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statistics for engineers and scientists

Introduction To Probability And Statistics For Engineers And Scientists 6th Edition Sheldon M Ross - Solutions

65. The breaking strength of a certain type of cloth is to be measured for 10 specimens. The underlying distribution is normal with unknown mean θbut with a standard deviation equal to 3 psi. Suppose also that based on previous experience we feel that the unknown mean has a prior distribution that
64. Each item produced will, independently, be defective with probability p.If the prior distribution on p is uniform on (0, 1), compute the posterior probability that p is less than .2 givena. a total of 2 defectives out of a sample of size 10;b. a total of 1 defective out of a sample of size
63. The functional lifetimes in hours of computer chips produced by a certain semiconductor firm are exponentially distributed with mean 1/λ.Suppose that the prior distribution on λ is the gamma distribution with density functionIf the average life of the first 20 chips tested is 4.6 hours,
62. Suppose that the number of accidents occurring daily in a certain plant has a Poisson distribution with an unknown mean λ. Based on previous experience in similar industrial plants, suppose that a statistician’s initial feelings about the possible value of λ can be expressed by an
61. Consider two estimators d1 and d2 of a parameter θ. If E[d1] = θ, Var(d1) = 6 and E[d2] = θ + 2, Var(d2) = 2, which estimator should be preferred?
60. Consider two independent samples from normal populations having the same variance σ2, of respective sizes n and m. That is, X1, . . . , Xn and Y1, . . . , Ym are independent samples from normal populations each having variance σ2. Let S2 x and S2 y denote the respective sample variances.Thus
59. Let X1,X2, . . . , Xn denote a sample from a population whose mean valueθ is unknown. Use the results of Example 7.7.b to argue that among all unbiased estimators of θ of the formn i=1 λiXi ,n i=1 λi = 1, the one with minimal mean square error has λi ≡ 1/n, i = 1, . . . , n.
58. Determine 100(1 − α) percent one-sided upper and lower confidence intervals for θ in Problem 57.
57. Suppose the lifetimes of batteries are exponentially distributed with mean θ. If the average of a sample of 10 batteries is 36 hours, determine a 95 percent two-sided confidence interval for θ.
56. Derive 100(1 − α) percent lower and upper confidence intervals for p, when the data consist of the values of n independent Bernoulli random variables with parameter p.
55. Of 100 randomly detected cases of individuals having lung cancer, 67 died within 5 years of detection.a. Estimate the probability that a person contracting lung cancer will die within 5 years.b. How large an additional sample would be required to be 95 percent confident that the error in
54. A random sample of 100 items from a production line revealed 17 of them to be defective. Compute a 95 percent two-sided confidence interval for the probability that an item produced is defective. Determine also a 99 percent upper confidence interval for this value. What assumptions are you
53. In a recent study, 79 of 140 meteorites were observed to enter the atmosphere with a velocity of less than 25 miles per second. If we takeˆp=79/140 as an estimate of the probability that an arbitrary meteorite that enters the atmosphere will have a speed less than 25 miles per second, what can
52. A market research firm is interested in determining the proportion of households that are watching a particular sporting event. To accomplish this task, they plan on using a telephone poll of randomly chosen households.How large a sample is needed if they want to be 90 percent certain that
51. A recent newspaper poll indicated that Candidate A is favored over Candidate B by a 53 to 47 percentage, with a margin of error of ±4 percent.The newspaper then stated that since the 6-point gap is larger than the margin of error, its readers can be certain that Candidate A is the current
50. An airline is interested in determining the proportion of its customers who are flying for reasons of business. If they want to be 90 percent certain that their estimate will be correct to within 2 percent, how large a random sample should they select?
49. To estimate p, the proportion of all newborn babies that are male, the gender of 10,000 newborn babies was noted. If 5106 of them were male, determine (a) a 90 percent and (b) a 99 percent confidence interval estimate of p.
48. A random sample of 1200 engineers included 48 Hispanic Americans, 80 African Americans, and 204 females. Determine 90 percent confidence intervals for the proportion of all engineers who area. female;b. Hispanic Americans or African Americans.
47. A problem of interest in baseball is whether a sacrifice bunt is a good strategy when there is a man on first base and no outs. Assuming that the bunter will be out but will be successful in advancing the man on base, we could compare the probability of scoring a run with a player on first base
46. Two analysts took repeated readings on the hardness of city water. Assuming that the readings of analyst i constitute a sample from a normal population having variance σ2 i , i = 1, 2, compute a 95 percent two-sided confidence interval for σ2 1 /σ 2 2 when the data are as follows: Analyst 1
45. If X1, . . . , Xn is a sample from a normal population having known meanμ1 and unknown variance σ2 1 , and Y1, . . . , Ym is an independent sample from a normal population having known mean μ2 and unknown varianceσ2 2 , determine a 100(1− α) percent confidence interval for σ2 1 /σ 2 2 .
