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introduction to probability statistics
Introduction To Probability And Statistics For Engineers And Scientists 3rd Edition Sheldon M. Ross - Solutions
16. Surface defects have been counted on 25 rectangular steel plates, and the data are shown below. Set up a control chart. Does the process producing the plates appear to be in statistical control Plate Numbers Number of Defects Number of Plate Numbers Defects 2 14 10 11 x 33125025118
17. The following data represent 25 successive subgroup averages and moving-averages of span size 5 of these subgroup averages. The data are generated by a process that, when in control, produces normally distributed items having mean 30 and variance 40. The subgroups are of size 4. Would you judge
18. The data shown below give subgroup averages and moving averages of the values from Problem 17. The span of the moving averages is k = 8. When in control the subgroup averages are normally distributed with mean 50 and variance 5. What can you conclude? X 50.79806 46.21413 M 50.79806 48.50609
A manufacturer produces steel shafts having diameters that should be normally distributed with mean 3 mm and standard deviation .1 mm. Successive samples of four shafts have yielded the following sample averages in millimeters.What conclusion should be drawn? Sample X Sample 1 3.01 6 3.02 2 2.97 7
7. The following are X and S values for 20 subgroups of size 5.(a) Determine trial control limits for an X -control chart.(b) Determine trial control limits for an S-control chart.(c) Does it appear that the process was in control throughout?(d) If your answer in part (c) is no, suggest values for
Let us reconsider Example 13.2a under the new supposition that the process is just beginning and so μ and σ are unknown. Also suppose that the sample standard deviations were as follows:Since all the X i fall within these limits, we suppose that the process is in control with μ = 3.067 and σ =
The following are the X and S values for 20 subgroups of size 5 for a recently started process. Subgroup X S Subgroup X S Subgroup X S Subgroup X S 1 35.1 4.2 6 36.4 4.5 11 38.1 4.2 16 41.3 8.2 2 33.2 4.4 7 35.9 3.4 12 37.6 3.9 17 35.7 8.1 3 31.7 2.5 8 38.4 5.1 13 38.8 3.2 18 36.3 4.2 4 35.4 3.2 9
Successive samples of 50 screws are drawn from the hourly production of an automatic screw machine, with each screw being rated as either acceptable or defective.This is done for 20 such samples with the following data resulting.We can compute the trial control limits as follows: Subgroup
The following data represent the number of defects discovered at a factory on successive units of 10 cars each.Does it appear that the production process was in control throughout? Cars Defects Cars Defects Cars Defects Cars Defects 1 141 2 162 3 150 4 111 92 10 67890 74 11 63 16 68 85 12 74 17 95
Consider the data of Example 13.6a but now use an exponentially weighted moving-average control chart with α = 2/9. This gives rise to the following data set. t X W t Xt W 1 9.617728 9.915051 14 9.220009 10.84522 2 10.25437 9.990456 15 11.25206 10.93563 3 9.867195 9.963064 16 10.48662 10.83585 4
4. Samples of size 5 are taken at regular intervals from a production process, and the values of the sample averages and sample standard deviations are calculated.Suppose that the sum of the X and S values for the first 25 samples are given by(a) Assuming control, determine the control limits for
3. If Y has a chi-square distribution with n − 1 degrees of freedom, show thatNow make the transformation x = y/2. ) EY]=2 (n/2) T[(n - 1)/2] (Hint: Write E[] = (3) dy 0 00 = 0 y- e-y/2y(n-1)/2-1 dy 2(n-1)/2 (n-1) 2 e-y/2yn/2-1 dy 0 (n-1) 2(n-1)/2 2 = So
1. Assume that items produced are supposed to be normally distributed with mean 35 and standard deviation 3. To monitor this process, subgroups of size 5 are sampled. If the following represents the averages of the first 20 subgroups, does it appear that the process was in control? Subgroup No. X
72. A researcher wants to analyze the average yearly increase in a stock over a 20 year period. To do so, she plans to randomly choose 100 stocks from the listing of current stocks, discarding any that were not in existence 20 years ago. She will then compare the current price of each stock with
71. A scientist looking into the effect of smoking on heart disease has chosen a large random sample of smokers and of nonsmokers. She plans to study these two groups for 5 years to see if the number of heart attacks among the members of the smokers’ group is significantly greater than the number
70. For the following data, sample 1 is from a Poisson distribution with mean λ1 and sample 2 is from a Poisson distribution with mean λ2. Test the hypothesis thatλ1 = λ2. Sample 1 24, 32, 29, 33, 40, 28, 34, 36 Sample 2 42, 36, 41
69. The following table gives the number of fatal accidents of U.S. commercial airline carriers in the 16 years from 1980 to 1995. Do these data disprove, at the 5 percent level of significance, the hypothesis that the mean number of accidents in a year is greater than or equal to 4.5? What is the
68. Test the hypothesis, at the .05 level of significance, that the yearly number of earthquakes felt on a certain island has mean 52 if the readings for the last 8 years are 46, 62, 60, 58, 47, 50, 59, 49. Assume an underlying Poisson distribution and give an explanation to justify this assumption.
