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modern mathematical statistics with applications

Questions and Answers of Modern Mathematical Statistics With Applications

43. Use the results of Section 6.4 to show that the variable T on which the PI is based does in fact have a t distribution with n 1 df.
42. A more extensive tabulation of t critical values than what appears in this book shows that for the t distribution with 20 df, the areas to the right of the values.687, .860, and 1.064 are .25,
41. Here are the lengths (in minutes) of the 63 nineinning games from the rst week of the 2001 major league baseball season:194 160 176 203 187 163 162 183 152 177 177 151 173 188 179 194 149 165 186
39. A sample of 25 pieces of laminate used in the manufacture of circuit boards was selected and the amount of warpage (in.) under particular conditions was determined for each piece, resulting in a
38. A study of the ability of individuals to walk in a straight line ( CanWe ReallyWalk Straight? Amer.J. Phys. Anthropol., 1992: 19—27) reported the accompanying data on cadence (strides per
37. The n 26 observations on escape time given in Exercise 33 of Chapter 1 give a sample mean and sample standard deviation of 370.69 and 24.36, respectively.a. Calculate an upper con dence bound
36. Exercise 43 in Chapter 1 introduced the following sample observations on stabilized viscosity of asphalt specimens: 2781, 2900, 3013, 2856, 2888.A normal probability plot supports the assumption
35. A sample of 14 joint specimens of a particular type gave a sample mean proportional limit stress of 8.48 MPa and a sample standard deviation of .79 MPa( Characterization of Bearing Strength
34. Here is a sample ofACT scores (average of the Math, English, Social Science, and Natural Science scores)for students taking college freshman calculus:24.00 28.00 27.75 27.00 24.25 23.50 26.25
33. A sample of ten guinea pigs yielded the following measurements of body temperature in degrees Celsius (Statistical Exercises in Medical Research, New York:Wiley, 1979, p. 26):38.1 38.4 38.3 38.2
32. Determine the t critical value for a lower or an upper con dence bound for each of the situations described in Exercise 31.
31. Determine the t critical value for a two-sided con -dence interval in each of the following situations:a. Con dence level 95%, df 10b. Con dence level 95%, df 15c. Con dence level 99%,
29. Determine the values of the following quantities:a. t.1,15b. t.05,15c. t.05,25d. t.05,40e. t.005,40
28. Young people may feel they are carrying the weight of the world on their shoulders, when what they are actually carrying too often is an excessively heavy backpack. The article Effectiveness of a
27. Reconsider the CI (8.10) for p, and focus on a con dence level of 95%. Show that the con dence limits agree quite well with those of the traditional interval (8.11) once two successes and two
26. The superintendent of a large school district, having once had a course in probability and statistics, 23 pˆbelieves that the number of teachers absent on any given day has a Poisson
24. A sample of 56 research cotton samples resulted in a sample average percentage elongation of 8.17 and a sample standard deviation of 1.42 ( An Apparent Relation Between the Spiral Anglef, the
23. The article An Evaluation of Football Helmets Under Impact Conditions (Amer. J. Sports Medi., 1984: 233—237) reports that when each football helmet in a random sample of 37 suspension-type
22. Arandom sample of 487 nonsmokingwomen of normal weight (body mass index between 19.8 and 26.0)who had given birth at a large metropolitan medical centerwas selected ( The Effects of Cigarette
21. A random sample of 539 households from a certain midwestern city was selected, and it was determined that 133 of these households owned at least one rearm ( The Social Determinants of Gun
20. The Associated Press (October 9, 2002) reported that in a survey of 4722 American youngsters aged 6 to 19, 15% were seriously overweight (a body mass index of at least 30; this index is a measure
18. The article Ultimate Load Capacities of Expansion Anchor Bolts (J. Energy Engrg., 1993: 139—158)gave the following summary data on shear strength(kip) for a sample of 3/8-in. anchor bolts: n
17. A study was done on 41 rst-year medical students to see if their anxiety levels changed during the rst semester. One measure used was the level of serum cortisol, which is associated with
16. A sample of 66 obese adults was put on a lowcarbohydrate diet for a year. The average weight loss was 11 pounds and the standard deviation was x .67s/1n x 2.05s/1n x .84s/1n x 89.10 19
15. Determine the con dence level for each of the following large-sample one-sided con dence bounds:a. Upper bound:b. Lower bound:c. Upper bound:
14. The article Evaluating Tunnel Kiln Performance(Amer. Ceramic Soc. Bull., Aug. 1997: 59—63) gave the following summary information for fracture strengths (MPa) of n 169 ceramic bars red in a