44. The following are the daily numbers of company website visits resulting from advertisements on two different types of media.Type I Type II 481 572 526 537 506 561 511 582 527 501 556 605 661 487 542 558 501 524 491 578 Find a 99 percent confidence interval for the mean difference in daily
43. Do Problem 42 when it is known in advance that the population variances are 4 and 5.
42. Independent random samples are taken from the output of twomachines on a production line. The weight of each item is of interest. From the first machine, a sample of size 36 is taken, with sample mean weight of 120 grams and a sample variance of 4. From the second machine, a sample of size 64
41. A civil engineer wishes to measure the compressive strength of two different types of concrete. A random sample of 10 specimens of the first type yielded the following data (in psi)Type 1: 3,250, 3,268, 4,302, 3,184, 3,266 3,297, 3,332, 3,502, 3,064, 3,116 whereas a sample of 10 specimens of
40. If X1, . . . , Xn is a sample from a normal population, explain how to obtain a 100(1 − α) percent confidence interval for the population varianceσ2 when the population mean μ is known. Explain in what sense knowledge of μ improves the interval estimator compared with when it is
39. The amount of beryllium in a substance is often determined by the use of a photometric filtration method. If the weight of the beryllium is μ, then the value given by the photometric filtration method is normally distributed with mean μ and standard deviation σ. A total of eight independent
38. The following are independent samples from two normal populations, both of which have the same standard deviation σ.16, 17, 19, 20, 18 and 3, 4, 8 Use them to estimate σ.
37. Find a 95 percent two-sided confidence interval for the variance of the diameter of a rivet based on the data given here.6.68 6.66 6.62 6.72 6.76 6.67 6.70 6.72 6.78 6.66 6.76 6.72 6.76 6.70 6.76 6.76 6.74 6.74 6.81 6.66 6.64 6.79 6.72 6.82 6.81 6.77 6.60 6.72 6.74 6.70 6.64 6.78 6.70 6.70 6.75
36. The capacities (in ampere-hours) of 10 batteries were recorded as follows:140, 136, 150, 144, 148, 152, 138, 141, 143, 151a. Estimate the population variance σ2.b. Compute a 99 percent two-sided confidence interval for σ2.c. Compute a value v that enables us to state, with 90 percent
35. Verify the formulas given in Table 7.1 for the 100(1 − α) percent lower and upper confidence intervals for σ2.
34. The daily dissolved oxygen concentration for a water stream has been recorded over 30 days. If the sample average of the 30 values is 2.5 mg/liter and the sample standard deviation is 2.12 mg/liter, determine a value which, with 90 percent confidence, exceeds the mean daily concentration.
33. National Safety Council data show that the number of accidental deaths due to drowning in the United States in the years from 1990 to 1993 were (in units of one thousand) 5.2, 4.6, 4.3, 4.8. Use these data to give an interval that will, with 95 percent confidence, contain the number of such
32. Let X1, . . . , Xn+1 be a sample from a population with mean μ and varianceσ2. As noted in the text, the natural predictor of Xn+1 based on the data values X1, . . . , Xn is ¯Xn =n i=1Xi/n. Determine the mean square error of this predictor. That is, find E[(Xn+1 − ¯Xn)2].
31. A random sample of 16 professors at a large private university yielded a sample mean annual salary of $90,450 with a sample standard deviation of $9400. Determine a 95 percent confidence interval of the average salary of all professors at that university.
