Do by using a test based on the normal approximation to

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 be an appropriate significance level in this situation?

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 to weigh the mice as an indicator of whether the assistants selection constituted a random sample or whether it was made with some unconscious bias (perhaps the mice selected were the ones that were slowest in avoiding the assistant, which might indicate some inferiority about this group). If the sample mean of the 25 mice was 30.4, would this be significant evidence, at the 5 percent level of significance, against the hypothesis that the selection constituted a random sample?

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.

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 actual pH and with a standard deviation of .02. Suppose 10 independent measurements yielded the following pH values: 8.18 8.17 8.16 8.15 8.17 8.21 8.22 8.16 8.19 8.18 (a) What conclusion can be drawn at the = .10 level of significance? (b) What about at the = .05 level of significance?

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, 210 198 195 202 197.4 196 199 195.5 would you conclude, at the 5 percent level of significance, that the fiber is unacceptable? What about at the 10 percent level of significance?

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 conclude? Explain what assumptions you are making.

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 difference to exceed .95. (a) What test procedure should be used? (b) What is the required sample size? (c) If x = 8.31, what is your conclusion? (d) If the actual pH is 8.32, what is the probability of concluding that the pH is not 8.20, using the foregoing procedure?

Verify that the approximation in Equation 8.3.7 remains valid even when 1 < 0.

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 Administration of the validity of this claim, Glaxo conducted an experiment on a randomly chosen set of migraine sufferers. To prove its claim, what should they have taken as the null and what as the alternative hypothesis?

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 years crop is at least 7.6 pounds. Suppose a random sample of 16 fish yielded an average weight of 7.2 pounds. Is this strong enough evidence to reject the hatcherys claims at the (a) 5 percent level of significance; (b) 1 percent level of significance? (c) What is the p-value?

Consider a test of H0 : 100 versus H1 : > 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.

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. Assume that the standard deviation of the number of cavities of a child using this new toothpaste remains equal to 1. (a) Are these data strong enough, at the 5 percent level of significance, to establish the claim of the toothpaste advertisement? (b) Do the data convince you to switch to this new toothpaste?

There is some variability in the amount of phenobarbital 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 draw from these data? In particular, are the data strong enough evidence to discredit the claim of the manufacturer? Use the 5 percent level of significance.

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 mean is no longer equal to 24? Use the 5 percent level of significance.

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 themean response time? Use the 5 percent level of significance.

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 the distribution of X is normal with mean 0 and standard deviation 14. Use data from randomly chosen football games to test this hypothesis.

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 with a sample standard deviation of 1.1 degrees, does this prove her claim at the 5 percent level? What about at the 1 percent level?

Use the results of a Sundays 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.

Use the results of a Sundays 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.

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?

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 distributed, do the data indicate that the specifications are not being met?

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.

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, can we conclude that the companys claims are invalid?

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 sheet. The following random sample was drawn off the production line. Based on this sample, would the claim clearly be unjustified? 30.1 32.7 22.5 27.5 27.7 29.8 28.9 31.4 31.2 24.3 26.4 22.8 29.1 33.4 32.5 21.7 Assume normality and use the 5 percent level of significance.

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 significance if (a) the sample size is 25; (b) the sample size is 64?

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 manufacturers claim? Do they disprove them?

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 PCB were measured by a different technique than that used at lake A. The resultant data were Lake B: 11.8, 12.6, 12.2, 12.5, 11.7, 12.1, 10.4, 12.6 If it is known that the measuring technique used at lake A has a variance of .09 whereas the one used at lake B has a variance of .16, could you reject (at the 5 percent level of significance) a claim that the two lakes are equally contaminated?

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 source. If this were so, then the pH level of the solutions would be equal. To test the plausibility of this claim, 10 independent measurements were made of the pH level for both solutions, with the following data resulting. (a) Do the data disprove the scientists claim? Use the 5 percent level of significance. (b) What is the p-value?

The following are the values of independent samples from two different populations. Sample 1 122, 114, 130, 165, 144, 133, 139, 142, 150 Sample 2 108, 125, 122, 140, 132, 120, 137, 128, 138 Let 1 and 2 be the respective means of the two populations. Find the p-value of the test of the null hypothesis H0 : 1 2 versus the alternative H1 : 1 > 2 when the population standard deviations are 1 = 10 and (a) 2 = 5; (b) 2 = 10; (c) 2 = 20.

