal to 30.
Question #22 of 32
Question ID: 438795
Which of the following statements is incorrect regarding a Bernoulli distributed random variable. I. It only has three possible outcomes. II. It is commonly used for assessing whether or not a company defaults during a specified time period. ✗ A) Neither I nor II. ✓ B) I only. ✗ C) Both I and II. ✗ D) II only. Explanation Bernoulli distributed random variable only has two possible outcomes. The outcomes can be defined as either a "success" or a "failure."
Question #23 of 32
Question ID: 438803
A client will move his investment account unless the portfolio manager earns at least a 10% rate of return on his account. The rate of return for the portfolio that the portfolio manager has chosen has a normal probability distribution with an expected return of 19% and a standard deviation of 4.5%. What is the probability that the portfolio manager will keep this account? ✗ A) 1.000. ✗ B) 0.950. ✓ C) 0.977. ✗ D) 0.750. Explanation Since we are only concerned with values that are below a 10% return this is a 1 tailed test to the left of the mean on the normal curve. With μ = 19 and σ = 4.5, P(X ≥ 10) = P(X ≥ μ − 2σ) therefore looking up -2 on the cumulative Z table gives us a value of 0.0228, meaning that (1 − 0.0228) = 97.72% of the area under the normal curve is above a Z score of -2. Since the Z score of -2 corresponds with the lower level 10% rate of return of the portfolio this means that there is a 97.72% probability that the portfolio will earn at least a 10% rate of return.
Question #24 of 32
Question ID: 438792
A group of investors wants to be sure to always earn at least a 5% rate of return on their investments. They are looking at an investment that has a normally distributed probability distribution with an expected rate of return of 10% and a standard deviation of 5%. The probability of meeting or exceeding the investors' desired return in any given year is closest to:
✗ A) 34%.
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✗ B) 98%. ✓ C) 84%. ✗ D) 50%. Explanation The mean is 10% and the standard deviation is 5%. You want to know the probability of a return 5% or better. 10% - 5% = 5% , so 5% is one standard deviation less than the mean. Thirty-four percent of the observations are between the mean and one standard deviation on the down side. Fifty percent of the observations are greater than the mean. So the probability of a return 5% or higher is 34% + 50% = 84%.
Question #25 of 32
Question ID: 438786
Which of the following statements about the normal probability distribution is most accurate? ✗ A) Sixty-eight percent of the area under the normal curve falls between 0 and +1 standard deviations above the mean. ✗ B) The standardized normal distribution has a mean of zero and a standard deviation of 10. ✓ C) Five percent of the normal curve probability falls more than outside two standard deviations from the mean. ✗ D) The normal curve is asymmetrical about its mean. Explanation The normal curve is symmetrical with a mean of zero and a standard deviation of 1 with 34% of the area under the normal curve falling between 0 and +1 standard deviation above the mean. Ninety-five percent of the normal curve is within two standard deviations of the mean, so five percent of the normal curve falls outside two standard deviations from the mean.
Question #26 of 32
Question ID: 438811
Which one of the following distributions is described entirely by the degrees of freedom? ✗ A) Lognormal distribution. ✓ B) Student's t-distribution. ✗ C) Binomial distribution. ✗ D) Normal distribution. Explanation Student's t-distribution is defined by a single parameter known as the degrees of freedom.
Question #27 of 32
Question ID: 438791
A call center receives an average of two phone calls per hour. The probability that they will receive 20 calls in an 8-hour day is closest to: ✗ A) 3.66%.
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✗ B) 16.56%. ✗ C) 6.40%. ✓ D) 5.59%. Explanation To solve this question, we first need to realize that the expected number of phone calls in an 8-hour day is λ = 2 × 8 = 16. Using the Poisson distribution, we solve for the probability that X will be 20. P(X = x) = (λxe-λ) / x! P(X = 20) = (1620e-16) / 20! = 0.0559 = 5.59%
Question #28 of 32
Question ID: 438794
The probability of returns less than −10%, assuming a normal distribution with expected return of 6.5% and standard deviation of 10%, is: ✓ A) approximately 5%. ✗ B) not defined with only this information. ✗ C) approximately 10%. ✗ D) less than 2.5%. Explanation −10 is 16.5% below the mean return, (16.5 / 10) = 1.65 standard deviations, which leaves approximately 5% of the possible outcomes in the left tail below −10%.
Question #29 of 32
Question ID: 438807
Which statement best describes the properties of Student's t-distribution? The t-distribution is: ✗ A) symmetrical, defined by two parameters and is less peaked than the normal distribution. ✓ B) symmetrical, defined by a single parameter and is less peaked than the normal distribution. ✗ C) skewed, defined by a single parameter and is more peaked than the normal distribution. ✗ D) skewed, defined by a single parameter and is less peaked than the normal distribution. Explanation The t-distribution is symmetrical like the normal distribution but unlike the normal distribution is defined by a single parameter known as the degrees of freedom and is less peaked than the normal distribution.
Questions #30-31 of 32 A study of hedge fund investors found that their household incomes are normally distributed with a mean of $280,000 and a standard deviation of $40,000.
Question #30 of 32
Question ID: 438813
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The percentage of hedge fund investors that have incomes greater than $350,000 is closest to: ✗ A) 5.0%. ✗ B) 3.0%. ✓ C) 4.0%. ✗ D) 25.0%. Explanation z = (350,000 - 280,000)/40,000 = 1.75. Using the z-table, F(1.75) = 0.9599. So, the percentage greater than $350,000 = (1 0.9599) = 4.0%.
Question #31 of 32
Question ID: 438814
The percentage of hedge fund investors with income less than $180,000 is closest to: ✗ A) 6.48%. ✗ B) 1.15%. ✗ C) 2.50%. ✓ D) 0.62%. Explanation z = (180,000 - 280,000)/40,000 = -2.50. Using the z-table, F(-2.50) = (1 - 0.9938) = 0.62%.
Question #32 of 32
Question ID: 438801
A food retailer has determined that the mean household income of her customers is $47,500 with a standard deviation of $12,500. She is trying to justify carrying a line of luxury food items that would appeal to households with incomes greater than $60,000. Based on her information and assuming that household incomes are normally distributed, what percentage of households in her customer base has incomes of $60,000 or more?
✓ A) 15.87%. ✗ B) 34.13%. ✗ C) 2.50%. ✗ D) 5.00%. Explanation Z = ($60,000 - $47,500) / $12,500 = 1.0 From the table of areas under the normal curve, 84.13% of observations lie to the left of +1 standard deviation of the mean. So, 100% 84.13% = 15.87% with incomes of $60,000 or more.
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Topic 19: Hypothesis Testing and Confidence Intervals
Test ID: 8450948
Question #1 of 82
Question ID: 438835
The table below shows a sample of returns on two securities: Period
1
2
3
4
Mean
Security P
0.2%
0.5%
1.1% −0.6%
0.3%
Security Q
−0.3%
0.9%
1.5% −0.5%
0.4%
The sample covariance between the two securities' returns is closest to: ✗ A) 0.78. ✗ B) 1.86. ✗ C) 0.47. ✓ D) 0.62. Explanation Period
1
2
3
4
−0.1
0.2
0.8 −0.9
−0.7
0.5
1.1 −0.9
Sum
0.07 0.10 0.88 0.81 1.86
Question #2 of 82
Question ID: 438853
Robert Patterson, an options trader, believes that the return on options trading is higher on Mondays than on other days. In order to test his theory, he formulates a null hypothesis. Which of the following would be an appropriate null hypothesis? Returns on Mondays are:
✗ A) not less than returns on other days. ✗ B) less than returns on other days. ✓ C) not greater than returns on other days. ✗ D) greater than returns on other days. Explanation An appropriate null hypothesis is one that the researcher wants to reject. If Patterson believes that the returns on Mondays are greater than on other days, he would like to reject the hypothesis that the opposite is true-that returns on Mondays are not greater than returns on other days.
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Question #3 of 82
Question ID: 438878
Which of the following statements about hypothesis testing is most accurate? ✓ A) The probability of a Type I error is equal to the significance level of the test. ✗ B) To test the claim that X is greater than zero, the null hypothesis would be H0: X > 0. ✗ C) If you can disprove the null hypothesis, then you have proven the alternative hypothesis. ✗ D) The power of a test is one minus the probability of a Type I error. Explanation The probability of getting a test statistic outside the critical value(s) when the null is ture is the level of significance and is the probability of a Type I error. The power of a test is 1 minus the probability of a Type II error. Hypothesis testing does not prove a hypothesis, we either reject the null or fail to reject it. The appropriate null would be "X ≤ 0" with "X > 0" as the alternative hypothesis.
Questions #4-5 of 82 Margo Hinsdale is testing the null hypothesis that the population mean is less than or equal to 45. A random sample of 81 observations selected from this population produced a mean of 46.3. The population has a standard deviation of 4.5.
Question #4 of 82
Question ID: 438890
The value of the appropriate test statistic for the test of the population mean is: ✗ A) t = 4.60. ✗ B) z = -2.75. ✓ C) z = 2.60. ✗ D) t = 3.84. Explanation The population variance is known and the sample size is large. The test statistic is:
Question #5 of 82
Question ID: 438891
At a 1 percent level of significance, the correct decision is to: ✓ A) reject the null hypothesis. ✗ B) neither reject nor fail to reject the null hypothesis. ✗ C) fail to reject the null hypothesis. ✗ D) accept the null hypothesis. Explanation Decision rule: reject H0 if zcomputed > zcritical. Therefore, reject the null hypothesis because the computed test statistic of 2.60 (see the answer to Part 1) exceeds the critical z-value of 2.33.
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Questions #6-9 of 82 An analyst is testing to see if the mean of a population is less than 133. A random sample of 50 observations had a mean of 130. Assume a standard deviation of 5. The test is to be made at the 1% level of significance. z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.0
0.0000
0.0040
0.0080
0.0120
0.0160
0.0199
0.0239
0.1
0.0398
0.0438
0.0478
0.0517
0.0557
0.0596
0.0636
0.2
0.0793
0.0832
0.0871
0.0910
0.0948
0.0987
0.1026
0.3
0.1179
0.1217
0.1255
0.1293
0.1331
0.1368
0.1406
|
|
|
|
|
|
|
|
1.7
0.4554
0.4564
0.4573
0.4582
0.4591
0.4599
0.4608
1.8
0.4641
0.4649
0.4656
0.4664
0.4671
0.4678
0.4686
1.9
0.4713
0.4719
0.4726
0.4732
0.4738
0.4744
0.4750
2.0
0.4772
0.4778
0.4783
0.4788
0.4793
0.4798
0.4803
2.1
0.4821
0.4826
0.4830
0.4834
0.4838
0.4842
0.4846
2.2
0.4861
0.4864
0.4868
0.4871
0.4875
0.4878
0.4881
2.3
0.4893
0.4896
0.4898
0.4901
0.4904
0.4906
0.4909
2.4
0.4918
0.4920
0.4922
0.4925
0.4927
0.4929
0.4931
Question #6 of 82
Question ID: 438861
The null hypothesis is: ✗ A) µ > 133. ✓ B) µ ≥ 133. ✗ C) µ ≤ 133. ✗ D) µ = 133. Explanation The null hypothesis is the hypothesis that the researcher wants to reject. Here the hypothesis that is being looked for is that the mean of a population is less than 133. The null hypothesis is that the mean is greater than or equal to 133. The question is whether the null hypothesis will be rejected in favor of the alternative hypothesis that the mean is less than 133.
Question #7 of 82
Question ID: 438862
The calculated test statistic is: ✗ A) +1.33. ✗ B) -3.00. ✗ C) -1.33. ✓ D) -4.24. Explanation A test statistic is calculated by subtracting the hypothesized parameter from the parameter that has been estimated and dividing
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the difference by the standard error of the sample statistic. Here, the test statistic = (sample mean - hypothesized mean) / ((sample standard deviation / (sample size)1/2)) = (130 - 133) / (5 / 501/2) = (-3) / (5 / 7.0711) = -4.24.
Question #8 of 82
Question ID: 438863
The critical value is: ✗ A) -2.38. ✗ B) 2.47. ✓ C) -2.33. ✗ D) 2.17. Explanation This is a one-tailed test with a significance level of 0.01. The critical value for a one-tailed test at a 1% level of significance is -2.33.
Question #9 of 82
Question ID: 438864
You should: ✗ A) Cannot be determined with the information given. ✗ B) accept the null hypothesis. ✗ C) reject the alternative hypothesis. ✓ D) reject the null hypothesis. Explanation The calculated test statistic of -4.24 falls to the left of the z-statistic of -2.33, and is in the rejection region. Thus, the null hypothesis is rejected and the conclusion is that the population mean is less than 133.
Question #10 of 82
Question ID: 438852
Brian Ci believes that the average return on equity in the airline industry, µ, is less than 5%. What are the appropriate null (H0) and alternative (Ha) hypotheses to test this belief? ✗ A) H0: µ > 0.05 versus Ha: µ < 0.05. ✗ B) H0: µ < 0.05 versus Ha: µ > 0.05. ✓ C) H0: µ ≥ 0.05 versus Ha: µ < 0.05. ✗ D) H0: µ < 0.05 versus Ha: µ ≥ 0.05. Explanation The alternative hypothesis is determined by the theory or the belief. The researcher specifies the null as the hypothesis that he wishes to reject (in favor of the alternative). Note that this is a one-sided alternative because of the "less than" belief.
Question #11 of 82
Question ID: 438832
A sample covariance of two random variables is most commonly utilized to:
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✗ A) demonstrate either the presence or absence of spurious correlation of the variables. ✗ B) estimate the "pure" measure of the tendency of two variables to move together over a period of time. ✗ C) identify and measure strong nonlinear relationships between the two variables. ✓ D) calculate the correlation coefficient, which is a measure of the strength of their linear relationship. Explanation Since the actual value of a sample covariance can range from negative to positive infinity depending on the scale of the two variables, it is most commonly used to calculate a more useful measure, the correlation coefficient.
Question #12 of 82
Question ID: 438898
The test statistic for an F-test of the equality of two sample variances is the: ✗ A) product of the two sample standard deviations. ✓ B) ratio of the two sample variances. ✗ C) product of the two sample variances. ✗ D) ratio of the two sample standard deviations. Explanation The test statistic for an F-test of the equality of two sample variances is the ratio of the two sample variances.
Question #13 of 82
Question ID: 438849
The approximate 99% confidence interval for the population mean based on a sample of 60 returns with a mean of 7% and a sample standard deviation of 25% is closest to: ✗ A) 1.546% to 13.454%. ✗ B) 0.546% to 13.454%. ✓ C) -1.584% to 15.584%. ✗ D) 1.584% to 14.584%. Explanation The standard error for the mean = s / (n)0.5 = 25% / (60)0.5 = 3.227%. The critical value from the t-table should be based on 60 − 1 = 59 df. Since the standard tables do not provide the critical value for 59 df the closest available value is for 60 df. This leaves us with an approximate confidence interval. Based on 99% confidence and df = 60, the critical t-value is 2.660. Therefore the 99% confidence interval is approximately: 7% ± 2.660(3.227) or 7% ± 8.584% or -1.584% to 15.584%. If you use a z-statistic, the confidence interval is 7% ± 2.58(3.227) = -1.326% to 15.326%, which is closest to the correct choice.
Questions #14-18 of 82 An analyst is conducting a hypothesis test to determine if the mean time spent on investment research is different from 3 hours
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per day. The test is performed at the 5 percent level of significance and uses a random sample of 64 portfolio managers, where the mean time spent on research is found to be 2.5 hours. The population standard deviation is 1.5 hours.