13. The article Extravisual Damage Detection? De ning the Standard Normal Tree (Photogrammetric Engrg. Remote Sensing, 1981: 515—522) discusses the use of color infrared photography in identi
12. Arandom sample of 110 lightning ashes in a certain region resulted in a sample average radar echo duration of .81 sec and a sample standard deviation of .34 sec ( Lightning Strikes to an Airplane
11. Consider the next 1000 95% CIs for m that a statistical consultant will obtain for various clients.Suppose the data sets on which the intervals are based are selected independently of one
10. A random sample of n 15 heat pumps of a certain type yielded the following observations on lifetime(in years):2.0 1.3 6.0 1.9 5.1 .4 1.0 5.3 15.7 .7 4.8 .9 12.2 5.3 .6a. Assume that the
9.a. Under the same conditions as those leading to the interval (8.5),. Use this to derive a one-sided interval for m that has in nite width and provides a lower con dence bound on m. What is this
8. Let a1 0, a2 0, with a1 a2 a. Thena. Use this equation to derive a more general expression for a 100(1 a)% CI for m of which the interval (8.5) is a special case.b. Let a .05 and a1 a/4,
7. By how much must the sample size n be increased if the width of the CI (8.5) is to be halved? If the sample size is increased by a factor of 25, what effect will this have on the width of the
6. On the basis of extensive tests, the yield point of a particular type of mild steel reinforcing bar is known to be normally distributed with s100. The composition of the bar has been slightly
4. A CI is desired for the true average stray-load loss m (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that stray-load loss
2. Each of the following is a con dence interval for m true average (i.e., population mean) resonance frequency (Hz) for all tennis rackets of a certain type:(114.4, 115.6) (114.1, 115.9)a. What is
1. Consider a normal population distribution with the value of s known.a. What is the con dence level for the interval?b. What is the con dence level for the interval?c. What value of za/2 in the CI
63. Let x denote the number of items in an order and y denote time (min) necessary to process the order.Processing time may be determined by various factors other than order size. So for any
62. Let X, the payoff from playing a certain game, have pmfa. Verify that f (x; u) is a legitimate pmf, and determine the expected payoff. (Hint: Look back at the properties of a geometric random
61. The principle of unbiasedness (prefer an unbiased estimator to any other) has been criticized on the grounds that in some situations the only unbiased estimator is patently ridiculous. Here is
60. Here is a result that allows for easy identi cation of a minimal suf cient statistic: Suppose there is a function t(x1, . . . , xn) such that for any two sets of observations x1, . . . , xn and
59. Let X1, . . . , Xn be a random sample from a normal distribution with both m and s unknown. An unbiased estimator of u P(Xc) based on the jointly suf cient statistics is desired. Let and . Then
58. The fraction of a bottle that is lled with a particular liquid is a continuous random variable X with pdf f (x; u) uxu1 for 0 x 1 (where u 0).a. Obtain the method of moments estimator for
57. Let p denote the proportion of all individuals who are allergic to a particular medication. An investigator tests individual after individual to obtain a group of r individuals who have the
56. For 0 u1 consider a random sample from a uniform distribution on the interval from u to 1/u. Identify a suf cient statistic for u.
55. Each of n specimens is to be weighed twice on the same scale. Let Xi and Yi denote the two observed weights for the ith specimen. Suppose Xi and Yi are independent of one another, each normally
54. When the sample standard deviation S is based on a random sample from a normal population distribution, it can be shown that Use this to obtain an unbiased estimator for s of the form cS. What is
53. When the population distribution is normal, the statistic median can be used to estimate s. This estimator is more resistant to the effects of outliers (observations far from the bulk of the
52. Let X1, . . . , Xn be a random sample from a pdf that is symmetric about m. An estimator for m that has been found to perform well for a variety of underlying distributions is the
51. The mean square error of an estimator is MSE . If is unbiased, then MSE , but in general MSE (bias)2. Consider the estimator KS2, where S2sample variance. What value of K minimizes the
50.a. Let X1, . . . , Xn be a random sample from a uniform distribution on [0, u]. Then the mle of u isY max(Xi). Use the fact that Y y iff each Xi y to derive the cdf of Y. Then show that the pdf
49. At time t 0, there is one individual alive in a certain population. A pure birth process then unfolds as follows. The time until the rst birth is exponentially distributed with parameter l.
