Solved: For each of the following rejection regions,

Problem 1E Which hypothesis, the null or the alternative, is the statusquo hypothesis? Which is the research hypothesis?

Which element of a test of hypothesis is used to decide whether to reject the null hypothesis in favor of the alternative hypothesis?

Problem 3E What is the level of significance of a test of hypothesis?

Problem 145SE The Hot Tamale caper. “Hot Tamales” are chewy, cinnamon flavored candies. A bulk vending machine is known to dispense, on average, 15 Hot Tamales per bag. Chance (Fall 2000) published an article on a classroom project in which students were required to purchase bags of Hot Tamales from the machine and count the number of candies per bag. One student group claimed it purchased five bags that had the following candy counts: 25, 23, 21, 21, and 20. There was some question as to whether the students had fabricated the data. Use a hypothesis test to gain insight into whether or not the data collected by the students were fabricated. Use a level of significance that gives the benefit of the doubt to the students.

What is the difference between Type I and Type II errors in hypothesis testing? How do \(\alpha\) and \(\beta\) relate to Type I and Type II errors?

Problem 5E List the four possible results of the combinations of decisions and true states of nature for a test of hypothesis.

Problem 6E We reject the null hypothesis when the test statistic falls in the rejection region, but we do not accept the null hypothesis when the test statistic does not fall in the rejection region. Why?

If you test a hypothesis and reject the null hypothesis in favor of the alternative hypothesis, does your test prove that the alternative hypothesis is correct? Explain.

Problem 8E For each of the following rejection regions, sketch the sampling distribution for z and indicate the location of the rejection region. g. For each of the rejection regions specified in parts a–f, what is the probability that a Type I error will be made?

Problem 9E Effectiveness of online courses. The Sloan Survey of Online Learning, “Going the Distance: Online Education in the United States, 2011,” reported that 68% of college presidents believe that their online education courses are as good as or superior to courses that use traditional, face-to-face instruction. a. Give the null hypothesis for testing the claim made by the Sloan Survey. b. Give the rejection region for a two-tailed test conducted at

Problem 10E Play Golf America program. The Professional Golf Association (PGA) and Golf Digest have developed the Play Golf America program, in which teaching professionals at participating golf clubs provide a free 10-minute lesson to new customers. According to Golf Digest (July 2008), golf facilities that participate in the program gain, on average, $2,400 in greens, fees, lessons, or equipment expenditures. A teaching professional at a golf club believes that the average gain in greens fees, lessons, or equipment expenditures for participating golf facilities exceeds $2,400. a. In order to support the claim made by the teaching professional, what null and alternative hypotheses should you test? b. Suppose you select Interpret this value in the words of the problem. c. For specify the rejection region of a large-sample test.

Problem 11E Student loan default rate. The national student loan default rate has fluctuated over the last several years. A few years ago, the Department of Education reported the default rate (i.e., the proportion of college students who default on their loans) at .07. Set up the null and alternative hypotheses if you want to determine if the student loan default rate this year is less than .07.

Work travel policy. American Express Consulting reported in USA Today (June 15, 2001) that 80% of U.S. companies have formal, written travel and entertainment policies for their employees. Give the null hypothesis for testing the claim made by American Express Consulting.

Problem 13E Calories in school lunches. A University of Florida economist conducted a study of Virginia elementary school lunch menus. During the state-mandated testing period, school lunches averaged 863 calories (National Bureau of Economic Research, Nov. 2002). The economist claims that after the testing period ends, the average caloric content of Virginia school lunches drops significantly. Set up the null and alternative hypotheses to test the economist’s claim.

Libor interest rate. The interest rate at which London banks lend money to one another is called the London interbank offered rate, or Libor. The British Bankers Association regularly surveys international banks for the Libor rate. One recent report (Bankrate.com, Jan. 25, 2012) had the average Libor rate at 1.10% for 1-year loans—a value considered high by many Western banks. Set up the null and alternative hypotheses for testing the reported value.