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 variance of the logarithms is equal for the two populations, test, at the 5 percent level of significance, the hypothesis that the two population distributions are identical. Type 1 32, 84, 37, 42, 78, 62, 59, 74 Type 2 39, 111, 55, 106, 90, 87, 85

The viscosity of two different brands of car oil is measured and the following data resulted: Brand 1 10.62, 10.58, 10.33, 10.72, 10.44, 10.74 Brand 2 10.50, 10.52, 10.58, 10.62, 10.55, 10.51, 10.53 Test the hypothesis that the mean viscosity of the two brands is equal, assuming that the populations have normal distributions with equal variances.

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. Wire A Wire B .140 ohm .135 ohm .138 .140 .143 .136 .142 .142 .144 .138 .137 .140 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 population distributions are normal and have equal variances.

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 test the hypothesis that the mean blood pressures of smokers and nonsmokers are the same.

In a 1943 experiment (Whitlock and Bliss, A Bioassay Technique for Antihelminthics, Journal of Parasitology, 29, pp. 4858) 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, the rats were randomly divided into two groups of 5 each; each rat in the first group received a dose of .032 cc of carbon tetrachloride, whereas the dosage for each rat in the second group was .063 cc. Two days later the rats were killed, and the number of adult worms in each rat was determined. The numbers detected in the group receiving the .032 dosage were 421, 462, 400, 378, 413 whereas they were 207, 17, 412, 74, 116 for those receiving the .063 dosage. Do the data prove that the larger dosage is more effective than the smaller?

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 2006, were chosen and sample members were queried about their starting salaries. If the industrial engineers had a sample mean salary of $72,700 and a sample standard deviation of $2,400, and the civil engineers had a sample mean salary of $71,400 and a sample standard deviation of $2,200, has the professors claim been verified? Find the appropriate p-value.

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 clearly greater. The yield of both processes is assumed to be normally distributed. However, there has been insufficient time to ascertain their true standard deviations, although there appears to be no reason why they cannot be assumed equal. Cost considerations impose size limits on the size of samples that can be obtained. If a 1 percent significance level is all that is allowed, what would be your recommendation based on the following random samples? The numbers represent percent yield of crude oil. A 23.2, 26.6, 24.4, 23.5, 22.6, 25.7, 25.5 B 25.7, 27.7, 26.2, 27.9, 25.0, 21.4, 26.1

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 womans calories that came from fat was determined. Similar observations were made on a different randomly selected group of size 12 during the month of January. The results were as follows: July 32.2, 27.4, 28.6, 32.4, 40.5, 26.2, 29.4, 25.8, 36.6, 30.3, 28.5, 32.0 January 30.5, 28.4, 40.2, 37.6, 36.5, 38.8, 34.7, 29.5, 29.7, 37.2, 41.5, 37.0 Test the hypothesis that the mean fat percentage intake is the same for both months. Use the (a) 5 percent level of significance and (b) 1 percent level of significance.

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. Female Bats Male Bats n = 12 m = 10 X = 180 Y = 136 Sx = 92 Sy = 86 Test the hypothesis that the mean distance flown between feedings is the same for the populations of both male and of female bats. Use the 5 percent level of significance.

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. 18801920 Today Sample size: 30 100 Sample mean: 48.5 26.6 Sample standard deviation: 14.5 12.3 (a) Do the above data establish, at the 1 percent level of significance, that the mean lead content of human hair is less today than it was in the years between 1880 and 1920? Clearly state what the null and alternative hypotheses are. (b) What is the p-value for the hypothesis test in part (a)?

Sample weights (in pounds) of newborn babies born in two adjacent counties in western Pennsylvania yielded the following data. n = 53, m = 44 X = 6.8, Y = 7.2 S2 = 5.2, S2 = 4.9 Consider a test of the hypothesis that the mean weight of newborns is the same in both counties. What is the resulting p-value?

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, A., Silberg, S., Owens, W., Roberts, M., and Saah, M., Lead Absorption in Children of Employees in a Lead-Related Industry, American Journal of Epidemiology, 115, 549555, 1982). Each of these children was then matched by another child who was of similar age, lived in a similar neighborhood, had a similar exposure to traffic, but whose parent did not work with lead. The blood levels of the 33 cases (sample 1) as well as those of the 33 controls (sample 2) were then used to test the hypothesis that the average blood levels of these groups are the same. If the resulting sample means and sample standard deviations were .x 1 = .015, s1 = .004, .x2 = .006, s2 = .006 find the resulting p-value. Assume a common variance.