Question #14 of 82
Question ID: 438884
The appropriate null hypothesis for the described test is: ✗ A) H0: µ ≥ 3 hours. ✗ B) H0: µ ≠ 3 hours. ✓ C) H0: µ = 3 hours. ✗ D) H0: µ ≤ 3 hours. Explanation H0: µ = 3 hours.
Question #15 of 82
Question ID: 438885
Which type of test is this? ✓ A) Two-tailed test. ✗ B) Chi-square test. ✗ C) One-tailed test. ✗ D) Equity of variance test. Explanation This is a two-sided (tailed) test. We want to test if the mean "differs from" 3 hours (i.e., Ha: µ ≠ 3 hours).
Question #16 of 82
Question ID: 438886
The calculated z-statistic is: ✗ A) +2.67. ✗ B) -2.13. ✗ C) +0.33. ✓ D) -2.67. Explanation The normally distributed test statistic is:
Question #17 of 82
Question ID: 438887
The critical z-value(s) of the test statistic is (are): ✓ A) ± 1.96. ✗ B) -1.96. ✗ C) ± 2.58. ✗ D) +1.96. Explanation
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At α/2 = 0.025, the critical z-values are: ±zα/2 = ±z0.025 = ±1.96.
Question #18 of 82
Question ID: 438888
Which of the following decisions is the CORRECT decision for this study? ✗ A) Fail to reject the null hypothesis. ✓ B) Reject the null hypothesis. ✗ C) No decision is possible because the sample standard deviation was not given. ✗ D) The sample size is too small, so increase the sample size. Explanation Decision rule: reject H0 if zcomputed < -1.96, or zcomputed > +1.96. Since -2.67 < -1.96 , reject H0.
Question #19 of 82
Question ID: 438899
What is the computed value of the test statistic that follows an F-distribution when sample variances are equal and the level of significance is 0.10? ✗ A) 0.90. ✗ B) 0.05. ✓ C) 1.00. ✗ D) 0.10. Explanation The calculated value of the F-distributed test statistic, F, is s12 / s22. With equal variances, F = 1.
Question #20 of 82
Question ID: 438846
The average U.S. dollar/Euro exchange rate from a sample of 36 monthly observations is $1.00/Euro. The population variance is 0.49. What is the 95% confidence interval for the mean U.S. dollar/Euro exchange rate? ✗ A) $0.8657 to $1.1343. ✗ B) $0.5100 to $1.4900. ✓ C) $0.7713 to $1.2287. ✗ D) $0.8075 to $1.1925. Explanation The population standard deviation is the square root of the variance (√0.49 = 0.7). Because we know the population standard deviation, we use the z-statistic. The z-statistic reliability factor for a 95% confidence interval is 1.960. The confidence interval is $1.00 ± 1.960($0.7 / √36) or $1.00 ± $0.2287.
Question #21 of 82
Question ID: 438834
You observe the following pairs of annual returns on a sample of two stocks: [(0.03, 0.01), (0.06, 0.06), (0.02, 0.05), (0.13,
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0.08)]. The covariance of the returns on the two stocks is closest to: ✓ A) 0.0011. ✗ B) 0.0033. ✗ C) 0.00165. ✗ D) 0.0008. Explanation The means for the two stocks are: (0.03 + 0.06 + 0.02 + 0.13) / 4 = 0.06 and (0.01 + 0.06 + 0.05 + 0.08) / 4 = 0.05. The covariance is: [(0.03 − 0.06)(0.01 − 0.05) + (0.06 − 0.06)(0.06 − 0.05) + (0.02 − 0.06)(0.05 − 0.05) + (0.13 − 0.06)(0.08 − 0.05)] / [4 - 1] = 0.0011.
Question #22 of 82
Question ID: 438868
A Type I error occurs when the null hypothesis: ✗ A) fails to be rejected when it is false. ✓ B) is rejected when it is true. ✗ C) is rejected when it is false. ✗ D) fails to be rejected when it is true. Explanation A Type I error occurs when the null hypothesis is rejected when it is true. A Type II error occurs when the null hypothesis fails to be rejected when it is false.
Question #23 of 82
Question ID: 438859
An analyst conducts a two-tailed z-test to determine if small cap returns are significantly different from 10%. The sample size was 200. The computed z-statistic is 2.3. Using a 5% level of significance, which statement is most accurate? ✗ A) Fail to reject the null hypothesis and conclude that small cap returns are close enough to 10% that we cannot say they are significantly different from 10%. ✗ B) You cannot determine what to do with the information given. ✓ C) Reject the null hypothesis and conclude that small cap returns are significantly different from 10%. ✗ D) A sample size of 200 indicates that we should fail to reject the null. Explanation At the 5% level of significance the critical z-statistic for a two-tailed test is 1.96 (assuming a large sample size). The null hypothesis is H0: x = 10%. The alternative hypothesis is HA: x ≠ 10%. Because the computed z-statistic is greater than the critical z-statistic (2.33 > 1.96), we reject the null hypothesis and we conclude that small cap returns are significantly different than 10%.
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Question #24 of 82
Question ID: 438876
The mean monthly return for an equity portfolio over 60 months is 1.5 percent. The standard deviation is 3.0 percent. The value of the test statistic to test the hypothesis that mean monthly return is equal to zero is closest to: ✓ A) 3.87. ✗ B) 30.00. ✗ C) 2.19. ✗ D) 0.50. Explanation z = (1.5% - 0.0%) / [(3.0% / √60)] = 3.87
Question #25 of 82
Question ID: 438873
A survey is taken to determine whether the average starting salaries of CFA charterholders is equal to or greater than $59,000 per year. What is the test statistic given a sample of 135 newly acquired CFA charterholders with a mean starting salary of $64,000 and a standard deviation of $5,500? ✗ A) -0.91. ✗ B) -10.56. ✓ C) 10.56. ✗ D) 0.91. Explanation With a large sample size (135) the z-statistic is used. The z-statistic is calculated by subtracting the hypothesized parameter from the parameter that has been estimated and dividing the difference by the standard error of the sample statistic. Here, the test statistic = (sample mean - hypothesized mean) / (population standard deviation / (sample size)1/2) = (X − µ) / (σ / n1/2) = (64,000 59,000) / (5,500 / 1351/2) = (5,000) / (5,500 / 11.62) = 10.56.
Question #26 of 82
Question ID: 438882
Which of the following statements about hypothesis testing is most accurate? ✗ A) When the critical Z-statistic is greater than the sample Z-statistic in a two-tailed test, reject the null hypothesis and accept the alternative hypothesis. ✗ B) A Type I error is rejecting the null hypothesis when it is true, and a Type II error is accepting the alternative hypothesis when it is false. ✓ C) A hypothesized mean of 3, a sample mean of 6, and a standard error of the sampling means of 2 give a sample Z-statistic of 1.5. ✗ D) A two-tailed test on a large sample with a significance level of 0.01 has confidence intervals of ±1.96 standard errors. Explanation
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The 0.01 level of significance has confidence intervals of ± 2.58 standard errors. A Type II error is wrongly accepting the null hypothesis. The null hypothesis should be rejected when the sample Z-statistic is greater than the critical Z-statistic.
Question #27 of 82
Question ID: 438851
Susan Bellows is comparing the return on equity for two industries. She is convinced that the return on equity for the discount retail industry (DR) is greater than that of the luxury retail (LR) industry. What are the hypotheses for a test of her comparison of return on equity? ✓ A) H0: µDR ≤ µLR versus Ha: µDR > µLR. ✗ B) H0: µDR = µLR versus Ha: µDR ≠ µLR. ✗ C) H0: µDR ≠ µLR versus Ha: µDR = µLR. ✗ D) H0: µDR = µLR versus Ha: µDR < µLR. Explanation The alternative hypothesis is determined by the theory or the belief. The researcher specifies the null as the hypothesis that she wishes to reject (in favor of the alternative). Note that this is a one-sided alternative because of the "greater than" belief.
Questions #28-32 of 82 Austin Roberts wants to test if the mean price of houses in the area is greater than $145,000. A random sample of 36 houses in the area has a mean price of $149,750. The population standard deviation is $24,000, and Roberts wants to conduct hypothesis testing at a 1 percent level of significance.
Question #28 of 82
Question ID: 438893
The appropriate alternative hypothesis is: ✗ A) Ha: µ ± $145,000. ✓ B) Ha: µ > $145,000. ✗ C) Ha: µ < $145,000. ✗ D) Ha: µ &ge $145,000. Explanation Ha: µ > $145,000
Question #29 of 82
Question ID: 438894
The value of the calculated test statistic is closest to: ✗ A) z = 0.67. ✗ B) z = 8.13. ✗ C) z = 4.00. ✓ D) z = 1.19. Explanation
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Question #30 of 82
Question ID: 438895
Which of the following most accurately describes the appropriate test structure? ✓ A) One-tailed test. ✗ B) Two-tailed test. ✗ C) F-test. ✗ D) Chi-square test. Explanation The alternative hypothesis, Ha: µ > $145,000, only allows for values greater than the hypothesized value. Thus, this is a one-sided (one-tailed) test.
Question #31 of 82
Question ID: 438896
The critical value of the z-statistic is: ✗ A) z = ±2.33. ✗ B) z = ±1.96. ✓ C) z = +2.33. ✗ D) z = -2.33. Explanation For a one-tailed z-test at the 1% level of significance, the critical z-value is z0.01 = 2.33. Since the test is one-tailed on the upper end (i.e., Ha: µ > 145,000), we use a positive z-critical value
Question #32 of 82
Question ID: 438897
At a 1 percent level of significance, Roberts should: ✓ A) fail to reject the null hypothesis. ✗ B) reject the null hypothesis. ✗ C) accept the null hypothesis. ✗ D) neither reject nor fail to reject the null hypothesis. Explanation The decision rule is: reject H0 if z-computed > z-critical. Since 1.1875 < 2.33, Roberts will fail to reject the null.
Question #33 of 82
Question ID: 438821
Assume that the following returns are a sample of annual returns for firms in the clothing industry. Given the following sample of returns, what are the sample variance and standard deviation? Firm 1
Firm 2
Firm 3
Firm 4
Firm 5
15%
2%
5%
(7%)
0%
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Variance
Standard Deviation
✓ A) 64.5
8.0
✗ B) 51.6
7.2
✗ C) 22.0
4.7
✗ D) 32.4
5.7
Explanation The sample variance is found by taking the sum of all squared deviations from the mean and dividing by (n − 1). [(15 − 3)2 + (2 − 3)2 + (5 − 3)2 + (-7 − 3)2 + (0 − 3)2] / (5 − 1) = 64.5 The sample standard deviation is found by taking the square root of the sample variance. √64.5 = 8.03
Question #34 of 82
Question ID: 438870
Which of the following statements about testing a hypothesis using a Z-test is least accurate? ✗ A) If the calculated Z-statistic lies outside the critical Z-statistic range, the null hypothesis can be rejected. ✗ B) A Type I error is rejecting the null hypothesis when it is actually true. ✓ C) The calculated Z-statistic determines the appropriate significance level to use. ✗ D) The confidence interval for a two-tailed test of a population mean at the 5% level of significance is that the sample mean falls between ±1.96 σ/√n of the null hypothesis value. Explanation The significance level is chosen before the test so the calculated Z-statistic can be compared to an appropriate critical value.
Questions #35-37 of 82 John Starwall, CFA, is assigned the task of estimating the risk and return for security X, which is being considered as an addition to his current portfolio. If the investment is accepted, then the total portfolio will consist of five different equities that are not highly correlated. Security X has an estimated beta of 1.4. The probabilities of various states of the economy and the expected return for security X given each state are as follows:
State of
Pi
E(Ri )
Recession
0.3
0.01
Average
0.6
0.10
0.1
0.20
Economy
Economy Boom
Question #35 of 82
Question ID: 438823
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Based on the information above, what is the estimated expected return for security X?
✗ A) 15.5%. ✓ B) 8.3%. ✗ C) 10.0%. ✗ D) 8.0%. Explanation E(R) = 0.3(0.01) + 0.6(0.1) + 0.1(0.2) = 0.003 + 0.06 + 0.02 = 0.083 or 8.3%
Question #36 of 82
Question ID: 438824
Based on the information above, what is the estimated standard deviation for security X?
✗ A) 10.1%. ✓ B) 5.6%. ✗ C) 7.8%. ✗ D) 0.3%. Explanation Variance = 0.3(0.01 - 0.083)2 + 0.6(0.10 - 0.083)2 + 0.1(0.20 - 0.083)2 = 0.00160 + 0.00017 + 0.00137 = 0.00314 Standard Deviation = (0.00314)0.5 = 0.056 or 5.6%
Question #37 of 82
Question ID: 438825
Based on the information above, what are the appropriate measures for assessing whether or not this security should be added to the portfolio?
✓ A) Expected return and standard deviation of the security. ✗ B) Beta and standard deviation of the security. ✗ C) Expected return, beta, and standard deviation of the security. ✗ D) Expected return and beta of the security. Explanation Total risk is comprised of systematic and unsystematic risk. In a well-diversified portfolio only systematic risk is important. Systematic risk is measured by beta. However, if the portfolio is not well diversified, total risk as measured by standard deviation must be considered. The stand-alone risk for this security is important in the decision because not all of the unsystematic risk is diversified away. Therefore, the correct answer is expected return and standard deviation. Beta only applies to a well-diversified portfolio.
Question #38 of 82
Question ID: 438858
An analyst conducts a two-tailed test to determine if mean earnings estimates are significantly different from reported earnings. The sample size is greater than 25 and the computed test statistic is 1.25. Using a 5% significance level, which of the following statements is most accurate? ✗ A) To test the null hypothesis, the analyst must determine the exact sample size and calculate the degrees of freedom for the test.
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✗ B) The analyst should reject the null hypothesis and conclude that the earnings estimates are significantly different from reported earnings. ✓ C) The analyst should fail to reject the null hypothesis and conclude that the earnings estimates are not significantly different from reported earnings. ✗ D) The appropriate test to apply is a two-tailed chi-square test. Explanation The null hypothesis is that earnings estimates are equal to reported earnings. To reject the null hypothesis, the calculated test statistic must fall outside the two critical values. IF the analyst tests the null hypothesis with a z-statistic, the crtical values at a 5% confidence level are ±1.96. Because the calculated test statistic, 1.25, lies between the two critical values, the analyst should fail to reject the null hypothesis and conclude that earnings estimates are not significantly different from reported earnings. If the analyst uses a t-statistic, the upper critical value will be even greater than 1.96, never less, so even without the exact degrees of freedom the analyst knows any t-test would fail to reject the null.
Questions #39-42 of 82 Two samples were drawn from a normally distributed population. For the first sample, the mean was $50, and the standard deviation was $5. For the second sample, the mean was $55, and the standard deviation was $6. The first sample consists of 25 observations, and the second sample consists of 36 observations. (Note: In the following questions, the subscripts "1" and "2" indicate the first and second sample, respectively.)
Question #39 of 82
Question ID: 438905
Consider the hypotheses structured as H0: µ1 = $48 versus Ha: µ1 ≠ $48. At a 1 percent level of significance, the null hypothesis: ✗ A) should be rejected. ✗ B) cannot be tested using this sample information provided. ✗ C) should neither be rejected nor fail to be rejected. ✓ D) cannot be rejected. Explanation A two-tailed t-test is appropriate. The decision rule is to reject H0 if the t-statistic is outside the range defined by ±t at α/2 = 0.025 with df = 24. The t-statistic =
t24 = ±t24 at α/2 = p = 0.025 = ±2.797, therefore, H0 cannot be rejected.