FDA certification of new drugs. According to Chemical Marketing Reporter, pharmaceutical companies spend $15 billion per year on research and development of new drugs. The pharmaceutical company must subject each new drug to lengthy and involved testing before receiving the necessary permission from the Food and Drug Administration (FDA) to market the drug. The FDA’s policy is that the pharmaceutical company must provide substantial evidence that a new drug is safe prior to receiving FDA approval, so that the FDA can confidently certify the safety of the drug to potential consumers. a. If the new drug testing were to be placed in a test of hypothesis framework, would the null hypothesis be that the drug is safe or unsafe? The alternative hypothesis? b. Given the choice of null and alternative hypotheses in part a, describe Type I and Type II errors in terms of this application. Define \(\alpha\) and \(\beta\) in terms of this application. c. If the FDA wants to be very confident that the drug is safe before permitting it to be marketed, is it more important that \(\(\alpha\) and \(\beta\)\) be small? Explain.

Authorizing computer users. At high-technology industries, computer security is achieved by using a password—a collection of symbols (usually letters and numbers) that must be supplied by the user before the computer permits access to the account. The problem is that persistent hackers can create programs that enter millions of combinations of symbols into a target system until the correct password is found. The newest systems solve this problem by requiring authorized users to identify themselves by unique body characteristics. For example, a system developed by Palmguard, Inc. tests the hypothesis \(H_0\): The proposed user is authorized \(H_a\): The proposed user is unauthorized by checking characteristics of the proposed user’s palm against those stored in the authorized users’ data bank. a. Define a Type I error and Type II error for this test. Which is the more serious error? Why? b. Palmguard reports that the Type I error rate for its system is less than 1%, whereas the Type II error rate is .00025%. Interpret these error rates. c. Another successful security system, the EyeDentifyer, “spots authorized computer users by reading the one-of-a-kind patterns formed by the network of minute blood vessels across the retina at the back of the eye.” The EyeDentifyer reports Type I and II error rates of .01% (1 in 10,000) and .005% (5 in 100,000), respectively. Interpret these rates.

Problem 17E Jury trial outcomes. Sometimes, the outcome of a jury trial defies the “common sense” expectations of the general public (e.g., the O. J. Simpson verdict and the 2011 Casey Anthony verdict). Such a verdict is more acceptable if we understand that the jury trial of an accused murderer is analogous to the statistical hypothesis-testing process. The null hypothesis in a jury trial is that the accused is innocent. (The status-quo hypothesis in the U.S. system of justice is innocence, which is assumed to be true until proven beyond a reasonable doubt.) The alternative hypothesis is guilt, which is accepted only when sufficient evidence exists to establish its truth. If the vote of the jury is unanimous in favor of guilt, the null hypothesis of innocence is rejected, and the court concludes that the accused murderer is guilty. Any vote other than a unanimous one for guilt results in a “not guilty” verdict. The court never accepts the null hypothesis; that is, the court never declares the accused “innocent.” A “not guilty” verdict (as in the Casey Anthony) implies that the court could not find the defendant guilty beyond a reasonable doubt. a. Define Type I and Type II errors in a murder trial. b. Which of the two errors is the more serious? Explain. c. The court does not, in general, know the values of and but ideally, both should be small. One of these probabilities is assumed to be smaller than the other in a jury trial. Which one, and why? d. The court system relies on the belief that the value of is made very small by requiring a unanimous vote before guilt is concluded. Explain why this is so. e. For a jury prejudiced against a guilty verdict as the trial begins, will the value of increase or decrease? Explain. f. For a jury prejudiced against a guilty verdict as the trial begins, will the value of increase or decrease? Explain.

