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modern mathematical statistics with applications

Modern Mathematical Statistics With Applications 1st Edition Jay L Devore - Solutions

68. The error in a measurement is normally distributed with mean m and standard deviation 1. Consider a random sample of n errors, and show that the likelihood ratio test for H0: m0 versus Ha: m0 rejects the null hypothesis when either or .What is c for a test with a .05? How does the test
67. Suppose that each of n randomly selected individuals is classi ed according to his/her genotype with respect to a particular characteristic and that the three possible genotypes are AA, Aa, and aa with long-run proportions (probabilities) u2, 2u(1 u), and (1 u)2, respectively (0 u 1).
66. Consider a random sample of size n from the shifted exponential distribution with pdf for x u and 0 otherwise (the graph is that of the ordinary exponential pdf with l 1 shifted so that it begins its descent at u rather than at 0). Let Y1 denote the smallest order statistic, and show that the
65. Consider a random sample of n component lifetimes, where the distribution of lifetime is exponential with parameter l.a. Obtain a most powerful test for H0: l 1 versus Ha: l .5, and express the rejection region in terms of a simple statistic.b. Is the test found in (a) uniformly most
64. Suppose that X, the fraction of a container that is lled, has pdf f (x; u) uxu1 for 0 x 1 (where u0), and let X1, . . . , Xn be a random sample from this distribution.a. Show that the most powerful test for H0: u 1 versus Ha: u 2 rejects the null hypothesis if.b. Is the test
63. The error X in a measurement has a normal distribution with mean value 0 and variance s2. Consider testing H0: s2 2 versus Ha: s2 3 based on a random sample X1, . . . , Xn of errors.gxi c pˆ x/n .21 X x 74a. Show that a most powerful test rejects H0 when.b. For n 10, nd the value
62. For a random sample of n individuals taking a licensing exam, let Xi 1 if the ith individual in the sample passes the exam and Xi 0 otherwise (i 1, . . . , n).a. With p denoting the proportion of all exam takers who pass, show that the most powerful test of H0: p .5 versus Ha: p .75
61. Consider the large-sample level .01 test in Section 9.3 for testing H0: p .2 against Ha: p.2.a. For the alternative value p.21, compute b(.21)for sample sizes n 100, 2500, 10,000, 40,000, and 90,000.b. For ,compute the P-value when n 100, 2500, 10,000, and 40,000.c. In most situations,
60. Reconsider the paint-drying problem discussed in Example 9.2. The hypotheses were H0: m 75 versus Ha: m 75, with s assumed to have value 9.0.Consider the alternative value m 74, which in the context of the problem would presumably not be a practically signi cant departure from H0.a. For a
59. The relative conductivity of a semiconductor device is determined by the amount of impurity doped into the device during its manufacture. A silicon diode to be used for a speci c purpose requires an average cut-on voltage of .60 V, and if this is not achieved, the amount of impurity must be
58. A spectrophotometer used for measuring CO concentration[ppm (parts per million) by volume] is checked for accuracy by taking readings on a manufactured gas (called span gas) in which the CO concentration is very precisely controlled at 70 ppm. If the readings suggest that the spectrophotometer
57. A certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of a writing machine) is at least 10 hours. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting
56. The times of rst sprinkler activation for a series of tests with re prevention sprinkler systems using an aqueous lm-forming foam were (in sec)27 41 22 27 23 35 30 33 24 27 28 22 24(see Use of AFFF in Sprinkler Systems, Fire Tech., 1976: 5). The system has been designed so that true average
55. A random sample of soil specimens was obtained, and the amount of organic matter (%) in the soil was determined for each specimen, resulting in the accompanying data (from Engineering Properties of Soil, Soil Sci., 1998: 93—102).1.10 5.09 0.97 1.59 4.60 0.32 0.55 1.45 0.14 4.47 1.20 3.50 5.02
54. Many consumers are turning to generics as a way of reducing the cost of prescription medications. The article Commercial Information on Drugs: Confusing to the Physician? (J. Drug Issues, 1988: 245—257) gives the results of a survey of 102 doctors.Only 47 of those surveyed knew the generic
52. An aspirin manufacturer lls bottles by weight rather than by count. Since each bottle should contain 100 tablets, the average weight per tablet should be 5 grains. Each of 100 tablets taken from a very large lot is weighed, resulting in a sample average weight per tablet of 4.87 grains and a
51. Let m denote true average serum receptor concentration for all pregnant women. The average for all women is known to be 5.63. The article Serum Transferrin Receptor for the Detection of Iron De ciency in Pregnancy (Amer. J. Clin. Nutrit., 1991: 1077—1081) reports that P-value.10 for a test of
50. The paint used to make lines on roads must re ect enough light to be clearly visible at night. Let m denote the true average re ectometer reading for a new type of paint under consideration. A test of H0:m20 versus Ha: m 20 will be based on a random sample of size n from a normal population
49. Give as much information as you can about the P-value of a t test in each of the following situations:a. Upper-tailed test, df 8, t 2.0b. Lower-tailed test, df 11, t2.4c. Two-tailed test, df 15, t1.6d. Upper-tailed test, df 19, t.4e. Upper-tailed test, df 5, t 5.0f.