67. In the nineteen seventies, the U.S. Veterans Administration (Murphy, 1977) conducted an experiment comparing coronary artery bypass surgery with medical drug therapy as treatments for coronary artery disease. The experiment involved 596 patients, of whom 286 were randomly assigned to receive
66. The following data refer to Larry Bird’s results when shooting a pair of free throws in basketball. During two consecutive seasons in the National Basketball Association, Bird shot a pair of free throws on 338 occasions. On 251 occasions he made both shots; on 34 occasions he made the first
65. Patients suffering from cancer must often decide whether to have their tumors treated with surgery or with radiation. A factor in their decision is the 5-year survival rates for these treatments. Surprisingly, it has been found that patient’s decisions often seem to be affected by whether
64. Use the approximate test given in Problem 63 on the data of Problem 60.
63. Let X1 and X2 be binomial random variables with respective parameters n1, p1 and n2, p2. Show that when n1 and n2 are large, an approximate level α test of H0 : p1 = p2 versus H1 : p1 = p2 is as follows:Hint: (a) Argue first that when n1 and n2 are largewhere ∼˙ means “approximately has
62. Verify that Equation 8.6.5 follows from Equation 8.6.4.
61. Let X1 denote a binomial random variable with parameters (n1, p1) and X2 an independent binomial random variable with parameters (n2, p2). Develop a test, using the same approach as in the Fisher-Irwin test, of H0 : p1 ≤ p2 versus the alternative H1 : p1 > p2
60. In a recently conducted poll, 54 out of 200 people surveyed claimed to have a firearm in their homes. In a similar survey done earlier, 30 out of 150 people made that claim. Is it possible that the proportion of the population having firearms has not changed and the foregoing is due to the
59. Do Problem 58 by using a test based on the normal approximation to the binomial.
58. A standard drug is known to be effective in 75 percent of the cases in which it is used to treat a certain infection. A new drug has been developed and has been found to be effective in 42 cases out of 50. Based on this, would you accept, at the 5 percent level of significance, the hypothesis
57. An ambulance service claims that at least 45 percent of its calls involve lifethreatening emergencies. To check this claim, a random sample of 200 calls was selected from the service’s files. If 70 of these calls involved life-threatening emergencies, is the service’s claim believable at
56. According to the U.S. Bureau of the Census, 25.5 percent of the population of those age 18 or over smoked in 1990. A scientist has recently claimed that this percentage has since increased, and to prove her claim she randomly sampled 500 individuals from this population. If 138 of them were
55. Three independent news services are running a poll to determine if over half the population supports an initiative concerning limitations on driving automobiles in the downtown area. Each wants to see if the evidence indicates that over half the population is in favor. As a result, all three
54. A standard drug is known to be effective in 72 percent of the cases in which it is used to treat a certain infection. A new drug has been developed and testing has found it to be effective in 42 cases out of 50. Is this strong enough evidence to prove that the new drug is more effective than
53. In the study of Problem 52, it also resulted that 119 from the aspirin group and 98 from the control group suffered strokes. Are these numbers significant to show that taking aspirin changes the probability of having a stroke?
52. In a famous experiment to determine the efficacy of aspirin in preventing heart attacks, 22,000 healthy middle-aged men were randomly divided into two equal groups, one of which was given a daily dose of aspirin and the other a placebo that looked and tasted identical to the aspirin. The
51. The amount of surface wax on each side of waxed paper bags is believed to be normally distributed. However, there is reason to believe that there is greater variation in the amount on the inner side of the paper than on the outside.A sample of 75 observations of the amount of wax on each side
50. If X1, . . . , Xn is a sample from a normal population with variance σ2 x , and Y1, . . . , Yn is an independent sample from normal population with variance σ2 y , develop a significance level α test of Ho: 0 0