Question #40 of 82 Using a 5 percent level of significance and a hypothesis test structure of
Question ID: 438906
versus
, the null hypothesis:
✓ A) cannot be rejected. ✗ B) should neither be rejected nor fail to be rejected. ✗ C) should be rejected. ✗ D) cannot be tested using this sample information provided. Explanation
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The chi-square test is used to test hypotheses concerning a single population variance. Since this is a one-tailed test, the decision rule is to reject H0 if Χ2 > the critical chi-square value at α = 0.05 with df = 24. . The right-tail critical chi-square value is 36.415. Since Χ2 = 25 ≤ 36.415, H0 cannot be rejected.
Question #41 of 82
Question ID: 438907
Consider the hypotheses structured as H0: µ1 ≤ $48 versus Ha: µ1 > $48. At a 5 percent level of significance, the null hypothesis: ✗ A) cannot be tested using the sample information provided. ✗ B) cannot be rejected. ✗ C) should neither be rejected nor fail to be rejected. ✓ D) should be rejected. Explanation A one-tailed t-test is appropriate. The decision rule is to reject H0 if the computed t-statistic > t-critical at α = 0.05 with df = 24. The computed value of the t-statistic =
, and t-critical = t24 = 1.711. Since t > t-critical, H0 should be
rejected.
Question #42 of 82
Question ID: 438908
Using a 5 percent level of significance for a test of the null of H0: σ1 = σ2 versus the alternative of Ha: σ1 ¼ σ2, the null hypothesis: ✗ A) should neither be rejected nor fail to be rejected. ✗ B) should be rejected. ✗ C) cannot be tested using the sample information provided. ✓ D) cannot be rejected. Explanation The two-sided F-test is appropriate to test the equality of population variances. The decision rule is to reject H0 if the computed test statistic, F, exceeds the critical F-value at α/2. For the information provided, F =
= 36/25 = 1.44. At a 0.025 level of
significance with d1 = 35 and d2 = 24, F-critical = 1.915. Since F < F-critical (1.44 < 1.915), we fail to reject the null hypothesis.
Question #43 of 82
Question ID: 438819
A sample covariance for the common stock of the Earth Company and the S&P 500 is −9.50. Which of the following statements regarding the estimated covariance of the two variables is most accurate? ✗ A) The two variables will have a strong tendency to move in opposite directions. ✗ B) The two variables will have a slight tendency to move together. ✓ C) The relationship between the two variables is not easily predicted by the calculated covariance. ✗ D) The relationship between the two variables is highly predictable given the calculated covariance. Explanation The actual value of the covariance for two variables is not very meaningful because its measurement is extremely sensitive to the scale of the two variables, ranging from negative to positive infinity. Covariance can, however be converted into the
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correlation coefficient, which is more straightforward to interpret.
Question #44 of 82
Question ID: 438830
You observe the following annual returns on a stock portfolio for the last nine years: [5%, 7%, 20%, −4%, 2%, 5%, 13%, 18%, 4%]. Which measure of central tendency has the highest value? ✗ A) For these observations, the mean, median, and mode are equal. ✓ B) The mean. ✗ C) The mode. ✗ D) The median. Explanation Ordering the observations gives: [−4%, 2%, 4%, 5%, 5%, 7%, 13%, 18%, 20%]. The mode is the value that occurs most frequently; in this case, 5%. The median is the fifth ordered observation, 5%. The mean is: (−4% + 2% + 4% + 5% + 5% + 7% + 13% + 18% + 20%) / 9 = 7.8%.
Question #45 of 82
Question ID: 438848
The 95 percent confidence interval of the sample mean of the price earnings ratio for all traded stocks is 19 to 44. There are over 5,000 traded stocks and the sample size of this test is 100. Given that the expected value of the price earnings ratio is 31.5, the standard error of the ratio is closest to: ✗ A) 2.58. ✓ B) 6.38. ✗ C) 1.96. ✗ D) 12.50. Explanation The confidence interval is 31.5 ± 1.96x, where x is the standard error. If we take the upper bound, we know that 31.5 +/- 1.96x = 44, or 1.96x = 12.5. Hence, x = 6.38.
Question #46 of 82
Question ID: 438902
Abby Ness is an analyst for a firm that specializes in evaluating firms involved in mineral extraction. Ness believes that the earnings of copper extracting firms are more volatile than those of bauxite extraction firms. In order to test this, Ness examines the volatility of returns for 31 copper firms and 25 bauxite firms. The standard deviation of earnings for copper firms was $2.69, while the standard deviation of earnings for bauxite firms was $2.92. Ness's Null Hypothesis is σ12 = σ22. Based on the samples, can we reject the null hypothesis at a 95% confidence level using an F-statistic and why? Null is: ✗ A) rejected. The F-value exceeds the critical value by 0.71. ✗ B) rejected. The F-value exceeds the critical value by 0.849. ✗ C) not rejected. The critical value exceeds the F-value by 0.194. ✓ D) not rejected. The critical value exceeds the F-value by 0.71.
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Explanation F = s12 / s22 = $2.922 / $2.692 = 1.18 From an F table, the critical value with numerator df = 24 and denominator df = 30 is 1.89.
Question #47 of 82
Question ID: 438845
What is the appropriate test statistic for constructing confidence intervals for the population mean of a nonnormal distribution when the population variance is unknown and the sample size is large (n ≥ 30)? ✗ A) The z-statistic at α with n degrees of freedom. ✗ B) The t-statistic at α/2 with n degrees of freedom. ✓ C) The z-statistic or the t-statistic. ✗ D) The t-statistic at α with 29 degrees of freedom. Explanation When the sample size is large, and the central limit theorem can be relied upon to assure a sampling distribution that approaches normal, either the t-statistic or the z-statistic is acceptable for constructing confidence intervals for the population mean. However, the t-statistic will provide a more conservative range (wider) at a given level of significance.
Question #48 of 82
Question ID: 438901
You have collected monthly returns for a mutual fund and want to test the null hypothesis that the standard deviation exceeds the advertised standard deviation of 3.5 percent. The most appropriate test statistic is based on a: ✓ A) chi-square test. ✗ B) t-test. ✗ C) F-test. ✗ D) z-test. Explanation The chi-square test is used to test hypotheses about a single population variance.
Question #49 of 82
Question ID: 438877
A goal of an "innocent until proven guilty" justice system is to place a higher priority on: ✓ A) avoiding type I errors. ✗ B) the null hypothesis. ✗ C) avoiding type II errors. ✗ D) avoiding type III errors. Explanation In an "innocent until proven guilty" justice system, the null hypothesis is that the accused is innocent. The hypothesis can only be
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rejected by evidence proving guilt beyond a reasonable doubt, favoring the avoidance of type I errors.
Question #50 of 82
Question ID: 438866
Maria Huffman is the Vice President of Human Resources for a large regional car rental company. Last year, she hired Graham Brickley as Manager of Employee Retention. Part of the compensation package was the chance to earn one of the following two bonuses: if Brickley can reduce turnover to less than 30%, he will receive a 25% bonus. If he can reduce turnover to less than 25%, he will receive a 50% bonus (using a significance level of 10%). The population of turnover rates is normally distributed. The population standard deviation of turnover rates is 1.5%. A recent sample of 100 branch offices resulted in an average turnover rate of 24.2%. Which of the following statements is most accurate? ✗ A) For the 50% bonus level, the critical value is -1.65 and Huffman should give Brickley a 50% bonus. ✗ B) Brickley should not receive either bonus. ✓ C) For the 50% bonus level, the test statistic is -5.33 and Huffman should give Brickley a 50% bonus. ✗ D) For the 25% bonus level, the test statistic is -10.66. Explanation Using the process of Hypothesis testing: Step 1: State the Hypothesis. For 25% bonus level - Ho: m ≥ 30% Ha: m < 30%; For 50% bonus level - Ho: m ≥ 25% Ha: m < 25%. Step 2: Select Appropriate Test Statistic. Here, we have a normally distributed population with a known variance (standard deviation is the square root of the variance) and a large sample size (greater than 30.) Thus, we will use the z-statistic. Step 3: Specify the Level of Significance. α = 0.10. Step 4: State the Decision Rule. This is a one-tailed test. The critical value for this question will be the z-statistic that corresponds to an α of 0.10, or an area to the left of the mean of 40% (with 50% to the right of the mean). Using the z-table (normal table), we determine that the appropriate critical value = -1.28 (Remember that we highly recommend that you have the "common" z-statistics memorized!) Thus, we will reject the null hypothesis if the calculated test statistic is less than -1.28. Step 5: Calculate sample (test) statistics. Z (for 50% bonus) = (24.2 - 25) / (1.5 / √ 100) = −5.333. Z (for 25% bonus) = (24.2 30) / (1.5 / √ 100) = −38.67. Step 6: Make a decision. Reject the null hypothesis for both the 25% and 50% bonus level because the test statistic is less than the critical value. Thus, Huffman should give Soberg a 50% bonus. The other statements are false. The critical value of -1.28 is based on the significance level, and is thus the same for both the 50% and 25% bonus levels.
Question #51 of 82
Question ID: 438820
The mean equity risk premium over a 40-year period is equal to 8.0 percent. The standard deviation of the sample is 12 percent. The standard error of the sample mean is closest to:
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✗ A) 1.26%. ✗ B) 8.00%. ✗ C) 0.30%. ✓ D) 1.90%. Explanation Note the size of the sample here is the number of years. σX = 12 / √40 = 1.90%
Question #52 of 82
Question ID: 438843
A nursery sells trees of different types and heights. Suppose that 75 pine trees are sold for planting at City Hall. These 75 trees average 60 inches in height with a standard deviation of 16 inches. Using this information, construct a 95% confidence interval for the mean height of all trees in the nursery. ✗ A) 0.8 + 1.96(16). ✓ B) 60 + 1.96(1.85). ✗ C) 0.8 + 1.96(1.85). ✗ D) 60 + 1.96(16). Explanation Because we know the population standard deviation, we use the z-statistic. A 95% confidence level is constructed by taking the population mean and adding and subtracting the product of the z-statistic reliability (zα/2) factor times the known standard deviation of the population divided by the square root of the sample size: x ± zα/2 × ( σ / n1/2) = 60 ± (1.96) × (16 / 751/2) = 60 ± (1.96) × (16 / 8.6603) = 60 ± (1.96) × (1.85).
Question #53 of 82
Question ID: 438844
What is the appropriate test statistic for constructing confidence intervals for the population mean of a normal distribution when the population variance is unknown? ✗ A) The z-statistic at α with n degrees of freedom. ✗ B) The t-statistic at α/2 with n degrees of freedom. ✗ C) The z-statistic with n - 1 degrees of freedom. ✓ D) The t-statistic at α/2 with n - 1 degrees of freedom. Explanation Use the t-statistic at α/2 and n - 1 degrees of freedom when the population variance is unknown, regardless of sample size.
Question #54 of 82
Question ID: 438869
If the null hypothesis is innocence, then the statement "It is better that the guilty go free, than the innocent are punished" is an example of
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preferring a:
✗ A) two tailed test over a one tailed test. ✗ B) higher level of significance. ✓ C) type II error over a type I error. ✗ D) type I error over a type II error. Explanation The statement shows a preference for not rejecting the null hypothesis when it is false (a type II error), over rejecting it when it is true (a type I error).
Question #55 of 82
Question ID: 438867
A bottler of iced tea wishes to ensure that an average of 16 ounces of tea is in each bottle. In order to analyze the accuracy of the bottling process, a random sample of 150 bottles is taken. Using a t-distributed test statistic of -1.09 and a 5% level of significance, the bottler should: ✗ A) reject the null hypothesis and conclude that bottles do not contain an average of 16 ounces of tea. ✗ B) reject the null hypothesis and conclude that bottles contain an average 16 ounces of tea. ✗ C) not reject the null hypothesis and conclude that bottles do not contain an average of 16 ounces of tea. ✓ D) not reject the null hypothesis and conclude that bottles contain an average 16 ounces of tea. Explanation Ho: µ = 16; Ha: µ ≠ 16. Do not reject the null since |t| = 1.09 < 1.96 (critical value).
Question #56 of 82
Question ID: 438871
A Type I error: ✗ A) fails to reject a true null hypothesis. ✓ B) rejects a true null hypothesis. ✗ C) fails to reject a false null hypothesis. ✗ D) rejects a false null hypothesis. Explanation A Type I Error is defined as rejecting the null hypothesis when it is actually true. The probability of committing a Type I error is the significance level or alpha risk.
Question #57 of 82
Question ID: 438865
Which of the following statements about hypothesis testing is least accurate? ✗ A) The null hypothesis is a statement about the value of a population parameter.
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✗ B) A Type I error is rejecting the null hypothesis when it is actually true. ✓ C) If the alternative hypothesis is Ha: µ > µ0, a two-tailed test is appropriate. ✗ D) A Type II error is failing to reject a false null hypothesis. Explanation The hypotheses are always stated in terms of a population parameter. Type I and Type II are the two types of errors you can make - reject a null hypothesis that is true or fail to reject a null hypothesis that is false. The alternative may be one-sided (in which case a > or < sign is used) or two-sided (in which case a ≠ is used).
Question #58 of 82
Question ID: 438857
In the process of hypothesis testing, what is the proper order for these steps? ✗ A) Specify the level of significance. State the hypotheses. Make a decision. Collect the sample and calculate the sample statistics. ✗ B) State the hypotheses. Collect the sample and calculate the sample statistics. Make a decision. Specify the level of significance. ✗ C) Collect the sample and calculate the sample statistics. State the hypotheses. Specify the level of significance. Make a decision. ✓ D) State the hypotheses. Specify the level of significance. Collect the sample and calculate the test statistics. Make a decision. Explanation The hypotheses must be established first. Then the test statistic is chosen and the level of significance is determined. Following these steps, the sample is collected, the test statistic is calculated, and the decision is made.
Question #59 of 82
Question ID: 438875
A return series with 250 observations has a sample mean of 10 percent and a standard deviation of 15 percent. The standard error of the sample mean is closest to: ✗ A) 0.06. ✗ B) 15.80. ✗ C) 3.87. ✓ D) 0.95. Explanation The standard error of the sample mean is the standard deviation of the sample divided by the square root of the number of observations in the sample. In this case, (15 / √250) = 0.95.
Question #60 of 82
Question ID: 438836
The mean and variance of a sample average help define the distributional characteristics of a sample distribution and allow assumptions to be made about this distribution when the sample size is large. Which of the following choices represent the
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standard deviation of a sample average? ✗ A) σ(X)2 ✗ B) 1/n. ✓ C) σ(X)/√n. ✗ D) σ(X)2/ n. Explanation The central limit theorem states that for a population with a mean μ and a finite variance σ2, the sampling distribution of the sample mean of all possible samples of size n will be approximately normally distributed with a mean equal to μ and a variance equal to σ2/n. Thus, the standard deviation of the sample average is equal to: σ/√n. This standard deviation measure is known as the standard error.
Question #61 of 82
Question ID: 438881
The mean monthly return for an equity portfolio over 60 months is 1.5%. The standard deviation is 3.0%. The value of the test statistic to test the hypothesis that mean monthly return is equal to zero is closest to: ✓ A) 3.87. ✗ B) 0.50. ✗ C) 2.19. ✗ D) 30.00 Explanation
Question #62 of 82
Question ID: 438855
Which of the following statements least describes the procedure for testing a hypothesis?
✗ A) Select the level of significance, compute the test statistic, and make a decision. ✗ B) Select the level of significance, formulate the decision rule, and make a decision. ✓ C) Compute the sample value of the test statistic, set up a rejection (critical) region, and make a decision. ✗ D) Develop a hypothesis, compute the test statistic, and make a decision. Explanation Depending upon the author there can be as many as seven steps in hypothesis testing which are: 1. Stating the hypotheses. 2. Identifying the test statistic and its probability distribution. 3. Specifying the significance level. 4. Stating the decision rule. 5. Collecting the data and performing the calculations. 6. Making the statistical decision. 7. Making the economic or investment decision.
