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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
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
7. It is believed that drinking has some bad effects on the human reproduction system.To study this, some evaluations of placenta tissue of 16 randomly selected drinking mothers were made that yielded the following values (recorded to the nearest whole number):18 17 22 21 15 21 22 22 14 20 14 16 13
6. The weights of a random sample of 49 university male first-year students yielded a mean of 165 pounds and a standard deviation of 6.5 pounds. Determine a 90%confidence interval for the mean weight of all university male first-year students.
5. The following data give the drying time (in hours) for 10 randomly selected concrete slabs:9.06 9.17 9.11 8.16 9.10 9.98 8.89 9.02 9.32 8.12 Assuming that drying times are normally distributed, determine a 95% confidence interval for the mean drying time for the slabs.
4. A study was undertaken to see if the length of slide pins used in the front disc brake assembly met with specifications. To this end, measurements of the lengths of 16 slide pins, selected at random, were made. The average value of 16 lengths was 3.15, with a sample standard deviation of 0.2.
3. An insurance company is interested in determining the average postoperative length of stay (in days) in hospitals for all patients who have bypass surgery. The following data give length of stay of 50 randomly selected patients who had bypass surgery:6 10 10 9 9 12 7 12 7 8 10 7 8 8 10 12 10 7 7
2. Suppose that in Problem 1 only 25 of 36 randomly selected family caregivers responded, so we have the following data:55 53 47 47 49 43 47 40 48 41 44 51 48 43 50 49 47 42 42 47 47 49 46 46 43 Assuming that these data come from a population that has a normal distribution,(a) Determine a 99%
1. The following data give the ages of 36 randomly selected family caregivers of older parents in the United States:55 53 47 47 49 43 47 40 48 41 44 51 48 43 50 49 47 42 42 47 47 49 46 46 43 41 45 51 44 48 43 50 53 44 49 53 Assuming normality,(a) Determine a 95% confidence interval for the mean
10. Referring to Problem 9,(a) Show that U = (n + 1)ˆθ/n is unbiased for θ.(b) Find the variance of U.(c) Verify that 2¯X is unbiased for θ. Find the variance of 2¯X .(d) Determine the ratio V ar(U)/V ar(2¯X ). Which of the unbiased estimators would you prefer?
9. Suppose (X1, . . . , Xn) is a random sample of n independent observations from a population having p.d.f.f(x) =1θ , 0 ≤ x ≤ θ0, otherwise Find the MLE of θ. If ˆθ is the MLE of θ, then show that ˆθ is not unbiased for θ.
8. Events occur in time in such a way that the time interval between two successive events is a random variable t having the p.d.f. θe−θt, θ>0. Suppose that observations are made of the n successive time intervals for n + 1 events, yielding(t1, . . . , tn). Assuming these time intervals to be
7. If (X1, . . . , Xn) is a random sample of size n from a population having a Poisson distribution with unknown parameter λ, find the MLE for λ.
6. Suppose that a random sample of size n is taken from a gamma distribution with parameters γ and λ. Find the method of moments estimators of γ and λ.
5. The following data give the pull strength of 20 randomly selected solder joints on a circuit board. Assume that the pull strengths are normally distributed with meanμ and variance σ2.12 11 12 9 8 11 11 11 8 9 10 9 8 11 10 8 9 9 11 10(a) Determine the maximum likelihood estimate of the
4. Suppose that S2 1 and S2 2 are sample variances of two samples of n1 and n2 independent observations, respectively, from a population with mean μ and variance σ2. Determine an unbiased estimator of σ2 as a combination of S2 1 and S2 2 .8.3 Interval Estimators for the Mean μ of a Normal
3. Let X1,X2, and X3 be independent random variables with mean μ and varianceσ2. Suppose that ˆμ1 and ˆμ2 are two estimators of μ, where ˆμ1 = 2X1 − 2X2 + X3 and ˆμ2 = 2X1 − 3X2 + 2X3.(a) Show that both estimators are unbiased for μ.(b) Find the variance of each estimator and
2. The lengths of a random sample of 20 rods produced the following data:12.2 9.5 13.2 13.9 9.5 9.5 11.9 9.2 11.0 10.4 9.9 12.8 10.5 11.9 12.3 10.0 8.7 6.2 10.0 11.2 Determine the method of moments estimate of μ and σ2, assuming that the rod lengths are normally distributed with mean μ and