48. Let X1, X2, . . . , Xn be a random sample from a continuous distribution with pdf f (x; u). For large n, the variance of the sample median is approximately 1/{4n[ f( ; u)]2}. If X1, X2, . . . ,
47. Let X1, X2, . . . , Xn be a random sample from the normal distribution with known mean m but with the standard deviation s as the unknown parameter.a. Find the information in a single
46. Let X1, X2, . . . , Xn be a random sample from the normal distribution with known mean m but with the variance s2 as the unknown parameter.a. Find the information in a single observation and the
45. Let X1, X2, . . . , Xn be a random sample from the normal distribution with known standard deviation s.a. Find the mle of m.b. Find the distribution of the mle.c. Is the mle an ef cient
44. Component lifetimes have the exponential distribution with pdf f (x; l) lelx, x 0, and f (x; l) 0 otherwise, where l 0. However, we wish to estimate the mean m1/l based on the random
43. In Example 7.23 f (x; u) 1/u for 0 xu and 0 otherwise. Given a random sample, the maximum likelihood estimate is the largest observation.a. Letting , show that is unbiased and nd its
42. Assume that the number of defects in a car has a Poisson distribution with parameter l. To estimate l we obtain the random sample X1, X2, . . . , Xn.a. Find the Fisher information in a single
41. A particular quality characteristic of items produced using a certain process is known to be normally distributed with mean m and standard deviation 1. Let X denote the value of the
40. In Example 7.30, we started with U I(X1 0) and used a conditional expectation argument to obtain an unbiased estimator of the zero-defect probability based on the suf cient statistic.
39. The probability that any particular component of a certain type works in a satisfactory manner is p. If n of these components are independently selected, then the statistic X, the number among
38. Suppose that material strength X has a lognormal distribution with parameters m and s [which are the mean and standard deviation of ln(X), not of X itself].Are gXi and jointly suf cient for the
37. For u 0 consider a random sample from a uniform distribution on the interval from u to 2u (pdf 1/u for u x 2u), and use the factorization theorem to determine a suf cient statistic for u.
36. Suppose waiting time for delivery of an item is uniform on the interval from u1 to u2 [so f (x; u1, u2) 1/(u2 u1) for u1 x u2 and is 0 otherwise]. Consider a random sample of nwaiting
35. Identify a pair of jointly suf cient statistics for the two parameters of a gamma distribution based on a random sample of size n from that distribution.
34. Let X1, . . . , Xn be a random sample of component lifetimes from an exponential distribution with parameter l. Use the factorization theorem to show that gXi is a suf cient statistic for l.
33. Components of a certain type are shipped in batches of size k. Suppose that whether or not any particular component is satisfactory is independent of the condition of any other component, and
32. The long-run proportion of vehicles that pass a certain emissions test is p. Suppose that three vehicles are independently selected for testing. Let Xi 1 if the ith vehicle passes the test and
31. At time t 0, 20 identical components are put on test. The lifetime distribution of each is exponential with parameter l. The experimenter then leaves the test facility unmonitored. On his
30. Consider a random sample X1, X2, . . . , Xn from the shifted exponential pdf Taking u 0 gives the pdf of the exponential distribution considered previously (with positive density to the right
29. Let X1, X2, . . . , Xn represent a random sample from the Rayleigh distribution with density function given in Exercise 15. Determinea. The maximum likelihood estimator of u and then calculate
28. Let X1, . . . , Xn be a random sample from a gamma distribution with parameters a and b.a. Derive the equations whose solution yields the maximum likelihood estimators of a andb. Do you think
27. Refer to Exercise 26. Suppose we choose another bagel and weigh it. Let Xweight of the bagel. Use the given data to obtain the mle of P(X 113.4).(Hint: P(X 113.4) [(113.4 m)/s)].)