49. In Problem 31, test the hypothesis that the populations have the same variances.
48. The production of large electrical transformers and capacitators requires the use of polychlorinated biphenyls (PCBs), which are extremely hazardous when released into the environment. Two methods have been suggested to monitor the levels of PCB in fish near a large plant. It is believed that
47. A pharmaceutical house produces a certain drug item whose weight has a standard deviation of .5 milligrams. The company’s research team has proposed a new method of producing the drug. However, this entails some costs and will be adopted only if there is strong evidence that the standard
46. A gun-like apparatus has recently been designed to replace needles in administering vaccines. The apparatus can be set to inject different amounts of the serum, but because of random fluctuations the actual amount injected is normally distributed with a mean equal to the setting and with an
45. In Problem 44, explain how the test would be modified if the population mean μwere known in advance.
44. If X1, . . . , Xn is a sample from a normal population having unknown parametersμ and σ2, devise a significance level α test offor a given positive value σ2 0 . 2 versus the alternative H = 0 > 0
43. A question of medical importance is whether jogging leads to a reduction in one’s pulse rate. To test this hypothesis, 8 nonjogging volunteers agreed to begin a 1-month jogging program. After the month their pulse rates were determined and compared with their earlier values. If the data are
42. Ten pregnant women were given an injection of pitocin to induce labor. Their systolic blood pressures immediately before and after the injection were:Do the data indicate that injection of this drug changes blood pressure? Patient Before After Patient Before After 1 134 140 6 140 138 234 in 122
41. To verify the hypothesis that blood lead levels tend to be higher for children whose parents work in a factory that uses lead in the manufacturing process, researchers examined lead levels in the blood of 33 children whose parents worked in a battery manufacturing factory. (Morton, D., Saah,
40. Sample weights (in pounds) of newborn babies born in two adjacent counties in Western Pennsylvania yielded the following data.Consider a test of the hypothesis that the mean weight of newborns is the same in both counties. What is the resulting p-value? n = 53, X = 6.8, m = 44 Y = 7.2 S = 5.2,
39. The following data summary was obtained from a comparison of the lead content of human hair removed from adult individuals that had died between 1880 and 1920 with the lead content of present-day adults. The data are in units of micrograms, equal to one-millionth of a gram.(a) Do the above data
38. To learn about the feeding habits of bats, 22 bats were tagged and tracked by radio. Of these 22 bats, 12 were female and 10 were male. The distances flown(in meters) between feedings were noted for each of the 22 bats, and the following summary statistics were obtained.Test the hypothesis that
37. A study was instituted to learn how the diets of women changed during the winter and the summer. A random group of 12 women were observed during the month of July and the percentage of each woman’s calories that came from fat was determined.Similar observations were made on a different
36. In a certain experimental laboratory, a method A for producing gasoline from crude oil is being investigated. Before completing experimentation, a new method B is proposed. All other things being equal, it was decided to abandon A in favor of B only if the average yield of the latter was
35. A professor claims that the average starting salary of industrial engineering graduating seniors is greater than that of civil engineering graduates. To study this claim, samples of 16 industrial engineers and 16 civil engineers, all of whom graduated in 1993, were chosen and sample members
34. In a 1943 experiment (Whitlock and Bliss, “A Bioassay Technique for Antihelminthics,”Journal of Parasitology, 29, pp. 48–58) 10 albino rats were used to study the effectiveness of carbon tetrachloride as a treatment for worms. Each rat received an injection of worm larvae. After 8 days,
33. Twenty-five men between the ages of 25 and 30, who were participating in a wellknown heart study carried out in Framingham, Massachusetts, were randomly selected. Of these, 11 were smokers and 14 were not. The following data refer to readings of their systolic blood pressure.Use these data to
32. It is argued that the resistance of wire A is greater than the resistance of wire B.You make tests on each wire with the following results.What conclusion can you draw at the 10 percent significance level? Explain what assumptions you are making.In Problems 33 through 40, assume that the
31. The viscosity of two different brands of car oil is measured and the following data resulted:Test the hypothesis that the mean viscosity of the two brands is equal, assuming that the populations have normal distributions with equal variances. Brand 1 10.62, 10.58, 10.33, 10.72, 10.44, 10.74
30. The data below give the lifetimes in hundreds of hours of samples of two types of electronic tubes. Past lifetime data of such tubes have shown that they can often be modeled as arising from a lognormal distribution. That is, the logarithms of the data are normally distributed. Assuming that