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Question #63 of 82
Question ID: 438880
Given the following hypothesis: The null hypothesis is H0 : µ = 5 The alternative is H1 : µ ≠ 5 The mean of a sample of 17 is 7 The population standard deviation is 2.0 What is the calculated z-statistic? ✗ A) 4.00. ✓ B) 4.12. ✗ C) 8.25. ✗ D) 8.00. Explanation The z-statistic is calculated by subtracting the hypothesized parameter from the parameter that has been estimated and dividing the difference by the standard error of the sample statistic. Here, the test statistic = (sample mean − hypothesized mean) / (population standard deviation / (sample size)1/2 = (X − μ) / (σ / n1/2) = (7 − 5) / (2 / 171/2) = (2) / (2 / 4.1231) = 4.12.
Question #64 of 82
Question ID: 438847
A random sample of 100 technology stocks earned an average of 10%. Assuming the distribution of equity returns is normal and the population standard deviation is 5%, the 95% confidence interval for the population mean is: ✗ A) 9.91% to 10.90%. ✗ B) 9.50% to 10.50%. ✓ C) 9.02% to 10.98%. ✗ D) 5.00% to 15.00%. Explanation Zα / 2 = Z0.025 = 1.96. So, 0.1 +/−1.96(0.05 / 10) = 9.02% to 10.98%.
Question #65 of 82
Question ID: 438874
If the sample size is greater than 30 and population variance is unknown, the appropriate test for the sample mean is the: ✗ A) p-test or F-test. ✗ B) t-test. ✗ C) z-test. ✓ D) t-test or z-test. Explanation The central limit theorem makes it appropriate to use the z-test with an unknown variance if the sample size is large enough (n ≥
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30), regardless of the distribution of the population. Since the t- and the z-distributions converge as sample size increases, either test is appropriate, although the t-test is a more conservative estimate.
Question #66 of 82
Question ID: 438833
A sample of paired points A and B is shown below. What is the covariance between the values of A and B?
Sample A B 1
1 2
2
4 5
3
9 7
4
11 10
5
14 12
✗ A) 82.20. ✓ B) 20.55. ✗ C) 7.80. ✗ D) 29.76. Explanation
Sample
A
A-mean (A)
B
B-mean (B)
Product
1
1
−6.8
2
−5.2
35.36
2
4
−3.8
5
−2.2
8.36
3
9
1.2
7
−0.2
−0.24
4
11
3.2
10
2.8
8.96
5
14
6.2
12
4.8
29.76
mean
7.8
7.2
Sum
82.2
Cov = 82.20 / (5 - 1) = 20.55
Question #67 of 82
Question ID: 438856
The first step in the process of hypothesis testing is: ✗ A) the calculation of sample statistics. ✗ B) the collection of the sample. ✗ C) selecting the test statistic. ✓ D) to state the hypotheses. Explanation The researcher must state the hypotheses prior to the collection and analysis of the data. More importantly, it is necessary to
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know the hypotheses before selecting the appropriate test statistic.
Question #68 of 82
Question ID: 438826
For the investments shown in the table below, what are the mean, median, and mode of the returns? Investment Return (%) A
12
B
14
C
9
D
13
E
7
F
8
G
12
Mean
Median
Mode
✓ A) 10.71%
12%
12%
✗ B) 12.00%
12%
12%
✗ C) 11.20%
12%
12%
✗ D) 10.71%
9%
13%
Explanation The mean is the average return computed by summing the returns and dividing by the number of investments: 75 / 7 = 10.71%. The median is the mid-point or central number of returns arranged from highest to lowest or lowest to highest. In this case: 7, 8, 9, 12, 12, 13, 14. The median return is 12%. The mode is the return that occurs most frequently. In this case, 12% is also the mode.
Question #69 of 82
Question ID: 438850
A sample of size 25 is selected from a normal population. This sample has a mean of 15 and the population variance is 4. Using this information, construct a 95% confidence interval for the population mean, m. ✓ A) 15 ± 1.96(0.4). ✗ B) 15 ± 1.96(2). ✗ C) 15 ± 1.96(4). ✗ D) 15 ± 1.96(0.8). Explanation Because we can compute the population standard deviation, we use the z-statistic. A 95% confidence level is constructed by taking the population mean and adding and subtracting the product of the z-statistic reliability (zα/2) factor times the known standard deviation of the population divided by the square root of the sample size (note that the population variance is given and
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its positive square root is the standard deviation of the population): x ± zα/2 × ( σ / n1/2) = 15 ± 1.96 × (41/2 / 251/2) = 15 ± 1.96 × (0.4).
Questions #70-71 of 82 Annual Returns on ABC Mutual Fund 1991 1992 1993 1994 1995 1996 1997 1998 1999
2000
11.0% 12.5% 8.0% 9.0% 13.0% 7.0% 15.0% 2.0% -16.5% 11.0%
Question #70 of 82
Question ID: 438828
What is the arithmetic mean return for ABC Mutual Fund for the period 1991-2000? Arithmetic Mean ✗ A) 28.2% ✗ B) 11% ✓ C) 7.2% ✗ D) 8.2%
Explanation arithmetic mean = (11 + 12.5 + 8 + 9 + 13 + 7 + 15 + 2 − 16.5 + 11) / 10 = 7.20
Question #71 of 82
Question ID: 438829
Assuming a mean of 7.2%, what is the sample standard deviation of the returns for ABC Mutual Fund for the period 1991-2000? ✗ A) 7.8%. ✓ B) 9.1%. ✗ C) 10.2%. ✗ D) 9.8%. Explanation Standard deviation = [∑i (xi − X)2 / (n − 1)]1/2 = √(744.10 / 9) = √(82.68) = 9.1%
Question #72 of 82
Question ID: 438903
The sampling distribution of variances is described by which distribution? ✗ A) F-distribution. ✗ B) Student's t-distribution. ✗ C) Normal distribution.
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✓ D) Chi-square distribution. Explanation The sampling distribution of variances is described by the chi-square distribution.
Question #73 of 82
Question ID: 438854
Which of the following is the correct sequence of events for testing a hypothesis? ✗ A) State the hypothesis, select the level of significance, compute the test statistic, formulate the decision rule, and make a decision. ✓ B) State the hypothesis, select the level of significance, formulate the decision rule, compute the test statistic, and make a decision. ✗ C) State the hypothesis, formulate the decision rule, select the level of significance, compute the test statistic, and make a decision. ✗ D) State the hypothesis, formulate the decision rule, compute the test statistic, select the level of significance, and make a decision. Explanation Depending upon the author there can be as many as seven steps in hypothesis testing which are: 1. Stating the hypotheses. 2. Identifying the test statistic and its probability distribution. 3. Specifying the significance level. 4. Stating the decision rule. 5. Collecting the data and performing the calculations. 6. Making the statistical decision. 7. Making the economic or investment decision.
Question #74 of 82
Question ID: 438879
Which of the following statements is FALSE? ✗ A) The probability of committing a Type I error is the significance level of the test. ✓ B) If a person is presumed innocent unless proven otherwise, finding a guilty person innocent is an example of a Type I error. ✗ C) Failing to reject the null when it is false is an example of a Type II error. ✗ D) A Type I error is rejecting the null when it is actually true. Explanation This is a Type II error. The null hypothesis is that the defendant is innocent which is not rejected when it should have been.
Question #75 of 82
Question ID: 438900
In order to test if Stock A is more volatile than Stock B, prices of both stocks are observed to construct the sample variance of the two
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stocks. The appropriate test statistics to carry out the test is the: ✗ A) Chi-square test. ✗ B) Z test. ✓ C) F test. ✗ D) t test. Explanation The F test is used to test the differences of variance between two samples.
Question #76 of 82
Question ID: 438872
In a two-tailed test of a hypothesis concerning whether a population mean is zero, Jack Olson computes a t-statistic of 2.7 based on a sample of 20 observations where the distribution is normal. If a 5% significance level is chosen, Olson should: ✗ A) not make a conclusion pending additional observations. ✓ B) reject the null hypothesis and conclude that the population mean is significantly different from zero. ✗ C) reject the null hypothesis and conclude that the population mean is not significantly different from zero. ✗ D) fail to reject the null hypothesis that the population mean is not significantly different from zero. Explanation At a 5% significance level, the critical t-statistic using the Student's t-distribution table for a two-tailed test and 19 degrees of freedom (sample size of 20 less 1) is ± 2.093 (with a large sample size the critical z-statistic of 1.960 may be used). Because the critical t-statistic of 2.093 is to the left of the calculated t-statistic of 2.7, meaning that the calculated t-statistic is in the rejection range, we reject the null hypothesis and we conclude that the population mean is significantly different from zero.
Question #77 of 82
Question ID: 438831
The last three year-end returns for a stock are 5%, −2%, and 1%. Using an arithmetic mean, the sample standard deviation is closest to: ✗ A) 1.33%. ✗ B) 2.87%. ✗ C) 3.11%. ✓ D) 3.51%. Explanation The sample arithmetic mean is: X = (5 − 2 + 1) / 3 = 1.33% The sample standard deviation is:
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Questions #78-82 of 82 XYZ Corp. Annual Stock Prices 1995 1996 1997 1998 1999 2000 22% 5% -7% 11% 2% 11%
Question #78 of 82
Question ID: 438838
What is the mean return for XYZ stock? ✗ A) 8.0%. ✗ B) -7.0%. ✓ C) 7.3%. ✗ D) 11.0%. Explanation
Question #79 of 82
Question ID: 438839
Assuming that the distribution of XYZ stock returns is a population, what is the population variance (in %2)? ✗ A) 7.7. ✓ B) 80.2. ✗ C) 5.0. ✗ D) 6.8. Explanation The population variance, σ2, is found by taking the mean of all squared deviations from the mean.
Question #80 of 82
Question ID: 438840
Assuming that the distribution of XYZ stock returns is a population, what is the population standard deviation? ✓ A) 8.96%. ✗ B) 5.02%. ✗ C) 46.22%. ✗ D) 6.84%. Explanation The population standard deviation, σ, is found by taking the square root of the population variance.
Question #81 of 82
Question ID: 438841
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Assuming that the distribution of XYZ stock returns is a sample, what is the sample variance (in %2)? ✗ A) 7.4. ✗ B) 5.0. ✓ C) 96.3. ✗ D) 72.4. Explanation The sample variance, s2, uses n - 1 in the denominator:
Question #82 of 82
Question ID: 438842
Assuming that the distribution of XYZ stock returns is a sample, what is the sample standard deviation? ✗ A) 72.4%. ✗ B) 7.4%. ✗ C) 96.3%. ✓ D) 9.8%. Explanation The sample standard deviation is the square root of the sample variance:
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Topic 21: Linear Regression with One Regressor
Test ID: 8450958
Question #1 of 37
Question ID: 438933
Which of the following statements regarding the coefficient of determination is least accurate? The coefficient of determination: ✓ A) may range from −1 to +1. ✗ B) is the percentage of the total variation in the dependent variable that is explained by the independent variable. ✗ C) cannot decrease as independent variables are added to the model. ✗ D) is the ratio of explained variation to total variation. Explanation In a simple regression, the coefficient of determination is calculated as the correlation coefficient squared and ranges from 0 to +1.
Question #2 of 37
Question ID: 438929
Which of the following statements about linear regression analysis is most accurate? ✗ A) The coefficient of determination is defined as the strength of the linear relationship between two variables. ✗ B) A perfectly negative correlation can be depicted by a correlation coefficient of +1. ✓ C) An assumption of linear regression is that the residuals are independently distributed. ✗ D) When there is a strong relationship between two variables we can conclude that a change in one will cause a change in the other. Explanation Even when there is a strong relationship between two variables, we cannot conclude that a causal relationship exists. The coefficient of determination is defined as the percentage of total variation in the dependent variable explained by the independent variable.
Question #3 of 37
Question ID: 438934
A simple linear regression equation had a coefficient of determination (R2) of 0.8. What is the correlation coefficient between the dependent and independent variables and what is the covariance between the two variables if the variance of the independent variable is 4 and the variance of the dependent variable is 9?
Correlation coefficient ✗ A) 0.91
Covariance
4.80
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✗ B) 0.89
4.80
✗ C) 0.91
5.34
✓ D) 0.89
5.34
Explanation The correlation coefficient is the square root of the R2, r = 0.89. To calculate the covariance multiply the correlation coefficient by the product of the standard deviations of the two variables: COV = 0.89 × √4 × √9 = 5.34
Question #4 of 37
Question ID: 438924
Consider the regression results from the regression of Y against X for 50 observations:
Y = 5.0 - 1.5 X The standard error of the estimate is 0.40 and the standard error of the coefficient is 0.45. The predicted value of Y if X is 10 is: ✓ A) -10. ✗ B) 10. ✗ C) 4.5. ✗ D) 20. Explanation The predicted value of Y is: Y = 5.0 - [1.5 (10)] = 5.0 - 15 = -10
Question #5 of 37
Question ID: 438936
Unlike the coefficient of determination, the coefficient of correlation: ✓ A) indicates whether the slope of the regression line is positive or negative. ✗ B) indicates the percentage of variation explained by a regression model. ✗ C) measures the strength of association between the two variables more exactly. ✗ D) can have an absolute value greater than 1. Explanation In a simple linear regression the coefficient of determination (R2) is the squared correlation coefficient, so it is positive even when the correlation is negative.
Question #6 of 37
Question ID: 438915
The purpose of regression is to:
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✗ A) explain the variation in the independent variable. ✗ B) get the largest R2 possible. ✗ C) explain the mean of the independent variable. ✓ D) explain the variation in the dependent variable. Explanation The goal of a regression is to explain the variation in the dependent variable.
Question #7 of 37
Question ID: 438910
In a regression analysis, the effects from independent variables that are not included in the model are embodied in the: ✓ A) error term. ✗ B) slope coefficient. ✗ C) intercept. ✗ D) scattergram. Explanation The error term embodies the effect of variables omitted from the model.
Question #8 of 37
Question ID: 438931
Linear regression is based on a number of assumptions. Which of the following is least likely an assumption of linear regression? ✗ A) A linear relationship exists between the dependent and independent variables. ✓ B) There is at least some correlation between the error terms from one observation to the next. ✗ C) Values of the independent variable are not correlated with the error term. ✗ D) The variance of the error terms each period remains the same. Explanation When correlation exists, autocorrelation is present. As a result, residual terms are not normally distributed. This is inconsistent with linear regression.
Question #9 of 37
Question ID: 438921
A simple linear regression is run to quantify the relationship between the return on the common stocks of medium sized companies (Mid Caps) and the return on the S&P 500 Index, using the monthly return on Mid Cap stocks as the dependent variable and the monthly return on the S&P 500 as the independent variable. The results of the regression are shown below:
Coefficient Standard Error of Coefficient t-Value Intercept 1.71
2.950
0.58
S&P 500 1.52
0.130
11.69
R2 = 0.599
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Use the regression statistics presented above and assume this historical relationship still holds in the future period. If the expected return on the S&P 500 over the next period were 11%, the expected return on Mid Cap stocks over the next period would be: ✓ A) 18.4%. ✗ B) 33.8%. ✗ C) 20.3%. ✗ D) 25.6%. Explanation Y = intercept + slope(X) Mid Cap Stock returns = 1.71 + 1.52(11) =18.4%
Question #10 of 37
Question ID: 438938
An analyst performs two simple regressions. The first regression analysis has an R-squared of 0.40 and a beta coefficient of 1.2. The second regression analysis has an R-squared of 0.77 and a beta coefficient of 1.75. Which one of the following statements is most accurate?