Problem 18E Intrusion detection systems. Refer to the Journal of Research of the National Institute of Standards and Technology (Nov.–Dec. 2003) study of a computer intrusion detection system (IDS), Exercise 3.69 (p. 167). Recall that an IDS is designed to provide an alarm whenever unauthorized access (e.g., an intrusion) to a computer system occurs. The probability of the system giving a false alarm (i.e., providing a warning when no intrusion occurs) is defined by the symbol , while the probability of a missed detection (i.e., no warning given when an intrusion occurs) is defined by the symbol . These symbols are used to represent Type I and Type II error rates, respectively, in a hypothesis-testing scenario. a. What is the null hypothesis, H0? b. What is the alternative hypothesis, Ha? c. According to actual data on the EMERALD system collected by the Massachusetts Institute of Technology Lincoln Laboratory, only 1 in 1,000 computer sessions with no intrusions resulted in a false alarm. For the same system, the laboratory found that only 500 of 1,000 intrusions were actually detected. Use this information to estimate the values of and .

Consider the test of \(H_0: \mu=7\). For each of the following, find the p-value of the test: a. \(H_{\mathrm{a}}: \mu>7, z=1.20\) b. \(H_{\mathrm{a}}: \mu<7, z=-1.20\) c. \(H_{\mathrm{a}}: \mu \neq 7, z=1.20\)

If a hypothesis test were conducted using \(\alpha=.05\), for which of the following p-values would the null hypothesis be rejected? a. .06 b. .10 c. .01 d. .001 e. .251 f. .042

Problem 21E For each and observed significance level (p-value) pair, indicate whether the null hypothesis would be rejected.

Problem 22E In a test of the hypothesis a sample of n = 100 observations possessed mean = 49.4 and standard deviation s = 4.1. Find and interpret the p-value for this test.

In a test of \(H_{0}: \mu=100\) against \(H_{\mathrm{a}}: \mu>100\) the sample data yielded the test statistic z = 2.17. Find and interpret the p-value for the test.

Problem 24E In a test of the hypothesis versus a sample of n = 50 observations possessed mean and standard deviation s = 3.1. Find and interpret the p-value for this test.

Problem 25E In a test of against the sample data yielded the test statistic z = 2.17. Find the p-value for the test.

In a test of \(H_{0}: \mu=75\) performed using the computer, SPSS reports a two-tailed p-value of .1032. Make the appropriate conclusion for each of the following situations: a. \(H_{\mathrm{a}}: \mu<75, z=-1.63, \alpha=.05\) b. \(H_{\mathrm{a}}: \mu<75, z=1.63, \alpha=.10\) c. \(H_{\mathrm{a}}: \mu>75, z=1.63, \alpha=.10\) d. \(H_{\mathrm{a}}: \mu \neq 75, z=-1.63, \alpha=.01\) Text Transcription: H_0: mu = 75 H_a: mu < 75, z = -1.63, alpha = .05 H_a: mu < 75, z = 1.63, alpha = .10 H_a: mu > 75, z =1.63, alpha = .10 H_a: mu neq 75, z = -1.63, alpha = .01

Consider the test \(H_0: \mu=70\) versus \(H_{\mathrm{a}}: \mu>70\) using a large sample of size n = 400. Assume \(\sigma=20\). a. Describe the sampling distribution of \(\bar{x}\). b. Find the value of the test statistic if \(\bar{x}=72.5\). c. Refer to part b. Find the p-value of the test. d. Find the rejection region of the test for \(\alpha=.01\). e. Refer to parts c and d}. Use the p-value approach to make the appropriate conclusion. f. Repeat part e, but use the rejection region approach. g. Do the conclusions, parts e and f, agree?

Problem 29E Suppose you are interested in conducting the statistical test of and you have decided to use the following decision rule: Reject H0 if the sample mean of a random sample of 81 items is more than 270. Assume that the standard deviation of the population is 63. Express the decision rule in terms of z. b. Find ?, the probability of making a Type I error, by using this decision rule.

An analyst tested the null hypothesis \(\mu\ \geq\ 20\) against the alternative hypothesis that \(\mu\ <\ 20\). The analyst reported a p-value of .06. What is the smallest value of \(\alpha\) for which the null hypothesis would be rejected?

