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Statistics And Probability With Applications For Engineers And Scientists Using MINITAB R And JMP 2nd Edition Irwin Guttman, Kalanka P. Jayalath, Bhisham C. Gupta - Solutions
4. Use Table A.7 to find the following values: (a) F10,12,0.95, (b) F8,10,0.975, (c) F15,20,0.95,(d) F20,15,0.99. Hint: Use the formula Fm,n,1−α = 1/Fn,m,α.
3. Use Table A.7 to find the following values of upper percent points of various F-distributions: (a) F6,8,0.05, (b) F8,10,0.01, (c) F6,10,0.05, (d) F10,11,0.025
2. Use Table A.6 to find the following values of upper percent points of various tm-distributions: (a) t18,0.025, (b) t20,0.05, (c) t15,0.01, (d) t10,0.10, (d) t12,0.005.
1. If X is a chi-square random variable with 15 degrees of freedom, find the value of x such that (a) P(X ≥ x) = 0.05, (b) P(X ≥ x) = 0.975, (c) P(X ≤ x) = 0.025, (d)P(X ≥ x) = 0.95, (e) P(X ≤ x) = 0.05
8. The amount of beverage dispensed by a bottling machine is normally distributed with mean of 12 oz and a standard deviation of 1 oz. A random sample of n bottles is selected, and a sample average ¯X is calculated. Determine the following probabilities:(a) P(|¯X − 12| ≤ 0.25) for sample
7. In 1995, the median price of a PC was $1200. Suppose that a random sample of 100 persons who bought their PCs during that year recorded the amount spent (by each of them) on his/her PC. State the approximate sampling distribution of ˆp, the proportion of persons who spent more than $1200 on a
6. Let X be a random variable distributed as binomial B(n, p). State the approximate sampling distribution of the sample proportion ˆp when (a) n = 40, p = 0.4;(b) n = 50, p = 0.2; (c) n = 80, p = 0.1.
5. Suppose that the amount of a weekly grocery bill of all households in a metropolitan area is distributed with a mean of $140 and a standard deviation of $35. Let ¯X be the average amount of grocery bill of a random sample of 49 households selected from this metropolitan area. Find the
4. The weight of all cars traveling on interstate highway I-84 is normally distributed with a mean of 3000 lb and standard deviation of 100 lb. Let ¯X be the mean weight of a random sample of 16 cars traveling on I-84. Calculate the probability that ¯X falls between 2960 and 3040 lb.
3. Suppose that random samples are drawn from an infinite population. How does the standard error of ¯X change if the sample size is (i) increased from 36 to 64, (ii)increased from 100 to 400, (iii) increased from 81 to 324, (iv) increased from 256 to 576?
2. Suppose that random samples (without replacement) of size 5 are repeatedly drawn from a finite population of size N = 50. Suppose that the mean and the standard deviation of the population are 18 and 5, respectively. Find the mean and the standard deviation of the sampling distribution of ¯X.
1. Suppose that random samples of size 36 are repeatedly drawn from a population with mean 28 and standard deviation 9. Describe the sampling distribution of ¯X .
6. Refer to Problem 5. Suppose that the instructor of the class decided to give 10 extra points to every student. Find the mean and variance of the new data and comment on your result.
5. The following data give the scores on a midterm test of 20 randomly selected students.29 26 40 27 35 39 37 40 37 34 36 28 26 33 37 25 27 33 26 29 Find the mean and standard deviation for these data.
4. A manufacturing company has developed a new device for the army, obtaining a defense contract to supply 25,000 pieces of this device to the army. In order to meet the contractual obligations, the department of human resources wants to estimate the number of workers that the company would need to
3. Refer to Problem 2. Let T denote the total expenses for entertainment of all the students. Estimate the mean and variance of T.