48. Newly purchased tires of a certain type are supposed to be lled to a pressure of 30 lb/in2. Let m denote the true average pressure. Find the P-value associated with each given z statistic value for testing H0: m 30 versus Ha: m 30.a. 2.10b. 1.75c. .55d. 1.41e. 5.3
47. Let m denote the mean reaction time to a certain stimulus. For a large-sample z test of H0: m 5 versus Ha: m 5, nd the P-value associated with each of the given values of the z test statistic.a. 1.42b. .90c. 1.96d. 2.48e. .11
46. Pairs of P-values and signi cance levels,a, are given. For each pair, state whether the observed P-value would lead to rejection of H0 at the given signi cance level.a. P-value .084, a .05b. P-value .003, a .001c. P-value .498, a .05d. P-value .084, a .10e. P-value .039, a
45. For which of the given P-values would the null hypothesis be rejected when performing a level .05 test?a. .001b. .021c. .078d. .047e. .148
43. A manufacturer of plumbing xtures has developed a new type of washerless faucet. Let p P(a randomly selected faucet of this type will develop a leak within 2 years under normal use). The manufacturer has decided to proceed with production unless it can be determined that p is too large; the
42. Each of a group of 20 intermediate tennis players is given two rackets, one having nylon strings and the other synthetic gut strings. After several weeks of playing with the two rackets, each player will be asked to state a preference for one of the two types of strings. Let p denote the
40. The article Statistical Evidence of Discrimination(J. Amer. Statist. Assoc., 1982: 773—783) discusses the court case Swain v. Alabama (1965), in which it was alleged that there was discrimination against blacks in grand jury selection. Census data suggested that 25% of those eligible for
39. A university library ordinarily has a complete shelf inventory done once every year. Because of new shelving rules instituted the previous year, the head librarian believes it may be possible to save money by postponing the inventory. The librarian decides to select at random 1000 books from
37. A manufacturer of nickel—hydrogen batteries randomly selects 100 nickel plates for test cells, cycles them a speci ed number of times, and determines that 14 of the plates have blistered.a. Does this provide compelling evidence for concluding that more than 10% of all plates blister under
35. The industry standard for the amount of alcohol poured into many types of drinks (e.g., gin for a gin and tonic, whiskey on the rocks) is 1.5 oz. Each individual in a sample of 8 bartenders with at least ve years of experience was asked to pour rum for a rum and coke into a short, wide
34. For a xed alternative value m, show that b(m) S 0 as n S q for either a one-tailed or a two-tailed z test in the case of a normal population distribution with known s.
33. Show that for any 0, when the population distribution is normal and s is known, the two-tailed test satis es b(m0 ) b(m0 ), so that b(m)is symmetric about m0.