Problem 30E A random sample of 100 observations from a population with standard deviation 60 yielded a sample mean of 110. a. Test the null hypothesis that against the alternative hypothesis that Interpret the results of the test. b. Test the null hypothesis that against the alternative hypothesis that Interpret the results of the test. c. Compare the results of the two tests you conducted. Explain why the results differ.

Corporate sustainability of CPA firms. Refer to the Business and Society (March 2011) study on the sustainability behaviors of CPA corporations, Exercise 6.12 (p. 308). Recall that the level of support for corporate sustainability (measured on a quantitative scale ranging from 0 to 160 points) was obtained for each in a sample of 992 senior managers at CPA firms. The data (where higher point values indicate a higher level of support for sustainability) are saved in the accompanying file. The CEO of a CPA firm claims that the true mean level of support for sustainability is 75 points. a. Specify the null and alternative hypotheses for testing this claim. b. For this problem, what is a Type I error? A Type II error? c. The XLSTAT printout on the following page gives the results of the test. Locate the test statistic and p-value on the printout. d. At \(\alpha = .05\) give the appropriate conclusion. e. What assumptions, if any, about the distribution of support levels must hold true in order for the inference derived from the test to be valid? Explain. Text Transcription: alpha = .05

Problem 31E A random sample of 64 observations produced the following summary statistics: a. Test the null hypothesis that against the alternative hypothesis that using b. Test the null hypothesis that against the alternative hypothesis that using Interpret the result.

Problem 33E Packaging of a children’s health food. Junk foods (e.g., potato chips) are typically packaged to appeal to children. Can similar packaging of a healthy food product influence children’s desire to consume the product? This was the question of interest in an article published in the Journal of Consumer Behaviour (Vol. 10, 2011). A fictitious brand of a healthy food product—sliced apples—was packaged to appeal to children (a smiling cartoon apple was on the front of the package). The researchers showed the packaging to a sample of 408 school children and asked each whether he or she was willing to eat the product. Willingness to eat was measured on a 5-point scale, with 1 = “not willing at all” and 5 = “very willing.” The data are summarized as follows: s = 2.44. Suppose the researchers knew that the mean willingness to eat an actual brand of sliced apples (which is not packaged for children) is a. Conduct a test to determine whether the true mean willingness to eat the brand of sliced apples packaged for children exceeded 3. Use to make your conclusion. b. The data (willingness to eat values) are not normally distributed. How does this impact (if at all) the validity of your conclusion in part a? Explain.

Problem 35E Facial structure of CEOs. Refer to the Psychological Science (Vol. 22, 2011) study on using a chief executive officer’s facial structure to predict a firm’s financial performance, Exercise 6.20 (p. 310). Recall that the facial width-to-height ratio (WHR) for each in a sample of 55 CEOs at publicly traded Fortune 500 firms was determined. The sample resulted in and s = .15. An analyst wants to predict the financial performance of a Fortune 500 firm based on the value of the true mean facial WHR of CEOs. The analyst wants to use the value of = 2.2. Do you recommend he use this value? Conduct a test of hypothesis for to help you answer the question. Specify all the elements of the test: H0, Ha, test statistic, p-value, a, and your conclusion.

Problem 34E Accounting and Machiavellianism. Refer to the Behavioral Research in Accounting (Jan. 2008) study of Machiavellian traits in accountants, Exercise 6.19 (p. 310). A Mach rating score was determined for each in a random sample of 122 purchasing managers, with the following results: s = 12.6. Recall that a director of purchasing at a major firm claims that the true mean Mach rating score of all purchasing managers is 85. a. Suppose you want to test the director’s claim. Specify the null and alternative hypotheses for the test. b. Give the rejection region for the test using c. Find the value of the test statistic. d. Use the result, part c, to make the appropriate conclusion.