Ten pregnant women were given an injection of pitocin to induce labor. Their systolic blood pressures immediately before and after the injection were: Patient Before After Patient Before After 1 134 140 6 140 138 2 122 130 7 118 124 3 132 135 8 127 126 4 130 126 9 125 132 5 128 134 10 142 144 Do the data indicate that injection of this drug changes blood pressure?

A question of medical importance is whether jogging leads to a reduction in ones 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 as follows, can we conclude that jogging has had an effect on the pulse rates? Subject 1 2 3 4 5 6 7 8 Pulse Rate Before 74 86 98 102 78 84 79 70 Pulse Rate After 70 85 90 110 71 80 69 74

If X1, . . . , Xn is a sample from a normal population having unknown parameters and 2, devise a significance level test of H0 = 2 2 0 versus the alternative H1 = 2 > 2 0 for a given positive value 2 0 .

In Problem 44, explain how the test would be modified if the population mean were known in advance.

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 unknown variance 2. It has been decided that the apparatus would be too dangerous to use if exceeds .10. If a random sample of 50 injections resulted in a sample standard deviation of .08, should use of the new apparatus be discontinued? Suppose the level of significance is = .10. Comment on the appropriate choice of a significance level for this problem, as well as the appropriate choice of the null hypothesis.

A pharmaceutical house produces a certain drug item whose weight has a standard deviation of .5 milligrams. The companys 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 deviation of the weight of the items will drop to below .4 milligrams. If a sample of 10 items is produced and has the following weights, should the new method be adopted? 5.728 5.731 5.722 5.719 5.727 5.724 5.718 5.726 5.723 5.722

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 each method will result in a normal random variable that depends on the method. Test the hypothesis at the = .10 level of significance that both methods have the same variance, if a given fish is checked 8 times by each method with the following data (in parts per million) recorded. Method 1 6.2, 5.8, 5.7, 6.3, 5.9, 6.1, 6.2, 5.7 Method 2 6.3, 5.7, 5.9, 6.4, 5.8, 6.2, 6.3, 5.5

In Problem 31, test the hypothesis that the populations have the same variances.

If X1, . . . , Xn is a sample from a normal population with variance 2 x , and Y1, . . . , Yn is an independent sample from normal population with variance 2 y , develop a significance level test of H0 : 2 x < 2 y versus H1 : 2 x > 2 y

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 of these bags is obtained and the following data recorded. Wax in Pounds per Unit Area of Sample Outside Surface Inside Surface x = .948 y = .652 _x2 i = 91 _y2 i = 82 Conduct a test to determine whether or not the variability of the amount of wax on the inner surface is greater than the variability of the amount on the outer surface ( = .05).

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 experiment was halted at a time when 104 men in the aspirin group and 189 in the control group had had heart attacks. Use these data to test the hypothesis that the taking of aspirin does not change the probability of having a heart attack.

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?

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 the old one? Find the relevant p-value.

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 services will be testing H0 : p .5 versus H1 : p > .5 where p is the proportion of the population in favor of the initiative. (a) Suppose the first news organization samples 100 people, of which 56 are in favor of the initiative. Is this strong enough evidence, at the 5 percent level of significance, to reject the null hypothesis and so establish that over half the population favors the initiative? (b) Suppose the second news organization samples 120 people, of which 68 are in favor of the initiative. Is this strong enough evidence, at the 5 percent level of significance, to reject the null hypothesis? (c) Suppose the third news organization samples 110 people, of which 62 are in favor of the initiative. Is this strong enough evidence, at the 5 percent level of significance, to reject the null hypothesis? (d) Suppose the news organizations combine their samples, to come up with a sample of 330 people, of which 186 support the initiative. Is this strong enough evidence, at the 5 percent level of significance, to reject the null hypothesis?

It has been a long held belief that the proportion of California births of African America mothers that result in twins is about 1.32 percent. (The twinning rate appears to be influenced by the ethnicity of the mother; claims are that it is 1.05 for Caucasian Americans, and 0.72 percent for Asian Americans.) A public health scientist believes that this number is no longer correct and has decided to test the null hypothesis that the proportion is 1.32 percent by gathering data on the next 1, 000 recorded birthing events, where twin birds are regarded as a single birthing event, in California. (a) What is the minimal number of twin births that would have to be observed in order to reject the null hypothesis at the 5 percent level of significance? (b) What is the probability the null hypothesis will be rejected if the actual twinning rate is 1.80?

