ALLAMA IQBAL OPEN UNIVERSITY, ISLAMABAD (Department of Mathematics and Statistics) WARNING 1.
2.
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Course: Statistics & Probability (3447) Level: BS (CS)
Semester: Autumn, 2012 Total Marks: 100 Pass Marks: 40
ASSIGNMENT No. 1 Note: Attempt all questions and each question carries equal marks.
Q.1
Q.2 Q.2
Diamet Diameters ers of of 36 rivet rivet head headss in 1/100 1/100 of of an inch inch 6.72 6.77 6.82 6.70 6.78 6.70 6.62 6.73 6.80 6.72 6.76 6.76 6.68 6.66 6.76 6.67 6.70 6.72 6.74 6.81 6.79
6.66 6.76 6.76
6.64 6.70 6.76
6.76 6.78 6.72
a) b) c)
Comp Comput utee the the samp sample le mea mean n and and samp sample le sta stand ndar ard d devi deviat atio ion. n. Construct a relative frequency frequency distribution distribution and histogram of the data. data. Comme Comment nt on whe wheth ther er ther theree is any cle clear ar indi indica cati tion on or not not that that the sam sampl plee came from a population that depicts a bell shaped distribution.
a)
A deve develo lope perr of a new new subd subdiv ivis isio ion n offe offers rs a pro prosp spec ecti tive ve hom homee buye buyerr a choic choicee of 4 deigns, 3 different heating systems, a garage or car-porch and a patio or screened porch. How many different plans are available to this buyer? A contractor wishes to build 9 car-porch houses, each different in design. In how many ways can he place these houses on a street of 6 lots are on one side of the street and 3 lots are on opposite side? The The proba probabi bili lity ty that that an Amer Americ ican an indu indust stry ry will will loca locate te in Muni Munich ch is 0.7, 0.7, the the probability that it will locate Brussels is 0.4, and the probability that it will locate in either Munich or Brussels or both is 0.8. What is the probability that the industry will locate. i) In both cities? ii) In either city?
b)
c)
Q.3
6.75 6.66 6.62 6.72 6.78 6.66
a)
Consider th the de density fu function.
i)
Evaluate k,
ii)
Find F(x) and use it to evaluate P (0.3< X < 0 <0.6) .6) 1
b)
A privately owned liquor store operates both a drive-in facility and a walk-in facility. One a randomly selected day, let X and Y, respectively, be the proportions of the time that the drive-in and walk-in facilities are in use, and suppose that the joint density function of these random variables is:
i) ii) c)
Find the marginal density of X & Y Find the probability that the drive-in facility is busy less than one-half of the time.
The probability distribution of the discrete random variable X is:
Find then mean of X Q.4
a)
The amount of time that a drive-through bank teller spends on a customer is a random variable with a mean µ = 3.2 minutes and a standard deviation σ = 1.6 minutes. If a random sample of 64 customers is observed, find the probability that their mean time at the teller’s counter is: i) At most 2.7 minutes; ii) More than 3.5 minutes; iii) At least 3.2 minutes but less than 3.4 minutes.
b)
The random variable X , representing the number of cherries in a cherry puff, has the following probability distribution: X P(X=x)
4 0.2
5 0.4
6 0.3
7 0.1
i)
Fine the mean µ and the variance σ 2 of X
ii)
− Fine the mean µ x and the variance of the mean i.e. σ 2 for random X
samples 36 cherry puffs.
Q.5
c)
An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If a sample of 30 bulbs has an average life of 780 hours, find a 96% confidence interval for the population mean of all bulbs produced by this firm.
a)
A random sample of 100 automobile owners shows that, in the state of Virginia, an automobile is driven on the average 23,500 kilometers per year with a standard deviation of 3900 kilometers. Construct a 99% confidence interval for the average number of kilometers an automobile is driven annually in Virginia. A manufacturer claims that the average tensile strength of thread A exceeds the average tensile strength of thread B by at least 12 kilograms. To test his
b)
2
claims, 50 pieces of each type of thread are tested under similar condition. Type A thread had an average tensile strength of 86.7 kilograms with a standard deviation of 6.28 kilograms while type B thread has an average tensile strength of 77.8 kilograms with a standard deviation of 5.61 kilograms. Test the manufacturer’s claim using a 0.05 level of significance.
ASSIGNMENT No. 2 Total Marks: 100
Pass Marks: 40
Note: Attempt all questions and each question carries equal marks.