1. Define the appropriate population from which the following samples have been drawn:(a) Fifty employees from a manufacturing company are asked if they would like the company to have some training program for all employees.(b) A quality control engineer of a semiconductor company randomly selects
49. Referring to Problem 5 of Section 8.7, find a 99% confidence interval for the difference between the percentages of persons who favor a nuclear plant in their state. Compare the confidence intervals you obtained here to the one you obtained in Problem 5 of Section 8.7 and comment on the effect
48. Repeat Problem 4 of Section 8.7, using 98% and 99% confidence levels.
47. Referring to Problem 3 of Section 8.7, find a 99% confidence interval for the difference in proportions of the population favoring Brand X before and after the advertising campaign. Comment on your result.
46. Referring to Problem 2 of Section 8.7, find 90% and 95% confidence intervals for p, the probability of obtaining a head when this coin is tossed.Review Practice Problems 351
45. Referring to Problem 1 of Section 8.7, find a 99% confidence interval for p, the proportion of people in the population wearing glasses.
44. Repeat Problem 36, assuming that the two populations are normally distributed with unequal variances.
43. Two textile mills are manufacturing a type of utility rope. A consumer group wants to test the tensile strength of the rope manufactured by these mills. A random sample of 10 pieces of rope manufactured by mill I results in a sample mean of ¯X1 = 850 psi and a sample standard deviation of S1 =
42. Suppose in Problem 37 that it had been agreed to take equal sample sizes from the two populations. What size samples should be selected so that we can be 99% confident that the margin of error in estimating the difference of two population means is no more than two points? Use the sample
41. Referring to Problem 40, find by how much the sample size increases or decreases if we are willing to increase the margin of error from 0.025 to 0.05 with the same probability of 0.95.
40. Referring to Problem 38, how large a sample should be taken in order to be confident with probability 95% that the margin of error to estimate the fraction p of defective computer chips is 0.025?
39. A company with two car-repair centers in Orange County, California, is interested in estimating the percentage of car owners who are very happy with the service they received at each center. In a random sample of 120 car owners who have their cars serviced at service center I, 72 are quite
38. A semiconductor company wants to estimate the fraction p of defective computer chips it produces. Suppose that a random sample of 900 chips has 18 defective chips.Find a point estimate of p. Next, find a 95% confidence interval for the proportion p of defective chips, and also find a one-sided
37. At a certain university, the Electrical Engineering faculty decides to teach a course on robot intelligence using two different methods, one an existing method and the other a method using new strategies. The faculty randomly selects two sections of students who are scheduled to take this
36. Two samples of sizes n1 = 16 and n2 = 25 pieces of wool yarn are randomly taken from the production of two spindles and tested for tensile strength. These tests produce the following data:Sample I: 12.28 8.54 11.31 9.06 10.75 10.96 12.12 8.14 10.75 9.55 9.56 11.00 9.10 9.91 10.08 9.54 Sample
35. A recent study shows that the average annual incomes of cardiologists and gastroenterologists based on random samples of 100 and 121 are $295,000 and $305,000, respectively.Furthermore, these samples yield sample standard deviations of $10,600 and$12,800, respectively. Determine a 98%
34. Two types of copper wires used in manufacturing electrical cables are being tested for their tensile strength. From previous studies, it is known that the tensile strengths of these wires are distributed with unknown means μ1 and μ2 but known standard deviations σ1 = 6.0 psi and σ2 = 8.5
33. In a study of diameters of ball bearings manufactured by a newly installed machine, a random sample of 64 ball bearings is taken, yielding a sample mean of 12mm and a sample standard deviation of 0.6 mm. Compute a 99% confidence interval for the population mean μ.
32. A past study indicates that the standard deviation of hourly wages of workers in an auto industry is $4.00. A random sample of hourly wages of 49 workers yields an average of $55.00. Find (a) a point estimate of population mean wage, (b) the standard error of the point estimator calculated in
31. A manufacturing engineer wants to use the mean of a random sample of size 36 to estimate the average length of the rods being manufactured. If it is known that σ = 1.5 cm, find the margin of error at the 95% confidence level.Review Practice Problems 349
30. If (X1, . . . , Xn) is a random sample of size n from N(μ0, σ2), where μ0 is the known value of the population mean, find the maximum likelihood estimator of σ2. Is the maximum likelihood estimator unbiased or not for σ2? What is its distribution? What is its variance?