32. A sample of 12 radon detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows:105.6 90.9 91.2 96.9 96.5 91.3 100.1 105.0 99.6 107.7 103.3 92.4a. Does this data suggest that the population mean reading under these conditions
31. In an experiment designed to measure the time necessary for an inspector s eyes to become used to the reduced amount of light necessary for penetrant inspection, the sample average time for n 9 inspectors was 6.32 sec and the sample standard deviation was 1.65 sec. It has previously been
30. The recommended daily dietary allowance for zinc among males older than age 50 years is 15 mg/day.The article Nutrient Intakes and Dietary Patterns of Older Americans: A National Study (J. Gerontology, 1992: M145—150) reports the following summary data on intake for a sample of males age
29. The amount of shaft wear (.0001 in.) after a xed mileage was determined for each of n 8 internal combustion engines having copper lead as a bearing material, resulting in and s 1.25.a. Assuming that the distribution of shaft wear is normal with mean m, use the t test at level .05 to test
28. Minor surgery on horses under eld conditions requires a reliable short-term anesthetic producing good muscle relaxation, minimal cardiovascular and respiratory changes, and a quick, smooth recovery with minimal aftereffects so that horses can be left unattended. The article A Field Trial of
27. On the label, Pepperidge Farm bagels are said to weigh four ounces each (113 grams).Arandom sample of six bagels resulted in the following weights(in grams):117.6 109.5 111.6 109.2 119.1 110.8a. Based on this sample, is there any reason to doubt that the population mean is at least 113 grams?b.
26. In recent years major league baseball games have averaged three hours in duration. However, because games in Denver tend to be high-scoring, it might be expected that the games would be longer there. In 2001, the 81 games in Denver averaged 185.54 minutes with standard deviation 24.6 minutes.
25. Recall the rst-grade IQ scores of Example 1.2.Here is a random sample of 10 of those scores:107 113 108 127 146 103 108 118 111 119 The IQ test score has approximately a normal distribution with mean 100 and standard deviation 15 for the entire U.S. population of rst graders. Here we are
24. Reconsider the sample observations on stabilized viscosity of asphalt specimens introduced in Exercise 43 in Chapter 1 (2781, 2900, 3013, 2856, and 2888). Suppose that for a particular application, it is required that true average viscosity be 3000.Does this requirement appear to have been
23. Exercise 33 in Chapter 1 gave n 26 observations on escape time (sec) for oil workers in a simulated exercise, from which the sample mean and sample standard deviation are 370.69 and 24.36, respectively.Suppose the investigators had believed a priori that true average escape time would be at
22. The article The Foreman s View of Quality Control(Quality Engrg., 1990: 257—280) described an investigation into the coating weights for large pipes resulting from a galvanized coating process. Production standards call for a true average weight of 200 lb per pipe. The accompanying
21. The true average diameter of ball bearings of a certain type is supposed to be .5 in. A one-sample t test will be carried out to see whether this is the case.What conclusion is appropriate in each of the following situations?a. n 13, t 1.6, a .05b. n 13, t1.6, a .05c. n 25,
20. Lightbulbs of a certain type are advertised as having an average lifetime of 750 hours. The price of these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase arrangement unless it can be conclusively demonstrated that the true average lifetime is smaller
19. The melting point of each of 16 samples of a certain brand of hydrogenated vegetable oil was determined, resulting in . Assume that the distribution of melting point is normal with s 1.20.a. Test H0: m 95 versus Ha: m 95 using a twotailed level .01 test.b. If a level .01 test is used,
18. Reconsider the paint-drying situation of Example 9.2, in which drying time for a test specimen is normally distributed with s 9. The hypotheses H0: m 75 versus Ha: m 75 are to be tested using a random sample of n 25 observations.a. How many standard deviations (of ) below the null value
17. Answer the following questions for the tire problem in Example 9.7.a. If and a level a .01 test is used, what is the decision?b. If a level .01 test is used, what is b(30,500)?c. If a level .01 test is used and it is also required that b(30,500) .05, what sample size n is necessary?d. If ,
16. Let the test statistic T have a t distribution when H0 is true. Give the signi cance level for each of the following situations:a. Ha: m m0, df 15, rejection region t 3.733b. Ha: m m0, n 24, rejection region t2.500c. Ha: m m0, n 31, rejection region t 1.697 or t1.697
15. Let the test statistic Z have a standard normal distribution when H0 is true. Give the signi cance level for each of the following situations:a. Ha: m m0, rejection region z 1.88b. Ha: m m0, rejection region z2.75c. Ha: m m0, rejection region z 2.88 or z2.88
14. Reconsider the situation of Exercise 11 and suppose the rejection region is { : 10.1004 or 9.8940} {z: z 2.51 or z2.65}.a. What is a for this procedure?b. What is b when m10.1? When m9.9? Is this desirable?