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 services files. If 70 of these calls involved life-threatening emergencies, is the services claim believable at the (a) 5 percent level of significance; (b) 1 percent level of significance?

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 that the two drugs are of equal effectiveness? What is the p-value?

Do Problem 58 by using a test based on the normal approximation to the binomial.

In a study of the effect of two chemotherapy treatments on the survival of patients with multiple myeloma, each of 156 patients was equally likely to be given either one of the two treatments. As reported by Lipsitz, Dear, Laird, and Molenberghs in a 1998 paper in Biometrics, the result of this was that 39 of the 72 patients given the first treatment and 44 of the 84 patients given the second treatment survived for over 5 years. (a) Use these data to test the null hypothesis that the two treatments are equally effective. (b) Is the fact that 72 of the patients received one of the treatments while 84 received the other consistent with the claim that the determination of the treatment to be given to each patient was made in a totally random fashion?

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

Verify that Equation 8.6.5 follows from Equation 8.6.4.

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 test of H0 : p1 = p2 versus H1 : p1 _= p2 is as follows: reject H0 if |X1/n1 X2/n2| _X1 + X2 n1 + n2 _ 1 X1 + X2 n1 + n2 _ 1 n1 + 1 n2 > z/2 Hint: (a) Argue first that when n1 and n2 are large X1 n1 X2 n2 ( p1 p2) _p1(1 p1) n1 + p2(1 p2) n2 N(0, 1) where means approximately has the distribution. (b) Now argue that when H0 is true and so p1 = p2, their common value can be best estimated by (X1 + X2)/(n1 + n2).

Use the approximate test given in Problem 63 on the data of Problem 60.

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 patients decisions often seem to be affected by whether they are told the 5-year survival rates or the 5-year death rates (even though the information content is identical). For instance, in an experiment a group of 200 male prostate cancer patients were randomly divided into two groups of size 100 each. Each member of the first group was told that the 5-year survival rate for those electing surgery was 77 percent, whereas each member of the second group was told that the 5-year death rate for those electing surgery was 23 percent. Both groups were given the same information about radiation therapy. If it resulted that 24 members of the first group and 12 of the second group elected to have surgery, what conclusions would you draw?

The following data refer to Larry Birds 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 shot but missed the second one; on 48 occasions he missed the first shot but made the second one; on 5 occasions he missed both shots. (a) Use these data to test the hypothesis that Birds probability of making the first shot is equal to his probability of making the second shot. (b) Use these data to test the hypothesis that Birds probability of making the second shot is the same regardless of whether he made or missed the first one.

In the 1970s, 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 surgery, with the remaining 310 assigned to drug therapy. A total of 252 of those receiving surgery, and a total of 270 of those receiving drug therapy were still alive 3 years after treatment. Use these data to test the hypothesis that the survival probabilities are equal.

Test the hypothesis, at the 5 percent level of significance, that the yearly number of earthquakes felt on a certain island has mean 52 if the readings for the past 8 years are 46, 62, 60, 58, 47, 50, 59, 49. Assume an underlying Poisson distribution and give an explanation to justify this assumption.

In 1995, the Fermi Laboratory announced the discovery of the top quark, the last of six quarks predicted by the standard model of physics. The evidence for its existence was statistical in nature and involved signals created when antiprotons and protons were forced to collide. In a Physical Review Letters paper documenting the evidence, Abe, Akimoto, and Akopian (known in physics circle as the three As) based their conclusion on a theoretical analysis that indicated that the number of decay events in a certain time interval would have a Poisson distribution with a mean equal to 6.7 if a top quark did not exist and with a larger mean if it did exist. In a careful analysis of the data the three As showed that the actual count was 27. Is this strong enough evidence to prove the hypothesis that the mean of the Poisson distribution was greater than 6.7?

For the following data, sample 1 is from a Poisson distribution with mean 1 and sample 2 is from a Poisson distribution with mean 2. Test the hypothesis that 1 = 2. Sample 1 24, 32, 29, 33, 40, 28, 34, 36 Sample 2 42, 36, 41

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 among the nonsmokers. Such a result, the scientist feels, should be strong evidence of an association between smoking and heart attacks. Given that (a) older people are at greater risk of heart disease than are younger people; and (b) as a group, smokers tend to be somewhat older than nonsmokers, would the scientist be justified in her conclusion? Explain how the experimental design can be improved so that meaningful conclusions can be drawn.

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 its price 20 years ago to determine its percentage increase. Do you think this is a valid method to study the average increase in the price of a stock?