✓ A) The second regression equation has more explaining power than the first regression equation. ✗ B) The R-squared of the first regression indicates that there is a 0.40 correlation between the independent and the dependent variables.
✗ C) The first regression equation has more explaining power than the second regression equation. ✗ D) The beta coefficient of the 2nd regression indicates that this regression has more explaining power than the first.
Explanation The coefficient of determination (R-squared) is the percentage of variation in the dependent variable explained by the variation in the independent variable. The larger R-squared (0.77) of the second regression means that 77% of the variability in the dependent variable is explained by variability in the independent variable, while only 40% of that is explained in the first regression. This means that the second regression has more explaining power than the first regression. Note that the Beta is the slope of the regression line and doesn't measure explaining power.
Question #11 of 37
Question ID: 438913
In the scatter plot below, the correlation between the return on stock A and the market index is:
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✗ A) zero. ✗ B) not discernable using the scatter plot. ✓ C) positive. ✗ D) negative. Explanation In the scatter plot, higher values of the return on stock A are associated with higher values of the return on the market, i.e. a positive correlation between the two variables.
Question #12 of 37
Question ID: 438911
Which of the following statements regarding scatter plots is most accurate? Scatter plots: ✗ A) illustrate the scatterings of a single variable. ✗ B) are used to examine the third moment of a distribution (skewness). ✓ C) illustrate the relationship between two variables. ✗ D) are used to examine the fourth moment of a distribution (kurtosis). Explanation A scatter plot is a collection of points on a graph where each point represents the values of two variables. They are used to examine the relationship between two variables.
Question #13 of 37
Question ID: 424959
Which of the following is least likely an assumption of linear regression? The: ✗ A) expected value of the residuals is zero. ✓ B) residuals are mean reverting; that is, they tend towards zero over time. ✗ C) residuals are independently distributed. ✗ D) variance of the residuals is constant. Explanation The assumptions regarding the residuals are that the residuals have a constant variance, have a mean of zero, and are
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independently distributed.
Question #14 of 37
Question ID: 438940
Assume an analyst performs two simple regressions. The first regression analysis has an R-squared of 0.90 and a slope coefficient of 0.10. The second regression analysis has an R-squared of 0.70 and a slope coefficient of 0.25. Which one of the following statements is most accurate? ✗ A) The influence on the dependent variable of a one unit increase in the independent variable is 0.9 in the first analysis and 0.7 in the second analysis. ✗ B) Results of the second analysis are more reliable than the first analysis. ✗ C) The influence on the dependent variable of a one unit increase in the independent variable is 0.7 in the first analysis and 0.9 in the second analysis. ✓ D) The first regression has more explanatory power than the second regression. Explanation The coefficient of determination (R-squared) is the percentage of variation in the dependent variable explained by the variation in the independent variable. The larger R-squared (0.90) of the first regression means that 90 percent of the variability in the dependent variable is explained by variability in the independent variable, while 70 percent of that is explained in the second regression. This means that the first regression has more explanatory power than the second regression. Note that the Beta is the slope of the regression line and doesn't measure explanatory power.
Question #15 of 37
Question ID: 424957
Which of the following is least likely an assumption of a simple regression? ✓ A) The variance of the error term is one. ✗ B) The expected value of the error term is zero. ✗ C) There is a linear relationship between dependent and independent variables. ✗ D) The error term is normally distributed. Explanation There is no requirement that the variance of the error term should be equal to one.
Question #16 of 37
Question ID: 424960
Which of the following is least likely an assumption of linear regression? ✗ A) The variance of the residuals is constant. ✗ B) There is a linear relation between the dependent and independent variables. ✓ C) The independent variable is correlated with the residuals. ✗ D) The residuals are normally distributed. Explanation
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The assumption is that the independent variable is uncorrelated with the residuals.
Question #17 of 37
Question ID: 438916
The capital asset pricing model is given by: Ri =Rf + Beta ( Rm -Rf) where Rm = expected return on the market, Rf = risk-free market and Ri = expected return on a specific firm. The dependent variable in this model is: ✓ A) Ri. ✗ B) Rm. ✗ C) Rf. ✗ D) Rm - Rf. Explanation The dependent variable is the variable whose variation is explained by the other variables. Here, the variation in ri is explained by the variation in the other variables, rf and rm.
Question #18 of 37
Question ID: 438912
If the correlation between two variables is −1.0, the scatter plot would appear along a: ✗ A) a curved line running from southwest to northeast. ✗ B) straight line running from southwest to northeast. ✓ C) straight line running from northwest to southeast. ✗ D) a curved line centered in the scatter plot. Explanation If the correlation is −1.0, then higher values of the y-variable will be associated with lower values of the x-variable. The points would lie on a straight line running from northwest to southeast.
Question #19 of 37
Question ID: 438932
Assume you perform two simple regressions. The first regression analysis has an R-squared of 0.80 and a beta coefficient of 0.10. The second regression analysis has an R-squared of 0.80 and a beta coefficient of 0.25. Which one of the following statements is most accurate? ✗ A) Results of the second analysis are more reliable than the first analysis. ✗ B) Results from the first analysis are more reliable than the second analysis. ✗ C) The influence on the dependent variable of a one-unit increase in the independent variable is the same in both analyses. ✓ D) Results from both analyses are equally reliable. Explanation The coefficient of determination (R-squared) is the percentage of variation in the dependent variable explained by the variation in the independent variable. The R-squared (0.80) being identical between the first and second regressions means that 80 percent of the
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variability in the dependent variable is explained by variability in the independent variable for both regressions. This means that the first regression has the same explaining power as the second regression.
Question #20 of 37
Question ID: 438914
Joe Harris is interested in why the returns on equity differ from one company to another. He chose several company-specific variables to explain the return on equity, including financial leverage and capital expenditures. In his model: ✗ A) return on equity is the explanatory variable, and financial leverage and capital expenditure are the explained variables. ✗ B) return on equity, financial leverage, and capital expenditures are all independent variables. ✓ C) return on equity is the dependent variable, and financial leverage and capital expenditures are independent variables. ✗ D) return on equity is the independent variable, and financial leverage and capital expenditures are dependent variables. Explanation The dependent variable is return on equity. This is what he wants to explain. The variables he uses to do the explaining (i.e., the independent variables) are financial leverage and capital expenditures.
Question #21 of 37
Question ID: 438918
The independent variable in a regression equation is called all of the following EXCEPT: ✗ A) predicting variable. ✗ B) explanatory variable. ✗ C) exogenous variable. ✓ D) predicted variable. Explanation The dependent variable is the predicted variable.
Question #22 of 37
Question ID: 438935
Which term is least likely to apply to a regression model? ✗ A) Coefficient of determination. ✗ B) Goodness of fit. ✗ C) R2. ✓ D) Coefficient of variation. Explanation Goodness of fit, coefficient of determination and R2 are different names for the same concept. The coefficient of variation is not directly part of a regression model.
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Question #23 of 37
Question ID: 438927
Which of the following are included in a sample regression function? I. The intercept. II. The error term. III. The slope coefficient. IV. The independent variable. ✗ A) I, II, III, and IV. ✗ B) III and IV only. ✗ C) I and III only. ✓ D) I, III, and IV only. Explanation The sample regression function has a residual and not an error term. Although the residual and error term serve a similar purpose in their respective equations, there are distinctions.
Question #24 of 37
Question ID: 438917
A regression analysis has the goal of: ✗ A) measuring how the properties of the variables regress towards each other. ✓ B) estimating how changes in independent variables affect a dependent variable. ✗ C) estimating how changes in dependent variable affect an independent variable. ✗ D) measuring the tendency of both independent and dependent variables to regress towards their respective means. Explanation A regression analysis has the goal of measuring how changes in one variable, called a dependent or explained variable, can be explained by changes in one or more other variables called the independent or explanatory variables.
Question #25 of 37
Question ID: 424958
Which of the following is least likely an assumption of linear regression analysis? ✓ A) The Y values are all less than 3 standard deviations from the regression line. ✗ B) The error term is normally distributed. ✗ C) The expected value of the residuals is zero. ✗ D) The X values are uncorrelated with the error terms. Explanation In a normal distribution, approximately 99%, but not necessarily all, of the observations are within 3 standard deviations of the regression line.
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Question #26 of 37
Question ID: 438939
A simple linear regression is run to quantify the relationship between the return on the common stocks of medium sized companies (Mid Caps) and the return on the S&P 500 Index, using the monthly return on Mid Cap stocks as the dependent variable and the monthly return on the S&P 500 as the independent variable. The results of the regression are shown below: Standard Error Coefficient
t-Value of coefficient
Intercept
1.71
2.950
0.58
S&P 500
1.52
0.130
11.69
R2= 0.599
The strength of the relationship, as measured by the correlation coefficient, between the return on Mid Cap stocks and the return on the S&P 500 for the period under study was: ✗ A) 2.950. ✓ B) 0.774. ✗ C) 0.599. ✗ D) 0.130. Explanation You are given R2 or the coefficient of determination of 0.599 and are asked to find R or the coefficient of correlation. The square root of 0.599 = 0.774.
Question #27 of 37
Question ID: 438941
Sample regression coefficients are often estimated with a process known as: ✗ A) a population regression function. ✗ B) a scattergram. ✓ C) ordinary least squares. ✗ D) Ockham's razor. Explanation Ordinary least squares or OLS is a method for estimating population coefficients. It provides the line that minimizes the squared deviations of the dependent variables in the sample from the sample regression line.
Question #28 of 37
Question ID: 438937
What does the R2 of a simple regression of two variables measure and what calculation is used to equate the correlation coefficient to the coefficient of determination? R2 measures:
Correlation coefficient
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✗ A) percent of variability of the independent variable that is explained by the variability of
R2 = r2
the dependent variable ✗ B) percent of variability of the independent variable that is explained by the variability of the
R2 = r × 2
dependent variable ✓ C) percent of variability of the dependent variable that is explained by the variability of the
R2 = r2
independent variable ✗ D) percent of variability of the dependent variable that is explained by the variability of the
R2 = r × 2
independent variable
Explanation R2, or the Coefficient of Determination, is the square of the coefficient of correlation (r). The coefficient of correlation describes the strength of the relationship between the X and Y variables. The standard error of the residuals is the standard deviation of the dispersion about the regression line. The t-statistic measures the statistical significance of the coefficients of the regression equation. In the response: "percent of variability of the independent variable that is explained by the variability of the dependent variable," the definitions of the variables are reversed.
Question #29 of 37
Question ID: 438920
An analyst is examining the relationship between two random variables, RCRANTZ and GSTERN. He performs a linear regression that produces an estimate of the relationship: RCRANTZ = 61.4 − 5.9GSTERN Which interpretation of this regression equation is least accurate? ✗ A) The covariance of RCRANTZ and GSTERN is negative. ✗ B) In this regression, RCRANTZ is the dependent variable and GSTERN is the independent variable. ✓ C) If GSTERN increases by one unit, RCRANTZ should increase by 5.9 units. ✗ D) The intercept term implies that if GSTERN is zero, RCRANTZ is 61.4. Explanation The slope coefficient in this regression is -5.9. This means a one unit increase of GSTERN suggests a decrease of 5.9 units of RCRANTZ. The slope coefficient is the covariance divided by the variance of the independent variable. Since variance (a squared term) must be positive, a negative slope term implies that the covariance is negative.
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Question #30 of 37
Question ID: 438922
Given: Y = 2.83 + 1.5X What is the predicted value of the dependent variable when the value of an independent variable equals 2? ✗ A) 2.83 ✓ B) 5.83 ✗ C) 6.50 ✗ D) -0.55 Explanation Y = 2.83 + (1.5)(2) = 2.83 + 3 = 5.83
Question #31 of 37
Question ID: 438926
In the estimated regression equation Y = 0.78 - 1.5 X, which of the following is least accurate when interpreting the slope coefficient? ✗ A) The dependent variable declines by -1.5 units if X increases by 1 unit. ✗ B) The dependent variable increases by 1.5 units if X decreases by 1 unit. ✗ C) -1.5 is the elasticity of Y with respect to X. ✓ D) If the value of X is zero, the value of Y will be -1.5. Explanation The slope represents the change in the dependent variable for a one-unit change in the independent variable. If the value of X is zero, the value of Y will be equal to the intercept, in this case, 0.78.
Question #32 of 37
Question ID: 438942
As part of a regression analysis, an analyst finds that: Y - b1 × X = -1.8 and b1 = 3.2. Based upon these results, for every unit increase in the independent variable, on average the dependent variable increases by: ✗ A) 5.0. ✓ B) 3.2. ✗ C) 1.8. ✗ D) 1.4. Explanation b1 gives the slope coefficient which indicates the average change in the dependent variable given a unit change in the independent variable.
Question #33 of 37
Question ID: 438928
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Trudy Baker, FRM and Steven Phillips, FRM are planning to do a regression analysis. They discuss specifying the equation they wish to estimate. Baker proposes the specification E(Y i|Xi) = B0 + (B1) × (Xi2). Phillips proposes the specification (Yi|Xi) = B0 + (B1 × Xi)2. Which, if either, is appropriate when applying linear regression? ✓ A) The specification of Baker but not that of Phillips. ✗ B) Both the specification of Baker and Phillips. ✗ C) The specification of Phillips but not that of Baker. ✗ D) Neither the specification of Baker nor that of Phillips. Explanation Since the specification of Phillips would essentially be B0 + (B12) × (Xi2), this precludes the application of linear regression analysis because there is an exponent on B1, i.e., the specification is nonlinear with respect to the parameters. When we refer to a linear regression model we generally assume that the equation is linear in the parameters; it may or may not be linear in the variables.
Question #34 of 37
Question ID: 438923
Paul Frank is an analyst for the retail industry. He is examining the role of television viewing by teenagers on the sales of accessory stores. He gathered data and estimated the following regression of sales (in millions of dollars) on the number of hours watched by teenagers (TV, in hours per week):
Salest = 1.05 + 1.6 TVt The predicted sales if television watching is 5 hours per week is: ✗ A) $8.00 million. ✓ B) $9.05 million. ✗ C) $2.65 million. ✗ D) $1.05 million. Explanation The predicted sales are: Sales = $1.05 + [$1.6 (5)] = $1.05 + $8.00 = $9.05 million.
Question #35 of 37
Question ID: 438925
Sera Smith, a research analyst, had a hunch that there was a relationship between the percentage change in a firm's number of salespeople and the percentage change in the firm's sales during the following period. Smith ran a regression analysis on a sample of 50 firms, which resulted in a slope of 0.72, an intercept of +0.01, and an R2 value of 0.65. Based on this analysis, if a firm made no changes in the number of sales people, what percentage change in the firm's sales during the following period does the regression model predict? ✗ A) +0.65%. ✓ B) +1.00%. ✗ C) +0.72%. ✗ D) +0.10%. Explanation
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The slope of the regression represents the linear relationship between the independent variable (the percent change in sales people) and the dependent variable, while the intercept represents the predicted value of the dependent variable if the independent variable is equal to zero. In this case, the percentage change in sales is equal to: 0.72(0) + 0.01 = +0.01.
Question #36 of 37
Question ID: 438930
The assumptions underlying linear regression include all of the following EXCEPT the: ✓ A) independent variable is linearly related to the residuals (or disturbance term). ✗ B) disturbance term is normally distributed with an expected value of 0. ✗ C) dependent variable and independent variable are linearly related. ✗ D) disturbance term is homoskedastic and is independently distributed. Explanation The independent variable is uncorrelated with the residuals (or disturbance term). The other statements are true. The disturbance term is homoskedastic because it has a constant variance. It is independently distributed because the residual for one observation is not correlated with that of another observation. Note: The opposite of homoskedastic is heteroskedastic. For the examination, memorize the assumptions underlying linear regression!