1. Find a method of moments estimator of the mean of a Bernoulli distribution with parameter p.
20. Seven engineers in a manufacturing company are working on a project. Let random variables T1, . . . , T7 denote the time (in hours) needed by the engineers to finish the project. Suppose that T1, . . . , T7 are independently and identically distributed by the uniform distribution over an
19. Repeat Problems 17 and 18, by supposing that the lifetimes of the three components are independently and identically distributed as Weibull with α = 2, β = 0.5.
18. Suppose that in Problem 17, the components are in parallel, so that the system will fail only when all the components fail. Find the p.d.f. of T, and then find the probability P(T > 15).
17. A mechanical system has three components in series, so the system will fail when at least one of the components fails. The random variables X1,X2, and X3 represent the lifetime of these components. Suppose that the Xi(i = 1, 2, 3) are independently and identically distributed as exponential
16. In Problem 15, using MINITAB, R, or JMP, find the probabilities: (a) P(S2 > 100),(b) P(S2 > 130), (c) P(S2 > 140).
15. Suppose that the total cholesterol levels of the US male population between 50 and 70 years of age are normally distributed with mean 170 mg/dL and standard deviation 12 mg/dL. Let X1, . . . , X21 be the cholesterol levels of a random sample of 21 US males between the ages of 50 and 70 years.
14. Refer to Problem 13. Using MINITAB, R, or JMP, find the probabilities: (a)P(S2x/S2 y > 3.5), (b) P(2.0 < S2x/S2 y < 3.5).
13. Let X1, . . . , X16 and Y1, . . . , Y13 be two independent random samples from two normal populations with equal variances. Show that the p.d.f. of S2x/S2 y is distributed as Snedecor’s F15,12.
12. Find, using MINITAB/R/JMP, the value of x such that(a) P(5 ≤ χ2 16 ≤ x) = 0.95(b) P(10 ≤ χ2 20 ≤ x) = 0.90(c) P(0.5 ≤ F20,24 ≤ x) = 0.90(d) P(0.4 ≤ F12,15 ≤ x) = 0.85
11. Suppose that X(1), . . . , X(n) are the order statistics of a sample from a population having the rectangular distribution with p.d.f.f(x) =⎧⎪⎨⎪⎩0, x≤ 0 1θ , 0 < x ≤ θ0, x> θwhere θ is an unknown parameter. Show that for 0 < γ < 1, PX(n) ≤ θ ≤X(n)n√1 − γ= γ(Note:
10. In Problem 9, show that for 1 ≤ k ≤ n, the mean and variance of F(x(k)) are respectively kn + 1 and k(n − k + 1)(n + 1)2(n + 2)
9. If X(1), . . . , X(n) are the order statistics of a sample of size n from a population having a continuous c.d.f. F(x) and p.d.f. f(x), show that F(x(n)) has mean n/(n + 1) and variance n/[(n + 1)2(n + 2)].
8. A sample of n observations is taken at random from a population with p.d.f.f(x) =e−x, x≥ 0 0, x< 0 Find the p.d.f. of the smallest observation. What are its mean and variance? What is its c.d.f.?