Problem 36E Trading skills of institutional investors. Managers of stock portfolios make decisions as to what stocks to buy and sell in a given quarter. The trading skills of these institutional investors were quantified and analyzed in The Journal of Finance (April 2011). The study focused on “round-trip” trades, i.e., trades in which the same stock was both bought and sold in the same quarter. Consider a random sample of 200 round-trip trades made by institutional investors. Suppose the sample mean rate of return is 2.95% and the sample standard deviation is 8.82%. If the true mean rate of return of round-trip trades is positive, then the population of institutional investors is considered to have performed successfully. a. Specify the null and alternative hypotheses for determining whether the population of institutional investors performed successfully. b. Find the rejection region for the test using c. Interpret the value of a in the words of the problem. d. A Minitab printout of the analysis is shown below. Locate the test statistic and p-value on the printout. [Note: For large samples, z ? t.] e. Give the appropriate conclusion in the words of the problem.

Problem 37E Producer’s and consumer’s risk. In quality-control applications of hypothesis testing, the null and alternative hypotheses are frequently specified as H0: The production process is performing satisfactorily. Ha: The process is performing inan unsatisfactory manner. Accordingly, a is sometimes referred to as the producer’s risk, while b is called the consumer’s risk (Stevenson, Operations Management, 2008). An injection molder produces plastic golf tees. The process is designed to produce tees with a mean weight of .250 ounce. To investigate whether the injection molder is operating satisfactorily, 40 tees were randomly sampled from the last hour’s production. Their weights (in ounces) are listed in the following table. a. Write H0 and Ha in terms of the true mean weight of the golf tees, b. Access the data and find and s. c. Calculate the test statistic. d. Find the p-value for the test. e. Locate the rejection region for the test using f. Do the data provide sufficient evidence to conclude that the process is not operating satisfactorily? g. In the context of this problem, explain why it makes sense to call the producer’s risk and the consumer’s risk.

Time required to complete a task. When a person is asked, “How much time will you require to complete this task?” cognitive theory posits that people (e.g., a business consultant) will typically underestimate the time required. Would the opposite theory hold if the question was phrased in terms of how much work could be completed in a given amount of time? This was the question of interest to researchers writing in Applied Cognitive Psychology (Vol. 25, 2011). For one study conducted by the researchers, each in a sample of 40 University of Oslo students was asked how many minutes it would take to read a 32-page report. In a second study, 42 students were asked how many pages of a lengthy report they could read in 48 minutes. (The students in either study did not actually read the report.) Numerical descriptive statistics (based on summary information published in the article) for both studies are provided in the accompanying table. a. The researchers determined that the actual mean time it takes to read the report is \(\mu=48\) minutes. Is there evidence to support the theory that the students, on average, overestimated the time it would take to read the report? Test using \(\alpha =.10\). b. The researchers also determined that the actual mean number of pages of the report that is read within the allotted time is \(\mu=32\) pages. Is there evidence to support the theory that the students, on average, underestimated the number of report pages that could be read? Test using \(\alpha=.10\). c. The researchers noted that the distribution of both estimated time and estimated number of pages is highly skewed (i.e., not normally distributed). Does this fact impact the inferences derived in parts a and b? Explain.

Problem 40E Cooling method for gas turbines. During periods of high electricity demand, especially during the hot summer months, the power output from a gas turbine engine can drop dramatically. One way to counter this drop in power is by cooling the inlet air to the gas turbine. An increasingly popular cooling method uses high-pressure inlet fogging. The performance of a sample of 67 gas turbines augmented with high-pressure inlet fogging was investigated in the Journal of Engineering for Gas Turbines and Power (Jan. 2005). One measure of performance is heat rate (kilojoules per kilowatt per hour). Heat rates for the 67 gas turbines are listed in the table below. Suppose that a standard gas turbine has, on average, a heat rate of 10,000 kJ/kWh. Conduct a test to determine if the mean heat rate of gas turbines augmented with high-pressure inlet fogging exceeds 10,000 kJ/kWh. Use = .05.