13. Let X1, . . . , Xn denote a random sample from a normal population distribution with a known value of s.a. For testing the hypotheses H0: m m0 versus Ha:m m0 (where m0 is a xed number), show that the test with test statistic and rejection region has signi cance level .01.b. Suppose the
12. A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under speci ed conditions is known to be 120 ft. It is proposed that the new design be implemented only if sample data strongly indicates a
11. The calibration of a scale is to be checked by weighing a 10-kg test specimen 25 times. Suppose that the results of different weighings are independent of one another and that the weight on each trial is normally distributed with s.200 kg. Let m denote the true average weight reading on the
10. For healthy individuals the level of prothrombin in the blood is approximately normally distributed with mean 20 mg/100 mL and standard deviation 4 mg/100 mL. Low levels indicate low clotting ability.In studying the effect of gallstones on prothrombin, the level of each patient in a sample is
9. Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the rst company over the second. Consider testing H0: p .5 versus Ha: p .5 based on a random sample of 25 individuals. Let X denote
7. Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most 150F, there will be no negative effects on the river s ecosystem. To investigate whether
4. Let m denote the true average radioactivity level(picocuries per liter). The value 5 pCi/L is considered the dividing line between safe and unsafewater.Would you recommend testing H0: m 5 versus Ha:m 5 or H0: m 5 versus Ha: m 5? Explain your reasoning. (Hint: Think about the consequences
3. To determine whether the girder welds in a new performing arts center meet speci cations, a random H0: S21 S22, Ha: S21 S22 H: X Y 5 H: x ~ 45 sample of welds is selected, and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the
2. For the following pairs of assertions, indicate which do not comply with our rules for setting up hypotheses and why (the subscripts 1 and 2 differentiate between quantities for two different populations or samples):a. H0: m 100, Ha: m 100b. H0: s 20, Ha: s 20c. H0: p .25, Ha: p .25d.
1. For each of the following assertions, state whether it is a legitimate statistical hypothesis and why:a. H: s 100 b.c. H: s.20d. H: s1/s2 1 e.f. H: l.01, where l is the parameter of an exponential distribution used to model component lifetime
79. Consider 95% CI s for two different parameters u1 and u2, and let Ai (i 1, 2) denote the event that the value of ui is included in the random interval that results in the CI. Thus P(Ai) .95.a. Suppose that the data on which the CI for u1 is based is independent of the data used to obtain
78. The previous two exercises considered a CI for a population median based on the n order statistics from a random sample. Let s now consider a prediction interval for the next observation Xn1.a. What is P(Xn1 X1)? What is P({Xn1 X1}{Xn1 X2})?b. What is P(Xn1 Y1)? What is P(Xn1
77. Consider the situation described in the previous exercise.a. What is P({X1 } {X2} . . . {Xn}), that is, the probability that only the rst observation is smaller than the median?b. What is the probability that exactly one of the n observations is smaller than the median?c. What is P(
76. The one-sample t CI for m is also a con dence interval for the population median when the population distribution is normal.We now develop a CI for that is valid whatever the shape of the population distribution as long as it is continuous. Let X1, . . . , Xn be a random sample from the
75. The one-sample CI for a normal mean and PI for a single observation from a normal distribution were both based on the central t distribution. A CI for a particular percentile (e.g., the 1st percentile or the 95th percentile) of a normal population distribution is based on the noncentral t
74. Let u1 and u2 denote the mean weights for animals of two different species. An investigator wishes to estimate the ratio u1/u2. Unfortunately the species are extremely rare, so the estimate will be based on nding a single animal of each species. Let Xi denote the weight of the species i animal