7. Suppose that the diameter of ball bearings used in heavy equipment are manufactured in a certain plant and are normally distributed with mean 1.20 cm and a standard deviation 0.05 cm. What is the probability that the average diameter of a sample of size 25 will be(a) Between 1.18 and 1.22 cm?(b)
6. Suppose that n = 2m + 1 observations are taken at random from a population with probability density function f(x) =⎧⎪⎪⎨⎪⎪⎩0, x≤ a 1b−a, a b Find the distribution of the median of the observations and find its mean and variance.What is the probability that the median will exceed
5. Suppose that F(x) is the fraction of bricks in a very large lot having crushing strengths of x psi or less. If 100 such bricks are drawn at random from the lot:(a) What is the probability that the crushing strengths of all 100 bricks exceed x psi?Review Practice Problems 287(b) What is the
4. It is believed that the median annual starting salary of a fresh engineering graduate is$40,000. If we take a random sample of 100 recent engineering graduates and record their starting salary, then(a) Find the sampling distribution of ˆp, the proportion of fresh engineering graduates who
3. A manufacturer of car batteries finds that 80% of its batteries last more than five years without any maintenance. Suppose that the manufacturer took a random sample of 500 persons from those who bought those batteries and recorded the lifetimes of their batteries.(a) Find the sampling
2. Suppose that in a certain country, the ages of women at the time of death are distributed with mean 70 years and standard deviation 4 years. Find the approximate probability that the average age of a randomly selected group of 36 women will be (a)more than 75 years, (b) less than 70 years, (c)
1. The times taken by all students of a large university to complete a calculus test are distributed having a mean of 120 minutes and a standard deviation of 10 minutes.Calculate the approximate probability that the average time taken to complete their test by a random sample of 36 students will be
8. Consider a system of n identical components operating independently. Suppose the lifetime, in months, is exponentially distributed with mean 1/λ. These components are installed in series, so that the system fails as soon as the first component fails.Find the probability density function of the
7. In Problem 5, assume that n = 21.(a) Find the probability density function of the median time taken by the manager to drive from one plant to another.(b) Find the expected value of X(21).(c) Find the expected value of X(11), the median.
6. The lifetime, in years, X1,X2, · · · ,Xn of n randomly selected power steering pumps manufactured by a subsidiary of a car company is exponentially distributed with mean 1/λ. Find the probability density function of X(1) =Min(X1,X2, · · · ,Xn), and find its mean and variance.286 7
5. The time, in minutes, taken by a manager of a company to drive from one plant to another is uniformly distributed over an interval [15, 30]. Let X1,X2, . . . , Xn denote her driving times on n randomly selected days, and let X(n) =Max(X1,X2, · · · ,Xn).Determine(a) The probability density
4. Suppose F(x) is the fraction of objects in a very large lot having weights less than or equal to x pounds. If 10 objects are drawn at random from the lot:(a) What is the probability that the heaviest of 10 objects chosen at random without replacement will have a weight less than or equal to u
3. Assume that the cumulative distribution function of breaking strengths (in pounds)of links used in making a certain type of chain is given by F(x) =1 − e−λx, x> 0 0, x≤ 0 where λ is a positive constant. What is the probability that a 100-link chain made from these links would have a
2. If ten points are picked independently and at random on the interval (0, 1):(a) What is the probability that the point nearest 1 (i.e., the largest of the 10 numbers selected) will lie between 0.9 and 1.0?(b) The probability is 1/2 that the point nearest 0 will exceed what number?
1. A continuous random variable, say X, has the uniform distribution function on (0, 1)so that the p.d.f. of X is given by f(x) =⎧⎨⎩0, x≤ 0 1, 0 < x ≤ 1 0, x>1 If X(1),X(2), . . . , X(n) are the order statistics of n independent observations all having this distribution function, give the
7. Find the value of x such that (a) P(3.247 < χ2 10 < x) = 0.95, (b) P(8.260 < χ2 20
6. Suppose that the random variable T has the Student t-distribution with 24 degrees of freedom. Find the value of t such that (a) P(−1.318 < T < t) = 0.80, (b)P(−1.711 < T < t) = 0.85, (c) P(−2.064 < T < t) = 0.875.
5. Use MINITAB, R, or JMP to do the Problems 1, 2, 3, and 4 above.
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