30. An important issue for a retailer is to decide when to reorder stock from a supplier. A common policy used to make the decision is of a type called s,S: The retailer orders at the end of a period if the on-hand stock is less than s, and orders enough to bring the stock up to S. The appropriate
29. Suppose that U1,U2, . . . is a sequence of independent uniform (0,1) random variables, and define N by N = min{n : U1 +· · ·+Un > 1}That is, N is the number of uniform (0, 1) random variables that need to be summed to exceed 1. Use random numbers to determine the value of 36 random variables
28. A set of 10 determinations, by a method devised by the chemist Karl Fischer, of the percentage of water in a methanol solution yielded the following data..50, .55, .53, .56, .54,.57, .52, .60, .55, .58 Assuming normality, use these data to give a 95 percent confidence interval for the actual
27. Studies were conducted in Los Angeles to determine the carbon monoxide concentration near freeways. The basic technique used was to capture air samples in special bags and to then determine the carbon monoxide concentration by using a spectrophotometer. The measurements in ppm(parts per
26. The following are the daily number of steps taken by a certain individual in 20 weekdays.2,100 1,984 2,072 1,898 1,950 1,992 2,096 2,103 2,043 2,218 2,244 2,206 2,210 2,152 1,962 2,007 2,018 2,106 1,938 1,956 Assuming that the daily number of steps is normally distributed, construct(a) a 95
25. Verify the formula given in Table 7.1 for the 100(1 − α) percent lower confidence interval for μ when σ is unknown.
24. In Problem 23, find the smallest value v that “with 90 percent confidence,”exceeds the average debt per cardholder.
23. A random sample of 300 CitiBank VISA cardholder accounts indicated a sample mean debt of $1220 with a sample standard deviation of $840.Construct a 95 percent confidence interval estimate of the average debt of all cardholders.
22. Each of 20 science students independently measured the melting point of lead. The sample mean and sample standard deviation of these measurements were (in degrees centigrade) 330.2 and 15.4, respectively. Construct(a) a 95 percent and (b) a 99 percent confidence interval estimate of the true
21. A standardized test is given annually to all sixth-grade students in the state of Washington. To determine the average score of students in her district, a school supervisor selects a random sample of 100 students. If the sample mean of these students’ scores is 320 and the sample standard
20. A company self-insures its large fleet of cars against collisions. To determine its mean repair cost per collision, it has randomly chosen a sample of 16 accidents. If the average repair cost in these accidents is $2200 with a sample standard deviation of $800, find a 90 percent confidence
19. Suppose that a random sample of nine recently sold houses in a certain city resulted in a sample mean price of $222,000, with a sample standard deviation of $22,000. Give a 95 percent upper confidence interval for the mean price of all recently sold houses in this city.
18. The following are scores on IQ tests of a random sample of 18 students at a large eastern university.130, 122, 119, 142, 136, 127, 120, 152, 141, 132, 127, 118, 150, 141, 133, 137, 129, 142a. Construct a 95 percent confidence interval estimate of the average IQ score of all students at the
17. The following data resulted from 24 independent measurements of the melting point of lead.330◦C 322◦C 345◦C 328.6◦C 331◦ C 342◦C 342.4◦C 340.4◦C 329.7◦C 334◦C 326.5◦C 325.8◦C 337.5◦C 327.3◦C 322.6◦C 341◦C 340◦C 333◦C 343.3◦C 331◦ C 341◦ C 329.5◦C
16. Suppose that when sampling from a normal population having an unknown mean μ and unknown variance σ2, we wish to determine a sample size n so as to guarantee that the resulting 100(1 − α) percent confidence interval for μ will be of size no greater than A, for given valuesα and A.
15. In Problem 14, compute a value c for which we can assert “with 99 percent confidence” that c is larger than the mean nicotine content of a cigarette.
14. In Problem 13, suppose that the population variance is not known in advance of the experiment. If the sample variance is .04, compute a 99 percent two-sided confidence interval for the mean nicotine content.
13. A sample of 20 cigarettes is tested to determine nicotine content and the average value observed was 1.2 mg. Compute a 99 percent two-sided confidence interval for the mean nicotine content of a cigarette if it is known that the standard deviation of a cigarette’s nicotine content is σ =.2
12. If X1, . . . , Xn is a sample from a normal population whose mean μ is unknown but whose variance σ2 is known, show that (−∞,X+zασ/√n)is a 100(1−α) percent lower confidence interval for μ.
11. Let X1, . . . , Xn,Xn+1 be a sample from a normal population having an unknown mean μ and variance 1. Let ¯Xn =n i=1Xi/n be the average of the first n of them.a. What is the distribution of Xn+1 − ¯Xn?b. If ¯Xn = 4, give an interval that, with 90 percent confidence, will contain the
10. The standard deviation of test scores on a certain achievement test is 11.3.If a random sample of 81 students had a sample mean score of 74.6, find a 90 percent confidence interval estimate for the average score of all students.