73.a. Use the results of Example 8.5 to obtain a 95%lower con dence bound for the parameter l of an exponential distribution, and calculate the bound based on the data given in the example.b. If lifetime X has an exponential distribution, the probability that lifetime exceeds t is P(X t) elt. Use
72. Suppose x1, x2, . . . , xn are observed values resulting from a random sample from a symmetric but possibly heavy-tailed distribution. Let and fs denote the sample median and fourth spread, respectively.Chapter 11 of Understanding Robust and Exploratory Data Analysis (see the bibliography in
71. Let 0 ga. Then a 100(1 a)% CI for m when n is large is The choice g a/2 yields the usual interval derived in Section 8.2; if g a/2, this con dence interval is not symmetric about . The width of this interval is w s(zg zag)/ . Show that w is minimized for the choice g a/2, so that
70. Let X1, X2, . . . , Xn be a random sample from a uniform distribution on the interval [0, u], so that
69. Exercise 63 from Chapter 7 introduced regression through the origin to relate a dependent variable y to an independent variable x. The assumption there was that for any xed x value, the dependent variable is a random variable Y with mean value bx and variance s2 (so that Y has mean value zero
68. In Example 7.9, we introduced the concept of a censored experiment in which n components are put on test and the experiment terminates as soon as r of the components have failed. Suppose component lifetimes are independent, each having an exponential distribution with parameter l. Let Y1 denote
67. Chronic exposure to asbestos ber is a well-known health hazard. The article The Acute Effects of Chrysotile Asbestos Exposure on Lung Function(Envir. Res., 1978: 360—372) reports results of a study based on a sample of construction workers who had been exposed to asbestos over a prolonged
66. A journal article reports that a sample of size 5 was used as a basis for calculating a 95% CI for the true average natural frequency (Hz) of delaminated beams of a certain type. The resulting interval was(229.764, 233.504). You decide that a con dence level of 99% is more appropriate than the
65. It is important that face masks used by re ghters be able to withstand high temperatures because re ghters commonly work in temperatures of 200500F. In a test of one type of mask, 11 of 55 masks had lenses pop out at 250. Construct a 90%CI for the true proportion of masks of this type whose
64. Aphid infestation of fruit trees can be controlled either by spraying with pesticide or by inundation with ladybugs. In a particular area, four different groves of fruit trees are selected for experimentation.The rst three groves are sprayed with pesticides 1, 2, and 3, respectively, and the
63. There were 12 rst-round heats in the men s 100-meter race at the 1996 Atlanta Summer Olympics.Here are the reaction times in seconds (time to rst movement) of the top four nishers of each heat.The rst 12 are the 12 winners, then the secondplace nishers, and so on.1st .187 .152 .137 .175 .172
62. The nancial manager of a large department store chain selected a random sample of 200 of its credit card customers and found that 136 had incurred an interest charge during the previous year because of an unpaid balance.a. Compute a 90% CI for the true proportion of credit card customers who
61. A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull the pages from the binding. If this force is measured in pounds, how many books should be
60. For each of 18 preserved cores from oil-wet carbonate reservoirs, the amount of residual gas saturation after a solvent injection was measured at water ood-out. Observations, in percentage of pore volume, were 23.5 31.5 34.0 46.7 45.6 32.5 41.4 37.2 42.5 46.9 51.5 36.4 44.5 35.7 33.5 39.3 22.0
59. The reaction time (RT) to a stimulus is the interval of time commencing with stimulus presentation and ending with the rst discernible movement of a certain type. The article Relationship of ReactionTime and Movement Time in a Gross Motor Skill (Percept.Motor Skills, 1973: 453—454) reports
58. A triathlon consisting of swimming, cycling, and running is one of the more strenuous amateur sporting events. The article Cardiovascular and Thermal Response of Triathlon Performance (Med. Sci.Sports Exercise, 1988: 385—389) reports on a research study involving nine male triathletes.