29. If (X1, . . . , Xn) is a random sample of size n from N(μ, σ2 0), where σ2 0 is the known value of the population variance, find the maximum likelihood estimator of μ. Is the maximum likelihood estimator unbiased for μ? What is the distribution of the maximum likelihood estimator?
28. A random sample of 60 voters selected at random from a large city indicate that 70%will vote for candidate A in the upcoming mayoral election. Find the 99% confidence interval for the proportion of voters supporting candidate A.
27. Nine out of 15 students polled favored the holding of a demonstration on campus against “the war.” Using technology, find 95% confidence limits for the proportion of all students favoring this proposal.
26. The athletic department of a large school randomly selects two groups of 50 students each. The first group is chosen from students who voluntarily engage in athletics, the second group is chosen from students who do not engage in athletics. Their body weights are measured with the following
25. The effectiveness of two drugs is tested on two groups of randomly selected patients with the following results (in coded units):Group 1: n1 = 75, ¯X1 = 13.540, S1 = 0.476 Group 2: n2 = 45, ¯X2 = 11.691, S2 = 0.519 Assume normality.(a) Find a 95% confidence interval for σ2 1 = σ2 2.(b) On
24. A group of 121 students of a large university are retested on entrance to give an average score of 114 with a standard deviation of 19.6. Another group of 91 students 348 8 Estimation of Population Parameters who have spent one year at this university are given the same test; they performed
23. A sample of 100 workers in one large plant took an average time of 23 minutes to complete a task, with a standard deviation of 4 minutes. In another large, but similar, plant, a sample of 100 workers took an average time of 25 minutes to complete the same task, with a standard deviation of 6
22. A sample of 500 rounds of ammunition supplied by manufacturer A yields an average muzzle velocity of 2477 ft/s, with a standard deviation of 80 ft/s. A sample of 500 rounds made by another manufacturer, manufacturer B, yields an average muzzle velocity of 2422 ft/s, with a standard deviation of
21. Two groups of judges are asked to rate the tastiness of a certain product. The results are as follows:Group 1: n1 = 121, ¯X1 = 3.6, S2 1 = 1.96 Group 2: n2 = 121, ¯X2 = 3.2, S2 2 = 3.24(a) Find a 95% confidence interval for σ2 1/σ2 2, assuming normality.(b) On the basis of the interval
20. A sample of 70 employees selected at random from the employees of a large brewery yields an average disabled time of 41.8 hours during a fiscal year, with a standard deviation of 6.4 hours. Construct a 99% confidence interval for the mean disabled time of employees at this firm.
19. The heights of 105 male students of university X chosen randomly yield an average of 68.05 inches and a sample standard deviation of 2.79 in. Find a 95% confidence interval for the mean of the heights of male students attending the (rather large) university X.
18. A new spinning machine is installed and 64 test pieces of yarn it produces are tested for breaking strength. The observed data yield a sample average of 4.8 and standard deviation of 1.2. Find a 99% confidence interval for the true value of the breaking strength.
17. A large university wishes to estimate the mean expense account of the members of its staff who attend professional meetings. A random sample of 100 expense accounts yields a sample average of $515.87 with a sample standard deviation of $14.34. Find a 95% confidence interval for the mean expense
16. In Problems 9, 10, and 11, the assumption was made that the ratio of the variances of the populations being sampled is 1. In each of these problems, find 95% confidence intervals for the ratio of the variances of the populations being sampled and comment on the assumption.
