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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

10. The following data refer to the amounts by which the diameters of 14-inch ball bearings differ from 14-inch in units of .001 inches. The subgroup size is n = 5.(a) Set up trial control limits for
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
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
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
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
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
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 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
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 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
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
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
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
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
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
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
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,
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
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,
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
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
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.
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
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
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
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
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.
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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,
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
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
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
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)
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
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
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
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
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
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
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
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
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,
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)
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
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
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
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.
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
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
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
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
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
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
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
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

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