57. We would like to obtain a 95% con dence interval for the population standard deviation study time using the data in Exercise 49.a. Obtain a bootstrap sample of 1000 standard deviations and use it to form a 95% con dence interval for the population standard deviation using the percentile
56. The median might be a more meaningful statistic for the length-of-game data in Exercise 55. The median of all 2429 lengths is 175 minutes.a. Obtain a bootstrap sample of 1000 medians.b. Use the standard deviation for part (a) to get a 95% con dence interval for the population median.c.
55. If you go to a major league baseball game, how long do you expect the game to be? From the 2429 games played in 2001, here is a random sample of 25 times in minutes:352 150 164 167 225 159 142 182 229 163 188 197 189 235 161 195 177 166 195 160 154 130 189 188 225 This is one of those rare
54. We would like to obtain a 95% con dence interval for the population median temperature using the data in Exercise 53.a. Obtain a bootstrap sample of 1000 medians.b. Use the standard deviation for part (a) to get a 95% con dence interval for the population median.c. Investigate the distribution
53. Nine Australian soldiers were subjected to extreme conditions, which involved a 100-minutewalk with a 25-lb pack when the temperature was 40C (104F).One of them overheated (above 39C) and was removed from the study. Here are the rectal Celsius temperatures of the other eight at the end of
52. We would like to obtain a 95% con dence interval for the population median weight gain using the data in Exercise 51.a. Obtain a bootstrap sample of 1000 medians.b. Use the standard deviation for part (a) to get a 95% con dence interval for the population median.c. Investigate the distribution
51. Here are 68 weight gains in pounds for pregnant women from conception to delivery ( Classifying Data Displays with anAssessment of DisplaysFound in Popular Software, Teaching Statist., Autumn 2002: 96—101).25 14 20 38 21 22 36 38 35 37 35 24 31 28 25 32 23 30 39 26 38 20 21 11 35 42 31 25 59
50. We would like to obtain a 95% con dence interval for the population median of the study hours data in Exercise 49.a. Obtain a bootstrap sample of 1000 medians.b. Use the standard deviation for part (a) to get a95%con dence interval for the population median.c. Investigate the distribution of
49. In a survey, students gave their study time per week(hr), and here are the 22 values:15.0 10.0 10.0 15.0 25.0 7.0 3.0 8.0 10.0 10.0 11.0 7.0 5.0 15.0 7.5 7.5 12.0 7.0 10.5 6.0 10.0 7.5 We would like to get a 95% con dence interval for the population mean.a. Compute the t-based con dence
48. Refer to the baseball game times in Exercise 41.Calculate an upper con dence bound with con -dence level 95% for the population standard deviation of game time. Interpret your interval. Explore the issue of normality for the data and explain how this is relevant to your interval.
47. Here are the names of 12 orchestra conductors and their performance times in minutes for Beethoven s Ninth Symphony:Bernstein 71.03 Furtw ngler 74.38 Leinsdorf 65.78 Ormandy 64.72 Solti 74.70 Szell 66.22 Bohm 72.68 Karajan 66.90 Mazur 69.45 Rattle 69.93 Steinberg 68.62 Tennstedt 68.40a. Check
46. Exercise 34 gave a random sample of 20 ACT scores from students taking college freshman calculus.x2 x .995,25 2.99,25 x2 x .005,25 2.01,25 x2 x .1,25 2.1,15 Calculate a 99% CI for the standard deviation of the population distribution. Is this interval valid whatever the nature of the
45. Determine the following:a. The 95th percentile of the chi-squared distribution with n 10b. The 5th percentile of the chi-squared distribution with n 10c. P(10.98 x2 36.78), where x2 is a chisquared rv with n 22d. P(x2 14.611 or x2 37.652), where x2 is a chi-squared rv with n 25
44. Determine the values of the following quantities:a. b.c. d.e. f.
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, .20, and .15, respectively.What is the con dence level for each of the following three con dence
41. Here are the lengths (in minutes) of the 63 nineinning games from the rst week of the 2001 major league baseball season:194 160 176 203 187 163 162 183 152 177 177 151 173 188 179 194 149 165 186 187 187 177 187 186 187 173 136 150 173 173 136 153 152 149 152 180 186 166 174 176 198 193 218 173
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 sample mean warpage of .0635 and a sample standard deviation of .0065.a. Calculate a prediction for

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