15. For the data of Problem 8, find a 95% confidence interval for σ2A/σ2B, the ratio of the population variances. Comment on the assumption made in Problem 8, that the true value of the ratio is one.Review Practice Problems 347
14. Two analysts A and B each make 10 determinations of percentage chlorine in a batch of polymer. The sample variances S2A and S2B turn out to be 0.5419 and 0.6065, respectively. If σ2A and σ2B are the variances of the populations of A’s measurements and of B’s measurements respectively,
13. An experiment to determine the viscosity of two different types of gasoline (leaded versus nonleaded) give the following results:Leaded: n1 = 25, ¯X1 = 35.84, S2 1 = 130.4576 Nonleaded: n2 = 25, ¯X2 = 30.60, S2 2 = 53.0604 Assuming normality, find a 95% confidence interval for the difference
12. Determinations of atomic weights of carbon from two preparations I and II yield the following results:Preparation I: 12.0072 12.0064 12.0054 12.0016 12.0077 Preparation II: 11.9583 12.0017 11.9949 12.0061 Assuming (approximate) normality, find a 95% confidence interval for μ1 − μ2, the
11. Breaking strengths (in psi) are observed on a sample of five test pieces of type-A yarn and nine test pieces of type-B yarn, with the following results:Type A (psi): 93 94 75 84 91 Type B (psi): 99 93 99 97 90 96 93 88 89 Assuming normality of breaking strengths for each type of yarn and that
10. Resistance measurements (in ohms) are made on a sample of four test pieces of wire from lot A and five from lot B, with the following results given:Lot A Resistance (ohms): 0.143 0.142 0.143 0.137 Lot B Resistance (ohms): 0.140 0.142 0.136 0.138 0.14 If μ1 and μ2 are the mean resistances of
9. The following data give the Vickers hardness number of ten shell castings from company A and of ten castings from company B:Company A (Hardness): 66.3 64.5 65.0 62.2 61.3 66.5 62.7 67.5 62.7 62.9 Company B (Hardness): 62.2 67.5 60.4 61.5 64.8 60.9 60.2 67.8 65.8 63.8 Find a 90% confidence
8. A sample of four tires is taken from a lot of brand-A tires. Another sample of four tires is taken from a lot of brand-B tires. These tires are tested for amount of wear, for 24,000 miles of driving on an eight-wheel truck, when the tires are rotated every 1000 miles.The tires are weighed before
7. In firing a random sample of nine rounds from a given lot of ammunition, the tester finds that the standard deviation of muzzle velocities of the nine rounds is 38 ft/s.Assuming that muzzle velocities of rounds in this lot are (approximately) normally distributed, find 95% confidence limits for
6. The following data give the yield point (in units of 1000 psi) for a sample of 20 steel castings from a large lot:64.5 66.5 67.5 67.5 66.5 65.0 73.0 63.5 68.5 70.0 71.0 68.5 68.0 64.5 69.5 67.0 69.5 62.0 72.0 70.0 Assuming that the yield points of the population of castings to be
5. A clock manufacturer wants to estimate the variability of the precision of a certain type of clock being manufactured. To do this, a sample of eight clocks is run for exactly 48 hours. The number of seconds each clock is ahead or behind after the 48 hours, as measured by a master clock, is
4. Ten determinations of percentage of water in a methanol solution yield ¯X = 0.552 and S = 0.037. If μ is the “true” percentage of water in the methanol solution, assuming Review Practice Problems 345 normality, find (a) a 90% confidence interval for μ and (b) a 95% confidence interval for
3. Four determinations of the pH of a certain solution are 7.90, 7.94, 7.91, and 7.93.Assuming normality of determinations with mean μ, standard deviation σ, find (a)99% confidence limits for μ and (b) 95% confidence limits for σ.
2. Two machines A and B are packaging 8-oz boxes of corn flakes. From past experience with the machines, it is assumed that the standard deviations of weights of the filling from the machines A and B are 0.04 oz and 0.05 oz, respectively. One hundred boxes filled by each machine are selected at
1. In estimating the mean of a normal distribution N(μ, σ2) having known standard deviation σ by using a confidence interval based on a sample of size n, what is the minimum value of n in order for the 99% confidence interval for μ to be of length not greater than L?
