HYPOTHESIS TESTS FOR DIFFERENCE BETWEEN T WO MEANS
− = (̅ − ̅) − = + − = (̅ − ̅) − = √ 1 + 1 = ( − 1) + +( −2 − 1) − = ≠ (̅ − ̅) − = + and
=
−− >< − ≠ −− >< − ≠
known
but unknown
<> − < >−// <> −,, < −> /,/, = + − 2 <> −,, < −> /,/,
Reject H0 if Reject H0 if Reject H0 if
Reject H0 if Reject H0 if Reject H0 if
where
where
−− >< − ≠
but unknown
and
where
s1 2 s 2 2 n n 2 1
2
/
s
=mean of population 1 =mean of population 2 =standard deviation of population 1 =standard deviation of population 2 =mean of sample taken from population 1 =mean of sample taken from population 2
and
Reject H0 if Reject H0 if Reject H0 if
̅̅
or
1
2
n
1
1 =standard deviation of sample 1 =standard deviation of sample 1 =hypothesized difference of and =pooled variance =degrees of freedom
2
s
2
n
n
1
2
/ 2
2
n
2
1
**For the equal-variance t test, the observations should be independent, random samples from normal distributions with the same population variance. For the unequal-variance t test, the observations should be independent, random samples from normal distributions.
̅
n 1. A new computer software package has been developed to help systems analysts in developing computer based information systems. The objective of the new software package is to reduce the time required to design, develop, and implement an information system. A random sample of 20 analysts was chosen and each analyst was given specifications for a hypothetical information system. Ten analysts used the current technology while the other ten used the new software package. Use a 0.05 level of significance. The following data were revealed:
10 325 hrs 40 hrs
s
10 295 hrs 44 hrs
2. Independent random samples from normal distributions with equal variances of auditor fees when an audit resulted in (1) a lawsuit or (2) no lawsuit gave the following results. Test to see whether the mean fee when lawsuit occurs is equal or unequal to the mean fee when no lawsuit occurs. Use 0.05 for the level of significance. Lawsuit No Lawsuit
17
11
5 11
15 13 7
9
9
5
3 4
3
7
6
4
4
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HYPOTHESIS TESTS FOR DIFFERENCE BETWEEN T WO MEANS 3. Suppose that the makers of Duracell batteries want to prove that their size AA battery lasts an average of
effective versus the alternative hypothesis that they are not equally effective.
at least 45 minutes longer than Duracell’s main
competitors, the Energizer. Two independent random samples of 100 batteries of each kind are selected, and the batteries are run continuously until there are no longer operational. The sample average life for Duracell is found to be minutes, and the sample standard deviation is minutes. The results for the Energizer batteries are minutes and minutes. Is there evidence to substantiate
̅ = 308 = 84 ̅ = 254 = 67 = 0.01
Duracell’s claim that their batteries last, on the average,
at least 45 minutes longer than Energizer batteries of the same size? Use . 4. The following data represent the running times of films produced by two different motion picture companies: Company A B
Time (in minutes) 102 68 98 109 92 81 165 97 143 92 78 114
Test hypothesis that the average running time of films produced by Company B exceeds the average running time of films produced by Company A by 10 minutes against the alternative that the difference is more than 10 minutes. Use a 0.05 level of significance and assume the distribution of times to be approximately normal and the population variances are equal. 5. Many companies that cater to teenagers have learned that young people respond to commercials that provide dance-beat music, adventure, and a fast pace, rather than words. In one test, a group of 128 teenagers were shown commercials featuring rock music, and their purchasing frequency of the advertised products over the following month were recorded as a single score for each person in the group. Then a group of 212 teenagers were shown commercials for the same products, but with the music replaced by verbal persuasion. The purchase frequency scores of this group were computed as well. The results for the music group were and ; and the results for the verbal group were and . Assume that the two groups were randomly selected from the entire teenager consumer population. Using the level of significance, test the null hypothesis that both methods are equally
12.18.20 = 10.5
̅ = 23.5 ̅ ==
= 0.01
6. The photography department of a glamour magazine needs to choose a camera. Of the two models the department is considering, one is made by Nikon and one by Minolta. The department contracts with an agency to determine if one of the two models gets a higher average performance rating by the professional photographers, or whether the average performance ratings of these two cameras are not statistically different. The agency asks 60 different professional photographers to rate one of the cameras (30 photographers rate each model). The ratings are on a scale of 1 to 10. The average sample rating for Nikon is 8.5, and the standard deviation is 2.1. For the Minolta sample, the mean is 7.8, and the standard deviation is 1.8. Is there a difference between the average population ratings of the two cameras? Use 0.05 level of significance. 7. Mark Pollard, financial consultant for Merrill Lynch, Pierce, Fenner & Smith, Inc., is quoted in national advertisements for Merrill Lynch as saying: “I’ve made
more money for clients by saying no than by saying yes.” Suppose that Mr. Pollard allowed you access to
his files so that you could conduct a statistical test of the correctness of his statement. Suppose further that you gathered a random sample of 15 clients to whom Mr. Pollard said yes when presented with their investment proposals, and you found that the clients’
average gain on investments was 12% and the standard deviation was 2.5%. Suppose you gathered a random sample of 15 clients to whom Mr. Pollard said no when asked about possible investments; the clients were then offered other investments, which they consequently made. For this sample, you found that the average return was 13.5% and the standard deviation was 1%. Test Mr. Pollard’s claim at . Assume equal variances.
0.025
=
8. The manufacturers of compact disc players want to test whether a small price reduction is enough to increase sales of their product. Randomly chosen data on 15 weekly sales totals at outlets in a given area before the price reduction show a sample mean of $6,598 and a sample standard deviation of $844. A random sample of 12 weekly sales totals after the small price reduction gives a sample mean of $6,870 and a sample standard deviation of $699. Is there evidence that a small price reduction is enough to increase sales of compact disc players? Assume equal variances. Page 2 of 2