26. Six Pepperidge Farm bagels were weighed, yielding the following data (grams):117.6 109.5 111.6 109.2 119.1 110.8(Note: 4 ounces 113.4 grams)a. Assuming that the six bagels are a random sample
25. Refer to Exercise 21. Instead of selecting n 20 helmets to examine, suppose we examine helmets in succession until we have found r 3 awed ones.If the 20th helmet is the third awed one (so
24. Two different computer systems are monitored for a total of n weeks. Let Xi denote the number of breakdowns of the rst system during the ith week, and suppose the Xi s are independent and drawn
23. Let X denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test. Suppose the pdf of X is where 1 u. A random sample of ten students
22. Let X have a Weibull distribution with parameters a andb, soa. Based on a random sample X1, . . . , Xn, write equations for the method of moments estimators of b anda. Show that, once the
21. A random sample of n bike helmets manufactured by a certain company is selected. Let X the number among the n that are awed, and let p P( awed). Assume that only X is observed, rather than the
20. Return to the problem of estimating the population proportion p and consider another adjusted estimator, namely The justi cation for this estimator comes from the Bayesian approach to point
19. An investigator wishes to estimate the proportion of students at a certain university who have violated the honor code. Having obtained a random sample of n students, she realizes that asking
18. Let X1, X2, . . . , Xn be a random sample from a pdf f (x) that is symmetric about m, so that is an unbiased estimator of m. If n is large, it can be shown that 1/{4n[ f (m)]2}. When the
17. In Chapter 3, we de ned a negative binomial rv as the number of failures that occur before the r th success in a sequence of independent and identical success/failure trials. The probability mass
16. Suppose the true average growth m of one type of plant during a 1-year period is identical to that of a second type, but the variance of growth for the rst type is s2, whereas for the second
15. Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution with pdfa. It can be shown that E(X2) 2u. Use this fact to construct an unbiased estimator of u based on and use
14. A sample of n captured Pandemonium jet ghters results in serial numbers x1, x2, x3, . . . , xn. The CIA knows that the aircraft were numbered consecutively at the factory starting with a and
13. Consider a random sample X1, . . . , Xn from the pdf where1 u1 (this distribution arises in particle physics). Show that is an unbiased estimator of u. [Hint: First determine m E(X) E( ).]
12. Suppose a certain type of fertilizer has an expected yield per acre of m1 with variance s2, whereas the expected yield for a second type of fertilizer is m2 with the same variance s2. Let and
11. Of n1 randomly selected male smokers, X1 smoked lter cigarettes, whereas of n2 randomly selected female smokers, X2 smoked lter cigarettes. Let p1 and p2 denote the probabilities that a randomly
10. Using a long rod that has length m, you are going to lay out a square plot in which the length of each side is m. Thus the area of the plot will be m2. However, you do not know the value of m, so
9. Each of 150 newly manufactured items is examined and the number of scratches per item is recorded (the items are supposed to be free of scratches), yielding the following data:Number of scratches
8. In a random sample of 80 components of a certain type, 12 are found to be defective.a. Give a point estimate of the proportion of all such components that are not defective.b. A system is to be
7.a. A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected and the amount of gas (therms) used during the month of January is determined for each
6. Consider the accompanying observations on stream ow (1000 s of acre-feet) recorded at a station in Colorado for the period April 1—August 31 over a 31-year span (from an article in the 1974
5. As an example of a situation in which several different statistics could reasonably be used to calculate a point estimate, consider a population of N invoices.Associated with each invoice is its
4. The data set of Exercise 1 also includes these thirdgrade verbal IQ observations for males:117 103 121 112 120 132 113 117 132 149 125 131 136 107 108 113 136 114 and females:114 102 113 131 124
3. Consider the following sample of observations on coating thickness for low-viscosity paint ( Achieving a Target Value for a Manufacturing Process: A Case Study, J. Qual. Tech., 1992: 22—26):.83
2. A sample of 20 students who had recently taken elementary statistics yielded the following information on brand of calculator owned (T Texas Instruments, H Hewlett-Packard, C Casio, S
1. The accompanying data on IQ for rst graders in a particular school was introduced in Example 1.2.82 96 99 102 103 103 106 107 108 108 108 108 109 110 110 111 113 113 113 113 115 115 118 118 119

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