9. The PCB concentration of a fish caught in Lake Michigan was measured by a technique that is known to result in an error of measurement that is normally distributed with a standard deviation of .08 ppm (parts per million). Suppose the results of 10 independent measurements of this fish are 11.2,
8. An electric scale gives a reading equal to the true weight plus a random error that is normally distributed with mean 0 and standard deviationσ = .1 mg. Suppose that the results of five successive weighings of the same object are as follows: 3.142, 3.163, 3.155, 3.150, 3.141.a. Determine a 95
7. Recall that X is said to have a lognormal distribution with parameters μand σ2 if log(X) is normal with mean μ and variance σ2. Suppose X is such a lognormal random variable.a. Find E[X].b. Find Var(X).Hint: Make use of the formula for the moment generating function of a normal random
5. Suppose that X1, . . . , Xn are normal with mean μ1; Y1, . . . , Yn are normal with mean μ2; and W1, . . . , Wn are normal with mean μ1 +μ2. Assuming that all 3n random variables are independent with a common variance, find the maximum likelihood estimators of μ1 and μ2.6. River floods are
4. Determine the maximum likelihood estimates of a and λ when X1, . . . , Xn is a sample from the Pareto density function hax+1), -(2+1) f(x)= 0, if xa if x
3. Let X1, . . . , Xn be a sample from a normal μ, σ2 population. Determine the maximum likelihood estimator of σ2 when μ is known. What is the expected value of this estimator?
2. Determine the maximum likelihood estimator of θ when X1, . . . , Xn is a sample with density function f (x) = 1 2 e−|x−θ|, −∞
1. Let X1, . . . , Xn be a sample from the distribution whose density functionDetermine the maximum likelihood estimator of θ. f(x)= e-(x-8) 0 0x otherwise
30. A certain component is critical to the operation of an electrical system and must be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its standard deviation is 30 hours, how many of the components must be in stock so that the probability that
29. The average salary of newly graduated students with bachelor’s degrees in chemical engineering is $53,600, with a standard deviation of $3200.Approximate the probability that the average salary of a sample of 12 recently graduated chemical engineers exceeds $55,000.
28. The sample mean and sample standard deviation of all San Francisco student scores on the most recent Scholastic Aptitude Test examination in mathematics were 517 and 120. Approximate the probability that a random sample of 144 students would have an average score exceedinga. 507;b. 517;c.
27. Today, roughly 10.5 percent of the labor force belong to a union. If five workers are randomly chosen, what is the probability that none of them belong to a union? Compare your answer to what it would have been in 1983 when 20.1 percent of the workforce belonged to a union.
26. The following table uses data concerning the percentages of teenage male and female full-time workers whose annual salaries fall in different salary groupings. Suppose random samples of 1000 men and 1000 women were chosen. Use the table to approximate the probability thata. at least half of the
25. (Use the table from Problem 23.) Suppose random samples of 300 women and of 300 men are chosen. Approximate the probability that more women than men rarely eat breakfast.
24. (Use the table from Problem 23.) Suppose a random sample of 300 women is chosen. Approximate the probability thata. at least 60 of them are overweight by 20 percent or more;b. fewer than 50 of them sleep 6 hours or less nightly.
23. The following table gives the percentages of individuals of a given city, categorized by gender, that follow certain negative health practices. Suppose a random sample of 300 men is chosen. Approximate the probability thata. at least 150 of them rarely eat breakfast;b. fewer than 100 of them
22. Fifty-two percent of the residents of a certain city are in favor of teaching evolution in high school. Find or approximate the probability that at least 50 percent of a random sample of size n is in favor of teaching evolution, whena. n = 10;b. n = 100;c. n = 1000;d. n = 10,000.
21. Twelve percent of the population is left-handed. Find the probability that there are between 10 and 14 left-handers in a random sample of 100 members of this population. That is, find P{10 ≤ X ≤ 14}, where X is the number of left-handers in the sample.
20. Consider two independent samples — the first of size 10 from a normal population having variance 4 and the second of size 5 from a normal population having variance 2. Compute the probability that the sample variance from the second sample exceeds the one from the first. (Hint:Relate it to
19. In Problem 18, how large a sample would be necessary to ensure that the probability in part (a) is at least .95?
18. The temperature at which a thermostat goes off is normally distributed with variance σ2. If the thermostat is to be tested five times, finda. P{S2/σ 2 ≤ 1.8}b. P{.85 ≤ S2/σ 2 ≤ 1.15}where S2 is the sample variance of the five data values.