9. In a random sample of 500 voters interviewed across the nation, 200 criticized the use of personal attacks in an election campaign. Determine a 90% confidence interval for the proportion of voters who criticize the use of personal attacks in the election campaign.
8. An instructor is interested in determining the percentage of the students who prefer multiple choice questions versus open-ended questions in an exam. In a random sample of 100 students, 40 favored multiple choice questions. Find a 95% confidence interval for the proportion of students who favor
7. In a manufacturing plant, two machines are used to produce the same mechanical part. A random sample of 225 parts produced by machine 1 showed that 12 of the parts are nonconforming, whereas a random sample 400 parts produced by the machine 2 showed that 16 of the parts are nonconforming.
6. A random sample of 800 homes from a metropolitan area showed that 300 of them are heated by natural gas. Determine a 95% confidence interval for the proportion of homes heated by natural gas.
5. During the past 50 years, more than 10 serious nuclear accidents have occurred worldwide. Consequently, the US population is divided over building more nuclear plants. Suppose two US states decide to determine the percentage of their population that would like to have a nuclear plant in their
4. An orange juice manufacturing company uses cardboard cans to pack frozen orange juice. A six sigma black belt quality engineer found that some of the cans supplied by supplier A do not meet the specifications, as they start leaking at some point. In a random sample of 400 packed cans, 22 were
3. A sample of 100 consumers showed 16 favoring Brand X of orange juice. An advertising campaign was then conducted. A sample of 200 consumers surveyed after the campaign showed 50 favoring Brand X. Find a 95% confidence interval for the difference in proportions of the population favoring Brand X
2. A new coin is tossed 50 times, and 20 of the tosses show heads and 30 show tails.Find a 99% confidence interval for the probability of obtaining a head when this coin is tossed again.
1. A random sample of 500 individuals contains 200 that wear eyeglasses. Find a 95%confidence interval for p, the proportion of people in the population wearing glasses.
8. Using the information and the data provided in Problems 6 and 7, compute a 95%confidence interval for the ratio σ2 1/σ2 2 of the two variances.
2. Find a one-sided 99% lower and upper one-sided confidence interval for the population variance σ2 2.
7. In Problem 6, the chemical production from the last 15 batches in which the existing catalyst was used is:31 33 28 28 30 31 30 32 30 33 35 34 28 32 33 Assuming that these chemical yields are normally distributed with mean μ2 and variance σ2 2, find a 95% confidence interval for the population
6. A new catalyst is used in 10 batches of a chemical production process. The final yield of the chemical in each batch produce the following data:27 28 25 30 27 26 25 27 30 30 Assuming that chemical yields are normally distributed with mean μ1 and varianceσ2 1, find a 95% confidence interval for
5. A random sample from a normal population with unknown mean μ and variance σ2 yields the following summary statistics: n = 25, ¯X = 28.6, S = 5.0(a) Find a 95% confidence interval for σ2 and a 95% confidence interval for σ.(b) Find 95% one-sided lower and upper confidence limits for σ, and
4. Two types of tires are tested with the following results:Type A: n1 = 121, ¯X1 = 27, 465 miles, S1 = 2500 miles Type B: n2 = 121, ¯X2 = 29, 527 miles, S2 = 3000 miles(a) Assuming normality, find a 99% confidence interval for σ2 1/σ2 2.(b) On the basis of the interval found in (a), find a 99%
3. A sample of 220 items turned out during a given week by a certain process has an average weight of 2.57 lb and standard deviation of 0.57 lb. During the next week, a different lot of raw material was used, and the average weight of 220 items produced that week was 2.66 lb, and the standard
2. Tensile strengths were measured for 15 test pieces of cotton yarn randomly taken from the production of a given spindle. The value of S for this sample of 15 tensile strengths is found to be 11.2 lb. Find 95% confidence limits for the standard deviationσ of the population. It is assumed that
1. A sample of size 12 from a population assumed to be normal with unknown varianceσ2 yields S2 = 86.2. Determine 95% confidence limits for σ2.