Question #37 of 37
Question ID: 438919
Paul Frank is an analyst for the retail industry. He is examining the role of television viewing by teenagers on the sales of accessory stores. He gathered data and estimated the following regression of sales (in millions of dollars) on the number of hours watched by teenagers (in hours per week):
Salest = 1.05 + 1.6 TVt Which of the following is the most accurate interpretation of the estimated results? If TV watching: ✗ A) goes up by one hour per week, sales of accessories increase by $1.60. ✗ B) is zero (that is, every teenager turns off the TV for a week), the expected sales of accessories is $0. ✓ C) goes up by one hour per week, sales of accessories increase by $1.6 million. ✗ D) changes, no change in sales is expected. Explanation The interpretation of the slope coefficient is the change in the dependent variable (sales in millions of dollars) for a given one-unit change in the independent variable (TV hours per week). The intercept of 1.05 means that 1.05 million dollars worth of accessories is expected to be sold even if TV watching is zero.
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Topic 22: Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals Test ID: 8450964
Question #1 of 17
Question ID: 438947
Consider the regression results from the regression of Y against X for 50 observations:
Y = 0.78 + 1.2 X The standard error of the estimate is 0.40 and the standard error of the coefficient is 0.45. Which of the following reports the correct value of the t-statistic for the slope and correctly evaluates its statistical significance with 95 percent confidence? ✗ A) t = 1.789; slope is not significantly different from zero. ✗ B) t = 1.200; slope not significantly different from zero. ✗ C) t = 3.000; slope is significantly different from zero. ✓ D) t = 2.667; slope is significantly different from zero. Explanation The test statistic is t = (1.2 - 0) / 0.45 = 2.667. The critical t-values for 48 degrees of freedom are ± 2.011. Therefore, the slope is different from zero.
Question #2 of 17
Question ID: 438959
If the variance of the residuals is not constant across all observations in the sample, the regression exhibits heteroskedasticity. Effects of heteroskedasticity include which of the following problems? I. The coefficient estimates in the regression model are affected. II. If the standard errors are too small, but the coefficient estimates themselves are not affected, the t-statistics may be too large and the null hypothesis of no statistical significance will be rejected too often. ✓ A) II only. ✗ B) I only. ✗ C) Both I and II. ✗ D) Neither I nor II. Explanation Effects of heteroskedasticity include: The standard errors are usually unreliable estimates. The coefficient estimates are not affected. If the standard errors are too small, but the coefficient estimates themselves are not affected, the t-statistics will be too large and the null hypothesis of no statistical significance is rejected too often. The opposite will be true if the standard errors are too large.
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Question #3 of 17
Question ID: 438946
An analyst is regressing fund returns against the return on the Wilshire 5000 to determine whether beta is equal to 1.0. The analyst is trying to determine whether the number of observations should be increased. Which of the following is a reason why the test will have higher power if the number of observations is increased? The:
✗ A) mean squared error of the regression will be lower. ✗ B) constant of the regression will be closer to zero. ✗ C) estimate of beta will be farther away from 1.0. ✓ D) standard error of the regression will be lower. Explanation A larger number of observations will decrease the standard error of the regression which will increase the size of the test statistic if beta is different than 1.0.
Question #4 of 17
Question ID: 438954
Consider the following graph of residuals and the regression line from a time-series regression:
These residuals exhibit the regression problem of:
✗ A) homoskedasticity. ✗ B) multicolinearity. ✗ C) autocorrelation. ✓ D) heteroskedasticity.
Explanation The residuals appear to be from two different distributions over time; in the earlier periods, the model fits rather well compared to the later periods.
Question #5 of 17
Question ID: 438958
Which expression best represents the condition homoskedasticity? (In the expressions assume σ2 > 0)
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✗ A) corr(Xi, εi) = 0. ✗ B) E(εi|Xi) = σ2. ✗ C) corr(εi, εi + j) = 0. ✓ D) V(εi|Xi) = σ2. Explanation Homoskedasticity means the variance of εi is constant and unrelated to the value of the independent variable.
Question #6 of 17
Question ID: 438945
A sample of 200 monthly observations is used to run a simple linear regression: Returns = b0 + b1Leverage + u. The t-value for the regression coefficient of leverage is calculated as t = - 1.09. A 5 percent level of significance is used to test whether leverage has a significant influence on returns. The correct decision is to:
✗ A) reject the null hypothesis and conclude that leverage significantly explains returns. ✗ B) do not reject the null hypothesis and conclude that leverage significantly explains returns. ✗ C) reject the null hypothesis and conclude that leverage does not significantly explain returns. ✓ D) do not reject the null hypothesis and conclude that leverage does not significantly explain returns. Explanation Do not reject the null since |-1.09|<1.96(critical t-value).
Question #7 of 17
Question ID: 438955
An analyst is estimating whether a fund's excess return for a quarter is related to interest rates and last quarter's excess return. The regression equation is found to have unconditional heteroskedasticity and serial correlation. Which of the following is most accurate? Parameter estimates will be:
✗ A) accurate but statistical inference about the parameters will not be valid. ✓ B) inaccurate and statistical inference about the parameters will not be valid. ✗ C) inaccurate but statistical inference about the parameters will be valid. ✗ D) accurate and statistical inference about the parameters will be valid. Explanation One of the independent variables is a lagged value of the dependent variable. This means that serial correlation will cause an inaccurate parameter estimate. Serial correlation always impacts the statistical inference about the parameters. Unconditional heteroskedasticity never impacts statistical inference or parameter accuracy.
Question #8 of 17
Question ID: 438944
Consider the following estimated regression equation: ROEt = 0.23 - 1.50 CEt
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The standard error of the slope coefficient is 0.40 and the number of observations is 32. The 95 percent confidence interval for the slope coefficient, b1, is: ✗ A) {-2.300 < b1 < -0.700}. ✗ B) {0.683 < b1 < 2.317}. ✗ C) {-3.542 < b1 < 0.542}. ✓ D) {-2.317 < b1 < -0.683}. Explanation The confidence interval is -1.50 ± 2.042 (0.40), or {-2.317 < b1 < -0.683}.
Questions #9-10 of 17 Assume you ran a multiple regression to gain a better understanding of the relationship between lumber sales, housing starts, and commercial construction. The regression uses lumber sales as the dependent variable with housing starts and commercial construction as the independent variables. The results of the regression are:
Coefficient
Standard Error
t-statistics
Intercept
5.37
1.71
3.14
Housing starts
0.76
0.09
8.44
Commercial construction
1.25
0.33
3.78
The level of significance for a 95% confidence level is 1.96
Question #9 of 17
Question ID: 438951
Construct a 95% confidence interval for the slope coefficient for Housing Starts. ✓ A) 0.76 ± 1.96(0.09). ✗ B) 1.25 ± 1.96(0.33). ✗ C) 0.76 ± 1.96(8.44). ✗ D) 1.25 ± 1.96(3.78). Explanation The confidence interval for the slope coefficient is b1 ± (tc × sb1).
Question #10 of 17
Question ID: 438952
Construct a 95% confidence interval for the slope coefficient for Commercial Construction. ✗ A) 0.76 ± 1.96(8.44). ✓ B) 1.25 ± 1.96(0.33). ✗ C) 0.76 ± 1.96(0.09). ✗ D) 1.25 ± 1.96(3.78).
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Explanation The confidence interval for the slope coefficient is b1 ± (tc × sb1).
Question #11 of 17
Question ID: 438953
An analyst has been assigned the task of evaluating revenue growth for an online education provider company that specializes in training adult students. She has gathered information about student ages, number of courses offered to all students each year, years of experience, annual income and type of college degrees, if any. A regression of annual dollar revenue on the number of courses offered each year yields the results shown below.
Coefficient Estimates Predictor
Coefficient Standard Error of the Coefficient
Intercept
0.10
0.50
Slope (Number of Courses)
2.20
0.60
Which statement about the slope coefficient is most correct, assuming a 5 percent level of significance and 50 observations? ✗ A) t-Statistic: 0.20. Slope: Significantly different from zero. ✓ B) t-Statistic: 3.67. Slope: Significantly different from zero. ✗ C) t-Statistic: 3.67. Slope: Not significantly different from zero. ✗ D) t-Statistic: 0.20. Slope: Not significantly different from zero. Explanation t = 2.20/0.60 = 3.67. Since the t-statistic is larger than an assumed critical value of about 2.0, the slope coefficient is statistically significant.
Question #12 of 17
Question ID: 438949
Consider the following estimated regression equation:
AUTOt = 0.89 + 1.32 PIt The standard error of the coefficient is 0.42 and the number of observations is 22. The 95 percent confidence interval for the slope coefficient, b1, is: ✗ A) {-0.766 < b1 < 3.406}. ✗ B) {0.900 < b1 < 1.740}. ✗ C) {0.480 < b1 < 2.160}. ✓ D) {0.444 < b1 < 2.196}. Explanation The degrees of freedom are found by n-k-1 with k being the number of independent variables or 1 in this case. DF = 22-1-1 = 20. Looking up 20 degrees of freedom on the student's t distribution for a 95% confidence level and a 2 tailed test gives us a critical value of 2.086. The confidence interval is 1.32 ± 2.086 (0.42), or {0.444 < b1 < 2.196}.
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Question #13 of 17
Question ID: 438956
Which of the following conditions will least likely affect the statistical inference about regression parameters by itself? ✗ A) Serial correlation. ✗ B) Conditional heteroskedasticity. ✗ C) Multicollinearity. ✓ D) Unconditional heteroskedasticity. Explanation Unconditional heteroskedasticity does not impact the statistical inference concerning the parameters.
Question #14 of 17
Question ID: 438957
Which of the following statements regarding heteroskedasticity is FALSE? ✗ A) Conditional heteroskedasticity is the case in which the residuals are correlated with the values of the independent variables. ✓ B) The assumption of linear regression is that the residuals are heteroskedastic. ✗ C) Heteroskedasticity may occur in cross-section or time-series analyses. ✗ D) Heteroskedasticity results in an estimated variance that is too large and, therefore, affects statistical inference.
Explanation The assumption of regression is that the residuals are homoskedastic (i.e., the residuals are drawn from the same distribution).
Question #15 of 17
Question ID: 438948
Consider the regression results from the regression of Y against X for 50 observations:
Y = 0.78 - 1.5 X The standard error of the estimate is 0.40 and the standard error of the coefficient is 0.45. Which of the following reports the correct value of the t-statistic for the slope and correctly evaluates H0: b1 ≥ 0 versus Ha: b1 < 0 with 95 percent confidence? ✓ A) t = -3.333; slope is significantly negative. ✗ B) t = -3.750; slope is significantly different from zero. ✗ C) t = 3.750; slope is significantly different from zero. ✗ D) t = 3.333; slope not significantly different from zero. Explanation
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The test statistic is t = (-1.5 - 0) / 0.45 = -3.333. The critical t-value for 48 degrees of freedom is +/- 1.667. However, in the Schweser Notes you should use the closest degrees of freedom number of 40 df. which is +/-1.684. Therefore, the slope is different from zero. We reject the null in favor of the alternative.
Question #16 of 17
Question ID: 438960
The Gauss-Markov theorem says that if the linear regression model assumptions are true and the regression errors display homoskedasticity, then the ordinary least squares (OLS) estimators exhibit which of the following properties? ✓ A) In repeated sampling, the averages of the coefficients from a sample will be distributed around the true population parameters. ✗ B) The OLS estimated coefficients are based on non-linear functions. ✗ C) The OLS estimated coefficients have the maximum variance compared to other methods of estimating the coefficients. ✗ D) The OLS estimate of the variance of the errors is biased. Explanation If the estimator is the best available (has the minimum variance), exhibits linearity, and is unbiased, it is said to be the best linear unbiased estimator. The OLS estimated coefficients are unbiased, which means that in repeated sampling the averages of the coefficients from the sample will be distributed around the true population parameters.
Question #17 of 17
Question ID: 438943
The estimated slope coefficient from a single linear regression model is 0.55 with a standard error of 0.30. Assuming the sample for this model has 1,000 observations, what can we conclude about the 95% confidence interval for the model's slope coefficient? ✗ A) The null hypothesis that the slope coefficient is different than zero should be accepted. ✗ B) The slope coefficient is significantly different from zero. ✓ C) The slope coefficient is not significantly different from zero. ✗ D) The null hypothesis that the slope coefficient is equal to zero should be accepted. Explanation The confidence interval for b1 is: b1 ± (tc × sb1) The critical two-tail t-values are roughly ± 1.96 (with a high number of observations). The 95% confidence interval is computed as: 0.55 ± (1.96 × 0.3) = 0.55 ± 0.588 or -0.038 to 1.138 Because this confidence interval includes zero, we can conclude that the slope coefficient is not significantly different from zero. In hypothesis testing, we either reject or fail to reject the null hypothesis.
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Topic 23 - Linear Regression with Multiple Regressors
Test ID: 8450981
Question #1 of 10
Question ID: 438968
A variable is regressed against three other variables, x, y, and z. Which of the following would NOT be an indication of multicollinearity? X is closely related to:
✗ A) 3y + 2z. ✗ B) 3. ✗ C) 9y, and x is closely related to 4z. ✓ D) y2. Explanation If x is related to y2, the relationship between x and y is not linear, so multicollinearity does not exist. If x is equal to a constant (3), it will be correlated with the intercept term.
Question #2 of 10
Question ID: 438961
Which of the following statements regarding the results of a regression analysis is FALSE? The: ✗ A) slope coefficient in a multiple regression is the change in the dependent variable for a one-unit change in the independent variable, holding all other variables constant. ✗ B) slope coefficients in the multiple regression are referred to as partial betas. ✗ C) intercept is the value that the dependent variable takes on if all the independent variables had a value of zero. ✓ D) slope coefficient in a multiple regression is the value of the dependent variable for a given value of the independent variable. Explanation The slope coefficient is the change in the dependent variable for a one-unit change in the independent variable.
Question #3 of 10
Question ID: 438966
Which of the following is a potential remedy for multicollinearity?
✗ A) Increase the sample size. ✗ B) Add dummy variables to the regression. ✗ C) Take first differences of the dependent variable. ✓ D) Omit one or more of the collinear variables.
Explanation
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The first differencing is not a remedy for the collinearity, nor is the inclusion of dummy variables. Increasing the sample size may improve the t-statistics associated with the estimates, but does is not a remedy to the problem. The best potential remedy is to attempt to eliminate highly correlated variables.
Question #4 of 10
Question ID: 438964
Assume that in a particular multiple regression model, it is determined that the error terms are uncorrelated with each other. Which of the following statements is most accurate? ✗ A) Serial correlation may be present in this multiple regression model, and can be confirmed only through a Durbin-Watson test. ✗ B) Unconditional heteroskedasticity present in this model should not pose a problem, but can be corrected by using robust standard errors. ✓ C) This model is in accordance with the basic assumptions of multiple regression analysis because the errors are not serially correlated. ✗ D) Multicollinearity exists in this multiple regression model, and can be corrected through the addition of a correlated variable. Explanation One of the basic assumptions of multiple regression analysis is that the error terms are not correlated with each other. In other words, the error terms are not serially correlated. Multicollinearity and heteroskedasticity are problems in multiple regression that are not related to the correlation of the error terms.
Question #5 of 10
Question ID: 438970
An analyst further studies the independent variables of a study she recently completed. The correlation matrix shown below is the result. Which statement best reflects possible problems with a multivariate regression?