Problem 38E Birth order, IQ, and earnings. Recent research suggests that your annual pay is linked to your birth order— firstborn individuals tend to earn higher salaries than non- firstborn individuals. The evidence on whether IQ is associated with birth order is not as conclusive. An international team of economists investigated the possible link between IQ and birth order in CESifo Economic Studies (Vol. 57, 2011). The data source for the research was the Medical Birth Registry of Norway. It is known that the mean IQ (measured in stanines) for all Norway residents is 5.2 points. In the study, a sample of 581 Norway residents who were the sixth-born or later in their families had a mean IQ score of 4.7 points with a standard deviation of 1.8 points. Is this sufficient evidence to conclude that the mean IQ score of all Norway residents who were the sixth-born or later in their families is lower than the country mean of 5.2 points? Use as a measure of reliability for your inference.

Revenue for a full-service funeral. According to the National Funeral Directors Association (NFDA), the nation’s 22,000 funeral homes collected an average of $6,500 per full-service funeral in 2009 (www.nfda.org). A random sample of 36 funeral homes reported revenue data for the current year. Among other measures, each reported its average fee for a full-service funeral. These data (in thousands of dollars) are shown in the following table. a. What are the appropriate null and alternative hypotheses to test whether the average full-service fee of U. S. funeral homes this year exceeds $6,500? b. Conduct the test at \(\alpha=.05\). Do the sample data provide sufficient evidence to conclude that the average fee this year is higher than in 2009? c. In conducting the test, was it necessary to assume that the population of average full-service fees was normally distributed? Justify your answer.

Problem 41E Point spreads of NFL games. During the National Football League (NFL) season, Las Vegas odds makers establish a point spread on each game for betting purposes. For example, the New England Patriots were established as 3.5-point favorites over the eventual champion New York Giants in the 2012 Super Bowl. The final scores of NFL games were compared against the final point spreads established by the odds makers in Chance (Fall 1998). The difference between the game outcome and point spread (called a point-spread error) was calculated for 240 NFL games. The mean and standard deviation of the point-spread errors are = -1.6 and s = 13.3. Use this information to test the hypothesis that the true mean point-spread error for all NFL games differs from 0. Conduct the test at = .01 and interpret the result.

Problem 43E Buy-side vs. sell-side analysts’ earnings forecasts. Refer to the Financial Analysts Journal (Jul./Aug. 2008) study of earnings forecasts of buy-side and sell-side analysts, Exercise 2.86 (p. 86). Buy-side analysts differ from sell-side analysts on a variety of factors, including scope of industry coverage, sources of information used, and target audience. Recall that data were collected on 3,526 forecasts made by buy-side analysts and 58,562 forecasts made by sell-side analysts, and the relative absolute forecast error was determined for each. A positive forecast error indicates that the analyst is overestimating earnings, while a negative forecast error implies that the analyst is underestimating earnings. Summary statistics for the forecast errors in the two samples are reproduced in the table below. a. Conduct a test (at = .01) to determine if the true mean forecast error for buy-side analysts is positive. Use the observed significance level (p-value) of the test to make your decision and state your conclusion in the words of the problem. b. Conduct a test (at = .01) to determine if the true mean forecast error for sell-side analysts is negative. Use the observed significance level (p-value) of the test to make your decision and state your conclusion in the words of the problem.

Problem 44E Solder-joint inspections. Current technology uses highresolution X-rays and lasers for inspection of solder-joint defects on printed circuit boards (PCBs) (Global SMT & Packaging, April 2008). A particular manufacturer of laser-based inspection equipment claims that its product can inspect on average at least 10 solder joints per second when the joints are spaced .1 inch apart. The equipment was tested by a potential buyer on 48 different PCBs. In each case, the equipment was operated for exactly 1 second. The number of solder joints inspected on each run follows: a. The potential buyer wants to know whether the sample data refute the manufacturer’s claim. Specify the null and alternative hypotheses that the buyer should test. b. In the context of this exercise, what is a Type I error? A Type II error? c. Conduct the hypothesis test you described in part a and interpret the test’s results in the context of this exercise. Use = .05.