Q.1
a)
A builder claims that heat pumps are installed in 70% of all homes being constructed today in the city of Richmond. Would you agree with this claim if a random survey of new homes in this city shows that 8 out of 15 had heat pumps installed? Use 0.10 level of significance.
b)
In a study conducted by the Department of Mechanical Engineering and analyzed by the Statistics Consulting Center at the Virginia Polytechnic Institute and State University, the steel rods supplied by two different companies were compared. The sample springs were made out of the steel rods supplied by each company and the “bounciness” was studied. The data are as follows: 9.3 8.8 11.0 9.8
Company A: Company B:
6.8 9.9
8.7 8.5 10.2 10.1
6.7 9.7
8.0 11.0
6.5 9.2 7.0 11.1 10.2 9.6
Can you conclude that there is virtually no difference in means between the steel rods supplied by the two companies? Use a P -value to reach your conclusion. Should variances be pooled here? Q.2
a)
Observation x1 (drive in time, min) Gain / hFE (y)
b)
Transistor gain in an integrated circuit device between emitter and collector (hFE) is related to two variables that can be controlled at the deposition process, emitter drive-in time ( x1 in minutes), fourteen samples were observed following deposition, and the resulting data shown in the table below. Fit and interpret a linear regression models using gain as the response and emitter drive-in time as the regressor variables. 1
2
3
4
5
6
7
8
9
10
11
12
13
14
195
255
195
255
225
225
225
195
255
225
225
225
225
340
1004
1636
852
1506
1272
1270
1269
9.3
1555
1260
1146
1276
1225
1321
In a linear relationship, that is, test Ho: = 0, where is the slope of the regressor variables is. An experiment was designed for the Department of materials Engineering at Virginia Polytechnic Institute and State University to study hydrogen 3
embrittlement properties based on electrolytic hydrogen pressure measurements. The solution used was 0.1 N NaOH, the material being a certain type of stainless steel. The catholic charging current density was controlled and varied at four levels. The effective hydrogen pressure was observed as the response. The data follow.
Run 1 2 3 4 5 6 7 8 9 10 i)
Q.3
Charging current Density, x (mA/cm2) 0.5 0.5 0.5 0.5 1.5 1.5 1.5 2.5 2.5 2.5
Effective Hydrogen Pressure, y (atm) 86.1 92.1 64.7 74.7 223.6 202.1 132.9 413.5 231.5 466.7
Calculate and interpret the simple linear regression parameters of y against x.
Six different machines are being considered for use in manufacturing rubber seals. The machines are being compared with respect to tensile strength of the product. A random sample of 4 seals from each machine is used to determine whether the mean tensile strength varies from machine to machine. The following are the tensile-strength measurements in kilograms per square centimeter x10 -1 1 17.5 16.9 15.8 18.6
2 16.4 19.2 17.7 15.4
3 20.3 15.7 17.8 18.9
4 14.6 16.7 20.8 18.9
5 17.5 19.2 16.5 20.5
6 18.3 16.2 17.5 20.1
Perform the analysis of variance at the 0.05 level of significance and indicate whether or not the mean tensile strength differ significantly for the 6 machines.
Q.4
a)
Sample of size 50 are taken every hour from a process producing a certain type of item that is either considered defective or not defective. Twenty samples are taken. 4
Sample
1 2 3 4 5 6 7 8 9 10
Q.5
Number of Defective items 4 3 5 3 2 2 2 1 4 3
Sample
11 12 13 14 15 16 17 18 19 20
Number of Defective items 2 4 1 2 3 1 1 2 3 1
i) ii)
Construct a control chart for control of proportion defective Does the process appear to be in control? Explain.
b)
Suppose that an attribute quality control chart is being designed with a value of p= 0.01for the in-control probability of a defective. What is the sample size per subgroup producing a probability of 0.5 that a process shift to p= p1 =0.05 will be detected? The resulting p-chart will involve 3 σ limits.
a)
In a shop study, a set of data was collected to determine whether or not the proportion of defective produced by workers was the same for the day, evening, or night shift worked. The following data were collected: Shift: Defectives Non defectives
Day 45 905
Evening 55 890
Night 70 870
Use a 0.025 level of significance to determine if the proportion of defectives is the same for all three shifts. b)
A random sample of 90 adults is classified according to gender and the number of hours they watch television during a week: Male 15 27
Over 25 hours Under 25 hours
Female 29 19
Use a 0.01 level of significance and test the hypothesis that the time spent watching television is independent of whether the viewer is male or female.
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