Age Age
Education Experience Income
1.00
Education 0.50
1.00
Experience 0.95
0.55
1.00
0.65
0.89
Income
0.60
1.00
✗ A) Income is not needed. ✓ B) Experience may be a redundant variable. ✗ C) Education may be unnecessary. ✗ D) Age should be excluded from the regression. Explanation The correlation coefficient of experience with age and income, respectively, is close to +1.00. This indicates a problem of multicollinearity and should be addressed by excluding experience as an independent variable.
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Question #6 of 10
Question ID: 438962
When interpreting the results of a multiple regression analysis, which of the following terms represents the value of the dependent variable when the independent variables are all equal to zero? ✓ A) Intercept term. ✗ B) t-value. ✗ C) p-value. ✗ D) Slope coefficient. Explanation The intercept term is the value of the dependent variable when the independent variables are set to zero.
Question #7 of 10
Question ID: 438969
An analyst runs a regression of portfolio returns on three independent variables. These independent variables are price-to-sales (P/S), price-to-cash flow (P/CF), and price-to-book (P/B). The analyst discovers that the p-values for each independent variable are relatively high. However, the F-test has a very small p-value. The analyst is puzzled and tries to figure out how the F-test can be statistically significant when the individual independent variables are not significant. What violation of regression analysis has occurred?
✗ A) unconditional heteroskedasticity. ✓ B) multicollinearity. ✗ C) serial correlation. ✗ D) conditional heteroskedasticity. Explanation An indication of multicollinearity is when the independent variables individually are not statistically significant but the F-test suggests that the variables as a whole do an excellent job of explaining the variation in the dependent variable.
Question #8 of 10
Question ID: 438967
Which of the following statements regarding multicollinearity is FALSE? ✗ A) If the t-statistics for the individual independent variables are insignificant, yet the F-statistic is significant, this indicates the presence of multicollinearity. ✗ B) Multicollinearity makes it difficult to determine the contribution to explanation of the dependent variable of an individual explanatory variable. ✓ C) Multicollinearity may be present in any regression model. ✗ D) Multicollinearity may be a problem even if the multicollinearity is not perfect. Explanation Multicollinearity is not an issue in simple regression.
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Question #9 of 10
Question ID: 438965
Which of the following statements least accurately describes one of the fundamental multiple regression assumptions? ✗ A) There is no exact linear relationship between any two or more independent variables. ✗ B) The error term is normally distributed. ✗ C) The independent variables are not random. ✓ D) The variance of the error terms is not constant (i.e., the errors are heteroskedastic). Explanation The variance of the error term IS assumed to be constant, resulting in errors that are homoskedastic.
Question #10 of 10
Question ID: 438963
One of the underlying assumptions of a multiple regression is that the variance of the residuals is constant for various levels of the independent variables. This quality is referred to as: ✗ A) a normal distribution. ✓ B) homoskedasticity. ✗ C) a linear relationship. ✗ D) serial correlation. Explanation Homoskedasticity refers to the basic assumption of a multiple regression model that the variance of the error terms is constant.
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Topic 24: Hypothesis Tests and Confidence Intervals in Multiple Regression Test ID: 8450984
Question #1 of 21
Question ID: 438976
A dependent variable is regressed against three independent variables across 25 observations. The regression sum of squares is 119.25, and the total sum of squares is 294.45. The following are the estimated coefficient values and standard errors of the coefficients.
Coefficient Value Standard error 1
2.43
1.4200
2
3.21
1.5500
3
0.18
0.0818
For which of the coefficients can the hypothesis that they are equal to zero be rejected at the 0.05 level of significance? ✗ A) 1 and 2 only. ✗ B) 2 and 3 only. ✓ C) 3 only. ✗ D) 1, 2, and 3. Explanation The values of the t-statistics for the three coefficients are equal to the coefficients divided by the standard errors, which are 2.43 / 1.42 = 1.711, 3.21 / 1.55 = 2.070, and 0.18 / 0.0818 = 2.200. The statistic has 25 − 3 − 1 = 21 degrees of freedom. The critical value for a p-value of 0.025 (because this is a two-sided test) is 2.080, which means only coefficient 3 is significant.
Question #2 of 21
Question ID: 438971
Consider the following estimated regression equation, with standard errors of the coefficients as indicated: Salesi = 10.0 + 1.25 R&Di + 1.0 ADVi - 2.0 COMPi + 8.0 CAPi where the standard error for R&D is 0.45, the standard error for ADV is 2.2, the standard error for COMP 0.63, and the standard error for CAP is 2.5.
The equation was estimated over 40 companies. Using a 5 percent level of significance, what are the hypotheses and the calculated test statistic to test whether the slope on R&D is different from 1.0?
✓ A) H0: bR&D = 1 versus Ha: bR&D≠ 1; t = 0.556. ✗ B) H0: bR&D = 1 versus Ha: bR&D≠1; t = 2.778. ✗ C) H0: bR&D ≠ 1 versus Ha: bR&D = 1; t = 0.556. ✗ D) H0: bR&D ≠ 1 versus Ha: bR&D = 1; t = 2.778.
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Explanation The test for "is different from 1.0" requires the use of the "1" in the hypotheses and requires 1 to be specified as the hypothesized value in the test statistic. The calculated t-statistic = (1.25-1)/.45 = 0.556
Question #3 of 21
Question ID: 438974
David Black wants to test whether the estimated beta in a market model is equal to one. He collected a sample of 60 monthly returns on a stock and estimated the regression of the stock's returns against those of the market. The estimated beta was 1.1, and the standard error of the coefficient is equal to 0.4. What should Black conclude regarding the beta if he uses a 5% level of significance? The null hypothesis that beta is:
✗ A) equal to one is rejected. ✗ B) equal to one cannot be rejected or accepted. ✗ C) not equal to one cannot be rejected. ✓ D) equal to one cannot be rejected.
Explanation The calculated t-statistic is t = (1.1 − 1.0) / 0.4 = 0.25. The critical t-value for (60 − 2) = 58 degrees of freedom is approximately 2.0. Therefore, the null hypothesis that beta is equal to one cannot be rejected.
Question #4 of 21
Question ID: 440959
In the multivariable regression model: Y = B0 + B1 × Xli + B2 × X2i + εi, the formula for the standard errors of the estimated coefficients includes the variance of εi, which is represented by: σ2. Since the term is: ✓ A)
not known with certainty, the expression σ2 is replaced with:
.
✗ B) not known with certainty, the standard errors of the coefficients cannot be estimated. ✗ C)
not known with certainty, the expression σ2 is replaced with:
.
✗ D) known with certainty, the standard errors of the coefficients are known with certainty. Explanation is an estimate of σ2 when there are two independent variables and n-3 degrees of freedom.
The expression:
Question #5 of 21
Question ID: 438993
Consider the following analysis of variance (ANOVA) table:
Source Regression
Sum of squares Degrees of freedom 20
1
Mean square 20
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Error
80
40
Total
100
41
2
The F-statistic for the test of the fit of the model is closest to: ✓ A) 10.00. ✗ B) 0.10. ✗ C) 0.25. ✗ D) 0.20.
Explanation F = MSR/MSE = 20 / 2 = 10.
Questions #6-11 of 21 Autumn Voiku is attempting to forecast sales for Brookfield Farms based on a multiple regression model. Voiku has constructed the following model:
sales = b 0 + (b1 × CPI) + (b2 × IP) + (b 3 × GDP) + εt Where: sales = $ change in sales (in 000's) CPI = change in the consumer price index IP = change in industrial production (millions) GDP = change in GDP (millions) All changes in variables are in percentage terms. Voiku uses monthly data from the previous 180 months of sales data and for the independent variables. The model estimates (with coefficient standard errors in parentheses) are: sales = 10.2 + (4.6 × CPI) + (5.2 × IP) + (11.7 × GDP) (5.4) (3.5)
(5.9)
(6.8)
The sum of squared errors is 140.3 and the total sum of squares is 368.7. Voiku calculates the unadjusted R2, the adjusted R2, and the standard error of estimate to be 0.592, 0.597, and 0.910, respectively. Voiku is concerned that one or more of the assumptions underlying multiple regression has been violated in her analysis. In a conversation with Dave Grimbles, CFA, a colleague who is considered by many in the firm to be a quant specialist. Voiku says, "It is my understanding that there are five assumptions of a multiple regression model:" Assumption 1:
There is a linear relationship between the dependent and independent variables.
Assumption 2:
The independent variables are not random, and there is no correlation between
Assumption 3:
The residual term is normally distributed with an expected value of zero.
any two of the independent variables.
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Assumption 4:
The residuals are serially correlated.
Assumption 5:
The variance of the residuals is constant.
Grimbles agrees with Miller's assessment of the assumptions of multiple regression. Voiku tests and fails to reject each of the following four null hypotheses at the 99% confidence interval: Hypothesis 1:
The coefficient on GDP is negative.
Hypothesis 2:
The intercept term is equal to -4.
Hypothesis 3:
A 2.6% increase in the CPI will result in an increase in sales of more than 12.0%.
Hypothesis 4:
A 1% increase in industrial production will result in a 1% decrease in sales.
Figure 1: Partial table of the Student's t-distribution (One-tailed probabilities)
df p = 0.10 p = 0.05 p = 0.025 p = 0.01 p = 0.005 170 1.287
1.654
1.974
2.348
2.605
176 1.286
1.654
1.974
2.348
2.604
180 1.286
1.653
1.973
2.347
2.603
Figure 2: Partial F-Table critical values for right-hand tail area equal to 0.05 df1 = 1 df1 = 3 df1 = 5 df2 = 170 3.90
2.66
2.27
df2 = 176 3.89
2.66
2.27
df2 = 180 3.89
2.65
2.26
Figure 3: Partial F-Table critical values for right-hand tail area equal to 0.025 df1 = 1 df1 = 3 df1 = 5 df2 = 170 5.11
3.19
2.64
df2 = 176 5.11
3.19
2.64
df2 = 180 5.11
3.19
2.64
Question #6 of 21
Question ID: 438986
Concerning the assumptions of multiple regression, Grimbles is: ✗ A) incorrect to agree with Voiku's list of assumptions because one of the assumptions is stated incorrectly. ✗ B) correct to agree with Voiku's statement of the assumptions. ✗ C) incorrect to agree with Voiku's list of assumptions because three of the assumptions are stated incorrectly. ✓ D) incorrect to agree with Voiku's list of assumptions because two of the assumptions are stated incorrectly. Explanation Assumption 2 is stated incorrectly. Some correlation between independent variables is unavoidable; high correlation results in multicollinearity. An exact linear relationship between linear combinations of two or more independent variables should not exist.
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Assumption 4 is also stated incorrectly. The assumption is that the residuals are serially uncorrelated (i.e., they are not serially correlated).
Question #7 of 21
Question ID: 438987
For which of the four hypotheses did Voiku incorrectly fail to reject the null, based on the data given in the problem? ✓ A) Hypothesis 2. ✗ B) Hypothesis 3. ✗ C) Hypothesis 1. ✗ D) Hypothesis 4. Explanation The critical values at the 1% level of significance (99% confidence) are 2.348 for a one-tail test and 2.604 for a two-tail test (df = 176). The t-values for the hypotheses are: Hypothesis 1: 11.7 / 6.8 = 1.72 Hypothesis 2: 14.2 / 5.4 = 2.63 Hypothesis 3: 12.0 / 2.6 = 4.6, so the hypothesis is that the coefficient is 4.6, and the t-stat of that hypothesis is 0. Hypothesis 4: (5.2 + 1) / 5.9 = 1.05 Hypotheses 1 and 3 are one-tail tests; 2 and 4 are two-tail tests. Only Hypothesis 2 exceeds the critical value, so only Hypothesis 2 should be rejected.
Question #8 of 21
Question ID: 438988
The most appropriate decision with regard to the F-statistic for testing the null hypothesis that all of the independent variables are simultaneously equal to zero at the 5 percent significance level is to: ✓ A) reject the null hypothesis because the F-statistic is larger than the critical F-value of 2.66. ✗ B) reject the null hypothesis because the F-statistic is larger than the critical F-value of 3.19. ✗ C) fail to reject the null hypothesis because the F-statistic is smaller than the critical F-value of 3.19. ✗ D) fail to reject the null hypothesis because the F-statistic is smaller than the critical F-value of 2.66. Explanation RSS = 368.7 - 140.3 = 228.4, F-statistic = (228.4 / 3) / (140.3 / 176) = 95.51. The critical value for a one-tailed 5% F-test with 3 and 176 degrees of freedom is 2.66. Because the F-statistic is greater than the critical F-value, the null hypothesis that all of the independent variables are simultaneously equal to zero should be rejected.
Question #9 of 21
Question ID: 438989
Regarding Voiku's calculations of R2 and the standard error of estimate, she is: ✗ A) correct in her calculation of the unadjusted R2 but incorrect in her calculation of the standard error of estimate. ✗ B) incorrect in her calculation of the unadjusted R2 but correct in her calculation of the standard error of estimate. ✗ C) correct in her calculation of both the unadjusted R2 and the standard error of estimate. ✓ D) incorrect in her calculation of both the unadjusted R2 and the standard error of estimate.
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Explanation SEE = √[140.3 / (180 − 3 − 1)] = 0.893 unadjusted R2 = (368.7 − 140.3) / 368.7 = 0.619
Question #10 of 21
Question ID: 459978
The multiple regression, as specified, most likely suffers from: ✓ A) multicollinearity. ✗ B) serial correlation of the error terms. ✗ C) heteroskedasticity. ✗ D) omitted variables. Explanation The regression is highly significant (based on the F-stat in Part 3), but the individual coefficients are not. This is a result of a regression with significant multicollinearity problems. The t-stats for the significance of the regression coefficients are, respectively, 1.89, 1.31, 0.88, 1.72. None of these are high enough to reject the hypothesis that the coefficient is zero at the 5% level of significance (two-tailed critical value of 1.974 from t-table).
Question #11 of 21
Question ID: 438991
A 90 percent confidence interval for the coefficient on GDP is: ✓ A) 0.5 to 22.9. ✗ B) -1.9 to 19.6. ✗ C) -4.4 to 20.8. ✗ D) -1.5 to 20.0. Explanation A 90% confidence interval with 176 degrees of freedom is coefficient ± tc(se) = 11.7 ± 1.654 (6.8) or 0.5 to 22.9.
Questions #12-13 of 21 You have been asked to forecast the level of operating profit for a proposed new branch of a tire store. This forecast is one component in forecasting operating profit for the entire company for the next fiscal year. You decide to conduct multiple regression analysis using "branch store operating profit" as the dependent variable and three independent variables. The three independent variables are "population within 5 miles of the branch," "operating hours per week," and "square footage of the facility." You used data on the company's existing 23 branches to develop the model (n=23).