29. The following are the values of independent samples from two different populationsLet μ1 and μ2 be the respective means of the two populations. Find the p-value of the test of the null hypothesisversus the alternativewhen the population standard deviations are σ1 = 10 and (a) σ2 = 5; (b)
28. A method for measuring the pH level of a solution yields a measurement value that is normally distributed with a mean equal to the actual pH of the solution and with a standard deviation equal to .05. An environmental pollution scientist claims that two different solutions come from the same
27. A sample of 10 fish were caught at lake A and their PCB concentrations were measured using a certain technique. The resulting data in parts per million were Lake A: 11.5, 10.8, 11.6, 9.4, 12.4, 11.4, 12.2, 11, 10.6, 10.8 In addition, a sample of 8 fish were caught at lake B and their levels of
26. A manufacturer of capacitors claims that the breakdown voltage of these capacitors has a mean value of at least 100 V. A test of 12 of these capacitors yielded the following breakdown voltages:96, 98, 105, 92, 111, 114, 99, 103, 95, 101, 106, 97 Do these results prove the manufacturer’s
25. It is claimed that a certain type of bipolar transistor has a mean value of current gain that is at least 210. A sample of these transistors is tested. If the sample mean value of current gain is 200 with a sample standard deviation of 35, would the claim be rejected at the 5 percent level of
24. A company supplies plastic sheets for industrial use. A new type of plastic has been produced and the company would like to claim that the average stress resistance of this new product is at least 30.0, where stress resistance is measured in pounds per square inch (psi) necessary to crack the
23. An oil company claims that the sulfur content of its diesel fuel is at most .15 percent. To check this claim, the sulfur contents of 40 randomly chosen samples were determined; the resulting sample mean and sample standard deviation were.162 and .040. Using the 5 percent level of significance,
22. Use the data of Example 2.3i of Chapter 2 to test the null hypothesis that the average noise level directly outside of Grand Central Station is less than or equal to 80 decibels.
21. A producer specifies that the mean lifetime of a certain type of battery is at least 240 hours. A sample of 18 such batteries yielded the following data.237 242 232 242 248 230 244 243 254 262 234 220 225 236 232 218 228 240 Assuming that the life of the batteries is approximately normally
20. A car is advertised as having a gas mileage rating of at least 30 miles/gallon in highway driving. If the miles per gallon obtained in 10 independent experiments are 26, 24, 20, 25, 27, 25, 28, 30, 26, 33, should you believe the advertisement?What assumptions are you making?
19. Use the results of a Sunday’s worth of major league baseball scores to test the hypothesis that the average number of runs scored by winning teams is at least 5.6.Use the 5 percent level of significance.
18. Use the results of a Sunday’s worth of NFL professional football games to test the hypothesis that the average number of points scored by winning teams is less than or equal to 28. Use the 5 percent level of significance.
17. A medical scientist believes that the average basal temperature of (outwardly)healthy individuals has increased over time and is now greater than 98.6 degrees Fahrenheit (37 degrees Celsius). To prove this, she has randomly selected 100 healthy individuals. If their mean temperature is 98.74
16. Suppose that team A and team B are to play a National Football League game and that team A is favored by f points. Let S(A) and S(B) denote the scores of teams A and B, and let X = S(A) − S(B) − f . That is, X is the amount by which team A beats the point spread. It has been claimed that
15. The mean response time of a species of pigs to a stimulus is .8 seconds. Twentyeight pigs were given 2 oz of alcohol and then tested. If their average response time was 1.0 seconds with a standard deviation of .3 seconds, can we conclude that alcohol affects the mean response time? Use the 5
14. Twenty years ago, entering male high school students of Central High could do an average of 24 pushups in 60 seconds. To see whether this remains true today, a random sample of 36 freshmen was chosen. If their average was 22.5 with a sample standard deviation of 3.1, can we conclude that the
13. There is some variability in the amount of phenobarbitol in each capsule sold by a manufacturer. However, the manufacturer claims that the mean value is 20.0 mg. To test this, a sample of 25 pills yielded a sample mean of 19.7 with a sample standard deviation of 1.3. What inference would you
12. An advertisement for a new toothpaste claims that it reduces cavities of children in their cavity-prone years. Cavities per year for this age group are normal with mean 3 and standard deviation 1. A study of 2,500 children who used this toothpaste found an average of 2.95 cavities per child.
11. Consider a test of H0 : μ ≤ 100 versus H1 : μ > 100. Suppose that a sample of size 20 has a sample mean of X = 105. Determine the p-value of this outcome if the population standard deviation is known to equal(a) 5; (b) 10; (c) 15.