Why do small firms export? What motivates small firms to export their products? To answer this question, California State University Professor Ralph Pope conducted a survey of 137 exporting firms listed in the California International Trade Register (Journal of Small Business Management, Vol. 40, 2002). Firm CEOs were asked to respond to the statement “Management believes that the firm can achieve economies of scale by exporting” on a scale of 1 (strongly disagree) to 5 (strongly agree). Summary statistics for the n = 137 scale scores were reported as = 3.85 and s = 1.5. In the journal article, the researcher hypothesized that if the true mean scale score exceeds 3.5, then CEOs at all California small firms generally agree with the statement. a. Conduct the appropriate test using = .05. State your conclusion in the words of the problem. b. Explain why the results of the study, although “statistically significant,” may not be practically significant. c. The scale scores for the sample of 137 small firms are unlikely to be normally distributed. Does this invalidate the inference you made in part a? Explain.

a. Consider testing \(H_0: \mu=80\). Under what conditions should you use the t-distribution to conduct the test? b. In what ways are the distributions of the z-statistic and t-test statistic alike? How do they differ?

For each of the following rejection regions, sketch the sampling distribution of t and indicate the location of the rejection region on your sketch: a. t > 1.440 where df = 6 b. t < -1.782 where df = 12 c. t < -2.060 or t > 2.060 where df = 25 d. For each of parts a-c, what is the probability that a Type I error will be made?

Problem 49E A random sample of n observations is selected from a normal population to test the null hypothesis that m = 10. Specify the rejection region for each of the following combinations of Ha, , and n:

Problem 50E A sample of five measurements, randomly selected from a normally distributed population, resulted in the following summary statistics: = 4.8, s = 1.3. a. Test the null hypothesis that the mean of the population is 6 against the alternative hypothesis, = .05. b. Test the null hypothesis that the mean of the population is 6 against the alternative hypothesis, = .05. c. Find the observed significance level for each test.

Problem 46E Salaries of postgraduates. The Economics of Education Review (Vol. 21, 2002) published a paper on the relationship between education level and earnings. The data for the research were obtained from the National Adult Literacy Survey of more than 25,000 respondents. The survey revealed that males with a postgraduate degree have a mean salary of $61,340 (with standard error = $2,185), while females with a postgraduate degree have a mean of $32,227 (with standard error = $932). a. The article reports that a 95% confidence interval for , the population mean salary of all males with postgraduate degrees, is ($57,050, $65,631). Based on this interval, is there evidence to say that differs from $60,000? Explain. b. Use the summary information to test the hypothesis that the true mean salary of males with postgraduate degrees differs from $60,000. Use = .05. c. Explain why the inferences in parts a and b agree. d. The article reports that a 95% confidence interval for the population mean salary of all females with postgraduate degrees, is ($30,396, $34,058). Based on this interval, is there evidence to say that differs from $33,000? Explain. e. Use the summary information to test the hypothesis that the true mean salary of females with postgraduate degrees differs from $33,000. Use = .05. f. Explain why the inferences in parts d and e agree.

Suppose you conduct a t-test for the null hypothesis \(H_0: \mu=1,000\) versus the alternative hypothesis \(H_{\mathrm{a}}: \mu>1,000\) based on a sample of 17 observations. The test results are t = 1.89 and p-value =.038. a. What assumptions are necessary for the validity of this procedure? b. Interpret the results of the test. c. Suppose the alternative hypothesis had been the two-tailed \(H_{\mathrm{a}}: \mu \neq 1,000\). If the t - statistic were unchanged, then what would the p-value be for this test? Interpret the p-value for the two-tailed test.