Regression of Operating Profit on Population, Operating Hours, and Square Footage
Dependent Variable Independent Variables Intercept
Operating Profit (Y) Coefficient Estimate
t-value
103,886
2.740
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Population within 5 miles (X1)
4.372
2.133
Operating hours per week (X2)
214.856
0.258
Square footage of facility (X3)
56.767
2.643
R2
0.983
Adjusted R2
0.980
F-Statistic
360.404
Standard error of the model
19,181
Correlation Matrix Y
X1
X2
X3
Y
1.00
X1
0.99
1.00
X2
0.69
0.67
1.00
X3
0.99
0.99
.71
1.00
Degrees of Freedom
.20
.10
.05
.02
.01
3
1.638
2.353
3.182
4.541
5.841
19
1.328
1.729
2.093
2.539
2.861
23
1.319
1.714
2.069
2.50
2.807
Question #12 of 21
Question ID: 438978
You want to evaluate the statistical significance of the slope coefficient of an independent variable used in this regression model. For 95 percent confidence, you should compare the t-statistic to the critical value from a t-table using:
✓ A) 19 degrees of freedom and 0.05 level of significance for a two-tailed test. ✗ B) 24 degrees of freedom and 0.05 level of significance for a two-tailed test. ✗ C) 24 degrees of freedom and 0.05 level of significance for a one-tailed test. ✗ D) 19 degrees of freedom and 0.05 level of significance for a one-tailed test. Explanation The degrees of freedom are [n - k - 1)]. Here, n is the number of observations in the regression (23) and k is the number of independent variables (3). df = [23 - 3 - 1)] = 19 Because the null hypothesis is that the slope coefficient is equal to zero, this is a two-tailed test.
Question #13 of 21
Question ID: 438979
The probability of finding a value of t for variable X1 that is as large or larger than |2.133| when the null hypothesis is true is:
✓ A) between 2% and 5%. ✗ B) between 1% and 2%. ✗ C) between 10% and 20%. ✗ D) between 5% and 10%. Explanation
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The degree of freedom is = (n - [k + 1]) = (23 - [3 + 1]) = 19 In the table above, for 19 degrees of freedom, the value 2.133 would lie between a 2% chance (alpha of 0.02) or 2.539 and a 5% chance (alpha of 0.05) or 2.093.
Question #14 of 21
Question ID: 438973
Seventy-two monthly stock returns for a fund between 1997 and 2002 are regressed against the market return, measured by the Wilshire 5000, and two dummy variables. The fund changed managers on January 2, 2000. Dummy variable one is equal to 1 if the return is from a month between 2000 and 2002. Dummy variable number two is equal to 1 if the return is from the second half of the year. There are 36 observations when dummy variable one equals 0, half of which are when dummy variable two also equals 0. The following are the estimated coefficient values and standard errors of the coefficients.
Coefficient
Value
Standard error
Market
1.43000
0.319000
Dummy 1
0.00162
0.000675
Dummy 2
−0.00132
0.000733
What is the p-value for a test of the hypothesis that the new manager outperformed the old manager? ✓ A) Lower than 0.01. ✗ B) Between 0.01 and 0.05. ✗ C) Between 0.05 and 0.10. ✗ D) Greater than 0.10. Explanation Dummy variable one measures the effect on performance of the change in managers. The t-statistic is equal to 0.00162 / 0.000675 = 2.400, which is higher than the t-value (with 72 - 3 - 1 = 68 degrees of freedom) of approximately 2.39 for a p-value of between 0.01 and 0.005 for a 1 tailed test.
Questions #15-16 of 21 In a recent analysis of salaries (in $1,000) of financial analysts, a regression of salaries on education, experience, and gender is run. Gender equals one for men and zero for women. The regression results from a sample of 230 financial analysts are presented below, with t-statistics in parenthesis. Salaries = 34.98 + 1.2 Education + 0.5 Experience + 6.3 Gender (29.11)
(8.93)
(2.98)
(1.58)
Question #15 of 21
Question ID: 438982
What is the expected salary (in $1,000) of a woman with 16 years of education and 10 years of experience?
✗ A) 54.98.
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✗ B) 61.28. ✓ C) 59.18. ✗ D) 65.48. Explanation 34.98 + 1.2(16) + 0.5(10) = 59.18
Question #16 of 21
Question ID: 438983
Holding everything else constant, do men get paid more than women? Use a 5% level of significance. No, since the t-value: ✗ A) exceeds the critical value of 1.96. ✗ B) exceeds the critical value of 1.65. ✗ C) does not exceed the critical value of 1.96. ✓ D) does not exceed the critical value of 1.65. Explanation H0: bgender ≤ 0 Ha: bgender > 0 t-value of 1.58 < 1.65 (critical value)
Question #17 of 21
Question ID: 438972
63 monthly stock returns for a fund between 1997 and 2002 are regressed against the market return, measured by the Wilshire 5000, and two dummy variables. The fund changed managers on January 2, 2000. Dummy variable one is equal to 1 if the return is from a month between 2000 and 2002. Dummy variable number two is equal to 1 if the return is from the second half of the year. There are 36 observations when dummy variable one equals 0, half of which are when dummy variable two also equals 0. The following are the estimated coefficient values and standard errors of the coefficients.
Coefficient Value Standard error Market
1.43000
0.319000
Dummy 1 0.00162
0.000675
Dummy 2 0.00132
0.000733
What is the p-value for a test of the hypothesis that performance in the second half of the year is different than performance in the first half of the year? ✓ A) Between 0.05 and 0.10. ✗ B) Between 0.01 and 0.05. ✗ C) Greater than 0.10. ✗ D) Lower than 0.01. Explanation The difference between performance in the second and first half of the year is measured by dummy variable 2. The t-statistic is
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equal to 0.00132 / 0.000733 = 1.800, which is between the t-values (with 63 − 3 − 1 = 59 degrees of freedom) of 1.671 for a p-value of 0.10, and 2.00 for a p-value of 0.05 (note that the test is a two-sided test).
Question #18 of 21
Question ID: 438994
When utilizing a proxy for one or more independent variables in a multiple regression model, which of the following errors is most likely to occur? ✓ A) Model misspecification. ✗ B) Serial correlation. ✗ C) Multicollinearity. ✗ D) Heteroskedasticity. Explanation By using a proxy for an independent variable in a multiple regression analysis, there is some degree of error in the measurement of the variable.
Question #19 of 21
Question ID: 438980
An analyst is investigating the hypothesis that the beta of a fund is equal to one. The analyst takes 60 monthly returns for the fund and regresses them against the Wilshire 5000. The test statistic is 1.97 and the p-value is 0.05. Which of the following is CORRECT? ✗ A) For a sample of 100 beta values, the expected number of times beta would be equal to 1 is less than or equal to 5%. ✗ B) The proportion of occurrences when the absolute value of the test statistic will be higher when beta is equal to 1 than when beta is not equal to 1 is less than or equal to 5%. ✗ C) If beta is equal to 1, the likelihood that the absolute value of the test statistic is equal to 1.97 is less than or equal to 5%. ✓ D) If beta is equal to 1, the likelihood that the absolute value of the test statistic would be greater than or equal to 1.97 is 5%. Explanation A statistical test computes the likelihood of a test statistic being higher than a certain value assuming the null hypothesis is true.
Question #20 of 21
Question ID: 438992
A dependent variable is regressed against three independent variables across 25 observations. The regression sum of squares is 119.25, and the total sum of squares is 294.45. The following are the estimated coefficient values and standard errors of the coefficients. Coefficient Value Standard error 1
2.43
1.4200
2
3.21
1.5500
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3
0.18
0.0818
What is the p-value for the test of the hypothesis that all three of the coefficients are equal to zero?
✗ A) Greater than 0.10. ✓ B) lower than 0.025. ✗ C) Between 0.05 and 0.10. ✗ D) Between 0.025 and 0.05. Explanation This test requires an F-statistic, which is equal to the ratio of the mean regression sum of squares to the mean squared error. The mean regression sum of squares is the regression sum of squares divided by the number of independent variables, which is 119.25 / 3 = 39.75. The residual sum of squares is the difference between the total sum of squares and the regression sum of squares, which is 294.45 − 119.25 = 175.20. The denominator degrees of freedom is the number of observations minus the number of independent variables, minus 1, which is 25 − 3 − 1 = 21. The mean squared error is the residual sum of squares divided by the denominator degrees of freedom, which is 175.20 / 21 = 8.34. The F-statistic is 39.75 / 8.34 = 4.76, which is higher than the F-values (with 3 numerator degrees of freedom and 21 denominator degrees of freedom) of 3.07 at the 5% level of significance and higher than the F-value of 3.82 at the 2.5% level of significance. The conclusion is that the p-value must be lower than 0.025. Remember the p-value is the probability that lies above the computed test statistic for upper tail tests or below the computed test statistic for lower tail tests.
Question #21 of 21
Question ID: 438975
Seventy-two monthly stock returns for a fund between 1997 and 2002 are regressed against the market return, measured by the Wilshire 5000, and two dummy variables. The fund changed managers on January 2, 2000. Dummy variable one is equal to 1 if the return is from a month between 2000 and 2002. Dummy variable number two is equal to 1 if the return is from the second half of the year. There are 36 observations when dummy variable one equals 0, half of which are when dummy variable two also equals zero. The following are the estimated coefficient values and standard errors of the coefficients.
Coefficient
Value
Standard error
Market
1.43000
0.319000
Dummy 1
0.00162
0.000675
Dummy 2
−0.00132
0.000733
What is the p-value for a test of the hypothesis that the beta of the fund is greater than 1? ✓ A) Between 0.05 and 0.10. ✗ B) Greater than 0.10. ✗ C) Lower than 0.01. ✗ D) Between 0.01 and 0.05. Explanation The beta is measured by the coefficient of the market variable. The test is whether the beta is greater than 1, not zero, so the t-statistic is equal to (1.43 − 1) / 0.319 = 1.348, which is in between the t-values (with 72 − 3 − 1 = 68 degrees of freedom) of 1.29 for a p-value of 0.10 and 1.67 for a p-value of 0.05.
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Topic 28 to 29
Test ID: 8450994
Question #1 of 8
Question ID: 438996
The midsquare technique was one of the first pseudorandom number generators. What are the next two random numbers in a sequence of random numbers between 0 and 1 that is created using the midsquare technique with a seed number of 5236? ✓ A) 1st = 0.4156; 2nd = 0.2723. ✗ B) 1st = 0.6325; 2nd = 0.6514. ✗ C) 1st = 0.2741; 2nd = 0.7153. ✗ D) 1st = 0.5696; 2nd = 0.4416. Explanation The first random number is determined by squaring the seed and taking the middle four digits. 52362 = 27415696; thus, the first random number is 0.4156. The second random number is determined by squaring the first generated number, 4156, and taking the middle four digits. 41562 = 17272336; thus, the second random number is 0.2723.
Question #2 of 8
Question ID: 439003
GARCH(1,1) (generalized autoregressive conditional heteroskedastic) models of volatility may be useful for option traders because they: ✗ A) are used in the Black-Scholes option pricing model. ✗ B) provide efficient estimates of past volatility. ✓ C) are useful in forecasting future volatility ✗ D) are the simplest volatility models to estimate. Explanation GARCH(1,1) models have been shown to be useful in forecasting future volatility, which may indicate to an option trader the relative value of an option price.
Question #3 of 8
Question ID: 439001
Consider the following four GARCH equations:
Equation 1: σ2n = 0.83 + 0.05μ2n-1 + 0.93σ2n-1 Equation 2: σ2n = 0.06 + 0.04μ2n-1 + 0.95σ2n-1 Equation 3: σ2n = 0.60 + 0.10μ2n-1 + 0.94σ2n-1 Equation 4: σ2n = 0.03 + 0.03μ2n-1 + 0.93σ2n-1 Which of the following statements regarding these equations is (are) CORRECT?
I. Equation 1 is a stationary model.
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II. Equation 2 shows no mean reversion III. Volatility will revert to a long run mean level faster with Equation 1 than it will for Equation 4. IV. Volatility will revert to a long run mean level faster with Equation 3 than it will for Equation 2.
✗ A) III only. ✗ B) II and IV only. ✓ C) I only. ✗ D) II and III only. Explanation The format of the GARCH equation is σ2n = ω + αμ2n-1 + βσ2n-1, where (α + β) = persistence. For a model to be stationary over time, the persistence must be less than one. A persistence of one means there is no mean reversion and the higher the persistence (given that it is less than one), the longer it will take for volatility to revert to a long run mean level following a large shock or movement. The persistence for Equation 2 is (0.04 + 0.95) = 0.99, which is less than one meaning there is mean reversion. The persistence for Equation 1 is higher than that of Equation 3, meaning mean reversion will take longer for Equation 1. Because the persistence for Equation 1 is less than one, Equation 1 is a stationary model. Equation 3 has a persistence greater than one, which mean the model shows no mean reversion. Only Statement I is correct.
Question #4 of 8
Question ID: 438995
There are four major advantages of simulation modeling when multiple input variables and compounding distributions are involved. Which of the following statements correctly identifies one of these advantages? Simulation models: ✗ A) can easily examine effects on input variables when changing strategies or scenarios. ✗ B) allow for correlations between output variables. ✓ C) create visible output probability distributions that can be analyzed when compounding probability distributions result from multiple input variables. ✗ D) simplify complex functions when the probability distribution of the output variable is identified. Explanation Simulation models create visible output probability distributions that can be analyzed when compounding probability distributions result from multiple input variables. Simulation models simplify complex functions because the probability distribution of the output variable does not need to be identified. Simulation models allow for correlations between input variables. Simulation models can easily examine effects on output variables when changing strategies or scenarios.
Question #5 of 8
Question ID: 438997
Which of the following statements is incorrect regarding the use of historical data and weighting schemes to estimate volatility? ✗ A) If an analyst's goal is to estimate the current level of volatility, she may want to weight recent data more heavily. ✗ B) An example of a volatility estimation weighting scheme is to assume a long-run variance level in addition to weighted squared return observations.
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✗ C) Straightforward volatility estimation approaches weight each observation equally in that more distant past returns have the same influence on estimated volatility as observations that are more recent. ✓ D) The volatility estimate from an autoregressive conditional heteroskedasticity model is a function of a short-run variance level and a series of squared variance observations. Explanation An extension to a weighting scheme that weighs recent data more heavily is to assume a long-run variance level in addition to weighted squared return observations. The most frequently used model for this extension is an autoregressive conditional heteroskedasticity model, ARCH(m).
Question #6 of 8
Question ID: 439004
Traditional covariance calculations assume that all past observations receive the same weighting. One alternative is to apply differential weightings to past observations and create GARCH-type models to take into account the time variability of covariance. Which of the following statements is true regarding these alternative covariance calculations? I. If one uses a particular weighting scheme for variance calculations, one should use the same weighting scheme for the covariance calculations. II. If one uses a particular weighting scheme variance calculations, one should use a different weighting scheme for the covariance calculations.
✗ A) Neither I nor II. ✓ B) I only. ✗ C) II only. ✗ D) Both I and II. Explanation The main factor to consider when calculating covariance is to maintain consistency in the calculation process. That is, if one uses a particular weighting scheme (e.g., EWMA or GARCH) for variance calculations, one should use the same weighting scheme and procedure for the covariance calculations. Otherwise, the estimates are inconsistent with each other.
Question #7 of 8
Question ID: 438999
A portfolio manager is using an exponentially weighted moving average (EWMA) model to forecast volatility for a particular market parameter. What is the implication of an EWMA weighting parameter value of 0.84? ✗ A) More information is required to determine the implication. ✓ B) A greater weight is placed on the previous volatility estimate than on the most recent change in parameter value. ✗ C) An equal weight is placed on the previous volatility estimate as on the most recent change in parameter value. ✗ D) A greater weight is placed on the most recent change in parameter value than on the previous volatility estimate. Explanation
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The EWMA weighting parameter of 0.84 indicates that a weighting of 0.84 will be placed on the previous volatility estimate and a weighting of 0.16 will be placed on the most recent change in the parameter value.
Question #8 of 8
Question ID: 439000
You estimate the following GARCH model:
σ2n = 0.04 + 0.30μ2t-1 + 0.50σ2n-1 If the most recent volatility estimate and error term are 0.15 and 0.02, respectively, the long-run average variance is closest to: ✗ A) 0.04. ✗ B) 0.23. ✓ C) 0.20. ✗ D) 0.16. Explanation
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