17. Use the text disk to compute P{X ≤ 10} when X is a binomial random variable with parameters n = 100, p = .1. Now compare this with its (a)Poisson and (b) normal approximation. In using the normal approximation, write the desired probability as P{X
16. Argue, based on the central limit theorem, that a Poisson random variable having mean λ will approximately have a normal distribution with mean and variance both equal to λ when λ is large. If X is Poisson with mean 100, compute the exact probability that X is less than or equal to 116 and
15. A club basketball team will play a 60-game season. Thirty-two of these games are against class A teams and 28 are against class B teams. The outcomes of all the games are independent. The team will win each game against a class A opponent with probability .5, and it will win each game against a
14. Teams 1, 2, 3, 4 are all scheduled to play with each of the other teams 10 times. Whenever team i plays team j, teami is the winner with probability Pi,j , and team j is the winner with probability Pj,i = 1− Pi,j. If P1,2 = .6, P1,3 = .7, P1,4 = .75, P2,3 = .6, P2,4 = .70, P3,4 = .5,a.
13. If X is binomial with parameters n=150, p=.6, compute the exact value of P{X ≤ 80} and compare with its normal approximation both (a) making use of and (b) not making use of the continuity correction.
12. An instructor knows from past experience that student exam scores have mean 77 and standard deviation 15. At present the instructor is teaching two separate classes — one of size 25 and the other of size 64.a. Approximate the probability that the average test score in the class of size 25
11. The lifetime (in hours) of a type of electric bulb has expected value 500 and standard deviation 80. Approximate the probability that the sample mean of n such bulbs is greater than 525 whena. n = 4;b. n = 16;c. n = 36;d. n = 64.
10. A tobacco company claims that the amount of nicotine in its cigarettes is a random variable with mean 2.2 mg and standard deviation .3 mg.However, the sample mean nicotine content of 100 randomly chosen cigarettes was 3.1 mg. What is the approximate probability that the sample mean would have
9. The lifetime of a certain electrical part is a random variable with mean 100 hours and standard deviation 20 hours. If 16 such parts are tested, find the probability that the sample mean isa. less than 104;b. between 98 and 104 hours.
8. The amount of time that a certain type of battery functions is a random variable with mean 5 weeks and standard deviation 1.5 weeks. Upon failure, it is immediately replaced by a new battery. Approximate the probability that 13 or more batteries will be needed in a year.
7. A six-sided die, in which each side is equally likely to appear, is repeatedly rolled until the total of all rolls exceeds 400. Approximate the probability that this will require more than 140 rolls.
6. Fifty numbers are rounded off to the nearest integer and then summed.If the individual roundoff errors are uniformly distributed between −.5 and .5, what is the approximate probability that the resultant sum differs from the exact sum by more than 3?
5. A highway department has enough salt to handle a total of 80 inches of snowfall. Suppose the daily amount of snow has a mean of 1.5 inches and a standard deviation of .3 inch.a. Approximate the probability that the salt on hand will suffice for the next 50 days.b. What assumption did you make in
4. A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you bet 1 on a specified number, you either win 35 if the roulette ball lands on that number or lose 1 if it does not. If you continually make such bets, approximate the probability thata. you are winning after 34 bets;b. you
3. Approximate the probability that the sum of 16 independent uniform (0, 1) random variables exceeds 10.
2. If 10 fair dice are rolled, approximate the probability that the sum of the values obtained (which ranges from 10 to 60) is between 30 and 40 inclusive.
1. Suppose that X1, X2, X3 are independent with the common probability mass function = P{X; 0) 2, P{X;= 1)=.3, P{X;=3)=.5, i = 1,2,3 a. Plot the probability mass function of X2 = x1 + x2 b. Determine E[X2] and Var(X2). c. Plot the probability mass function of X3 = d. Determine E[X3] and Var(X3). 2
50. Suppose that Y = αeX, where X is exponential with rate λ. Use the lack of memory property of the exponential to argue that the conditional distribution of Y given that Y >y0 >α is Pareto with parameters y0 and λ.
49. Suppose that Y has a Pareto distribution with minimal parameter α and index parameter λ.a. Find E[Y] when λ>1, and show that E[Y]=∞ when λ ≤ 1.b. Find Var(Y ) when λ>2.
48. Let be the standard normal distribution function. If, for constants a and b>0characterize the distribution of X. $ (x - a) b P{X x}=\
47. If Tn has a t-distribution with n degrees of freedom, show that T 2 n has an F-distribution with 1 and n degrees of freedom.
46. If T has a t-distribution with 8 degrees of freedom, find (a) P{T ≥ 1},(b) P{T ≤ 2}, and (c) P{−1
45. Show that (1/2) =√π (Hint: Evaluate ∞0 e−xx−1/2 dx by letting x =y2/2, dx = y dy.)

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