12.84 13.85 13.40 12.48 13.39 13.61 12.37 12.08 Assuming that the drying times of the two brands are normally distributed with equal variances, determine a 95% confidence interval for μI − μII.
11.28 11.98 11.22 11.97 10.47 10.79 11.98 10.03 Brand II: 12.10 13.91 13.32 13.58 12.04 12.00 13.05 13.70
10. Two different brands of an all-purpose joint compound are used in residential construction and their drying times, in hours, are recorded. Sixteen specimens for each joint compound were selected. Recorded drying times are:Brand I: 11.19 10.22 10.29 11.11 10.08 10.14 10.60 10.08
9. Suppose in Problem 8 that the variances of the two populations are known from past experience to be 16 and 25, respectively. Determine a 99% confidence interval for μ1 − μ2.
8. The following data give the LDL cholesterol (commonly known as bad cholesterol)levels of two groups I and II of young female adults. Each member of group I followed a very strict exercise regimen, whereas group II members did not do any exercise.GroupI: 85 84 76 88 87 89 80 87 71 78 74 80 89 79
7. Repeat Problem 6 above, now assuming that the two population variances are not equal.
6. Reconsider Problem 5 of Section 8.3. Suppose now that the drying time is measured at two different temperature settings T1 and T2. The data obtained are as follows:T1: 9.93 9.66 9.11 9.45 9.02 9.98 9.17 9.27 9.63 9.46 T2: 10.26 9.75 9.84 10.42 9.99 9.98 9.89 9.80 10.15 10.37 Assuming that the
5. A new catalyst is to be used in production of a plastic chemical. Twenty batches of chemical are produced, 10 batches with the new catalyst, 10 without. The results are as shown below:Catalyst present: 7.2 7.3 7.4 7.5 7.8 7.2 7.5 8.4 7.2 7.5 Catalyst absent: 7.0 7.2 7.5 7.3 7.1 7.1 7.3 7.0 7.0
4. Before and after tests are to be made on the breaking strengths (oz) of a certain type of yarn. Specifically, seven determinations of the breaking strength are made on test pieces of the yarn before a spinning machine is reset and five on test pieces after the machine is reset, with the
3. A light bulb company tested ten light bulbs that contained filaments of type A and ten that contained filaments of type B. The following results were obtained for the length of life in hours of the 20 light bulbs (Steele and Torrie):8.4 Interval Estimators for The Difference of Means of Two
2. Suppose that random samples of size 25 are taken from two large lots of light bulbs, say lot A and lot B, and the average lives for the two samples are found to be¯X A = 1580 hours, ¯XB = 1425 hours Assuming that the standard deviation of bulb life in each of the two lots is 200 hours, find
1. A sample of size 10 from N(μ1, 225) yields an average ¯X1 = 170.2, while an independent sample of size 12 from N(μ2, 256) yields a sample average ¯X2 = 176.7.Find a 95% confidence interval for μ1 − μ2.
11. A certain type of electronic condenser is manufactured by the ECA company, and over a large number of years, the lifetimes of the parts are found to be normally distributed with standard deviation σ = 225 h. A random sample of 30 of these condensers yielded an average lifetime of 1407.5 hours.
10. A sample of 25 bulbs is taken from a large lot of 40-watt bulbs, and the average of the sample bulb lives is 1410 hours. Assuming normality of bulb lives and that the standard deviation of bulb lives in the mass-production process involved is 200 hours, find a 95% confidence interval for the
9. Refer to Problem 8. Determine one-sided lower and one-sided upper 95% confidence intervals for the mean temperature for July where the hotel is located.
8. A hotel facility management company is interested in determining the average temperature during July at the location of one of their hotels. The temperatures of 49 randomly selected days in July during the past five years were as follows:95 84 87 81 84 89 80 83 82 90 82 87 90 81 83 85 94 92 92
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