10. The weights of salmon grown at a commercial hatchery are normally distributed with a standard deviation of 1.2 pounds. The hatchery claims that the mean weight of this year’s crop is at least 7.6 pounds. Suppose a random sample of 16 fish yielded an average weight of 7.2 pounds. Is this
23. Repeat Problem 18, this time using a cumulative sum control chart with d = 1 and B = 2.49.
22. Repeat Problem 17, this time using a cumulative sum control chart with(a) d = .25, B = 8;(b) d = .5, B = 4.77.
21. Explain why a moving-average control chart with span size k must use different control limits for the first k−1 moving averages, whereas an exponentially weighted moving-average control chart can use the same control limits throughout. [Hint:Argue that Var(Mt ) decreases in t , whereas Var(Wt
20. Analyze the data of Problem 18 with an exponential weighted moving-average control chart having α = 29.
19. Redo Problem 17 by employing an exponential weighted moving average control chart with α = 13.
15. The following data represent the number of defective chips produced on the last 15 days: 121, 133, 98, 85, 101, 78, 66, 82, 90, 78, 85, 81, 100, 75, 89. Would you conclude that the process has been in control throughout these 15 days? What control limits would you advise using for future
14. Suppose that when a process is in control each item will be defective with probability.04. Suppose that your control chart calls for taking daily samples of size 500.What is the probability that, if the probability of a defective item should suddenly shift to .08, your control chart would
9. Control charts for X and S are maintained on resistors (in ohms). The subgroup size is 4. The values of X and S are computed for each subgroup. After 20 subgroups, X i = 8,620 andSi = 450.(a) Compute the values of the limits for the X and S charts.(b) Estimate the value of σ on the assumption
8. Control charts for X and S are maintained on the shear strength of spot welds.After 30 subgroups of size 4,X i = 12,660 andSi = 500. Assume that the process is in control.(a) What are the X -control limits?(b) What are the S-control limits?(c) Estimate the standard deviation for the
6. In Problem 4, determine the control limits for an S-control chart.
5. Determine the revised X - and S-control limits for the data in Example 13.3a.
2. Suppose that a process is in control with μ = 14 and σ = 2. An X -control chart based on subgroups of size 5 is employed. If a shift in the mean of 2.2 units occurs, what is the probability that the next subgroup average will fall outside the control limits? On average, how many subgroups will
Suppose that the mean and standard deviation of a subgroup average areμ = 30 and σ/√n = 8, respectively, and consider the cumulative sum control chart with d = .5, B = 5. If the first eight subgroup averages are 29, 33, 35, 42, 36, 44, 43, 45 then the successive values of Yj = X j − 30 − 4
A repair shop will send a worker to a caller’s home to repair electronic equipment. Upon receiving a request, it dispatches a worker who is instructed to call in when the job is completed. Historical data indicate that the time from when the server is dispatched until he or she calls is a normal
When a certain manufacturing process is in control, it produces items whose values are normally distributed with mean 10 and standard deviation 2. The following simulated data represent the values of 25 subgroup averages of size 5 from a normal population with mean 11 and standard deviation 2. That
9. A British pharmaceutical company, Glaxo Holdings, has recently developed a new drug for migraine headaches. Among the claims Glaxo made for its drug, called somatriptan, was that the mean time it takes for it to enter the bloodstream is less than 10 minutes. To convince the Food and Drug
8. Verify that the approximation in Equation 8.3.7 remains valid even whenμ1 < μ0.
7. Suppose in Problem 4 that we wished to design a test so that if the pH were really equal to 8.20, then this conclusion will be reached with probability equal to .95.On the other hand, if the pH differs from 8.20 by .03 (in either direction), we want the probability of picking up such a
6. It is known that the average height of a man residing in the United States is 5 feet 10 inches and the standard deviation is 3 inches. To test the hypothesis that men in your city are “average,” a sample of 20 men have been chosen. The heights of the men in the sample follow:What do you
5. The mean breaking strength of a certain type of fiber is required to be at least 200 psi. Past experience indicates that the standard deviation of breaking strength is 5 psi. If a sample of 8 pieces of fiber yielded breakage at the following pressures,would you conclude, at the 5 percent level
4. In a certain chemical process, it is very important that a particular solution that is to be used as a reactant have a pH of exactly 8.20. A method for determining pH that is available for solutions of this type is known to give measurements that are normally distributed with a mean equal to the
3. A population distribution is known to have standard deviation 20. Determine the p-value of a test of the hypothesis that the population mean is equal to 50, if the average of a sample of 64 observations is(a) 52.5; (b) 55.0; (c) 57.5.
2. A colony of laboratory mice consists of several thousand mice. The average weight of all the mice is 32 grams with a standard deviation of 4 grams. A laboratory assistant was asked by a scientist to select 25 mice for an experiment.However, before performing the experiment the scientist decided
1. Consider a trial in which a jury must decide between the hypothesis that the defendant is guilty and the hypothesis that he or she is innocent.(a) In the framework of hypothesis testing and the U.S. legal system, which of the hypotheses should be the null hypothesis?(b) What do you think would
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