Lobster trap placement. Refer to the Bulletin of Marine Science (April 2010) observational study of lobster trap placement by teams fishing for the red spiny lobster in Baja California Sur, Mexico. Trap spacing measurements (in meters) for a sample of seven teams of red spiny lobster fishermen are reproduced in the accompanying table (and saved in the TRAPSPACE file). Let ? represent the average of the trap spacing measurements for the population of red spiny lobster fishermen fishing in Baja California Sur, Mexico. In Exercise 5.35 you computed the mean and standard deviation of the sample measurements to be = 89.9 meters and s = 11.6 meters, respectively. Suppose you want to determine if the true value of ? differs from 95 meters. From Shester, G. G. “Explaining catch variation among Baja California lobster fishers through spatial analysis of trap-placement decisions.” Bulletin of Marine Science, Vol. 86, No. 2, April 2010 ( Table 1 ), pp. 479–498. Reprinted with permission from the University of Miami – Bulletin of Marine Science. a. Specify the null and alternative hypothesis for this test. b. Since = 89.9 is less than 95, a fisherman wants to reject the null hypothesis. What are the problems with using such a decision rule? c. Compute the value of the test statistic. d. Find the approximate p -value of the test. e. Select a value of a , the probability of a Type I error. Interpret this value in the words of the problem. f. Give the appropriate conclusion, based on the results of parts d and e. g. What conditions must be satisfied for the test results to be valid? h. In Exercise 5.35 you found a 95% confidence interval for ? . Does the interval support your conclusion in part f ? Lobster trap placement. Strategic placement of lobster traps is one of the keys for a successful lobster fisherman. An observational study of teams fishing for the red spiny lobster in Baja California Sur, Mexico, was conducted and the results published in Bulletin of Marine Science (April, 2010). One of the variables of interest was the average distance separating traps—called trap spacing—deployed by the same team of fishermen. Trap spacing measurements (in meters) for a sample of seven teams of red spiny lobster fishermen are shown in the accompanying table (and saved in the TRAPSPACE file). Of interest is the mean trap spacing for the population of red spiny lobster fishermen fishing in Baja California Sur, Mexico. From Shester, G. G. “Explaining catch variation among Baja California lobster fishers through spatial analysis of trap-placement decisions.” Bulletin of Marine Science, Vol. 86, No. 2, April 2010 ( Table 1 ), pp. 479–498. Reprinted with permission from the University of Miami – Bulletin of Marine Science. a. Identify the target parameter for this study. b. Compute a point estimate of the target parameter. c. What is the problem with using the normal ( z ) statistic to find a confidence interval for the target parameter? d. Find a 95% confidence interval for the target parameter. e. Give a practical interpretation of the interval, part d . f. What conditions must be satisfied for the interval, part d , to be valid? Conclusions and Consequences for a Test of Hypothesis

Problem 54E new dental bonding agent. When bonding teeth, orthodontists must maintain a dry field. A new bonding adhesive (called Smartbond) has been developed to eliminate the necessity of a dry field. However, there is concern that the new bonding adhesive is not as strong as the current standard, a composite adhesive (Trends in Biomaterials & Artificial Organs, Jan. 2003). Tests on a sample of 10 extracted teeth bonded with the new adhesive resulted in a mean breaking strength (after 24 hours) of = 5.07 Mpa and a standard deviation of s = .46 Mpa. Orthodontists want to know if the true mean breaking strength of the new bonding adhesive is less than 5.70 Mpa, the mean breaking strength of the composite adhesive. a. Set up the null and alternative hypotheses for the test. b. Find the rejection region for the test using = .01. c. Compute the test statistic. d. Give the appropriate conclusion for the test. e. What conditions are required for the test results to be valid?

Radon exposure in Egyptian tombs. Refer to the Radiation Protection Dosimetry (December 2010) study of radon exposure in Egyptian tombs, Exercise 6.30 (p. 318). The radon levels—measured in becquerels per cubic meter \(Bq/m^3\)—in the inner chambers of a sample of 12 tombs are listed in the table (next page). For the safety of the guards and visitors, the Egypt Tourism Authority (ETA) will temporarily close the tombs if the true mean level of radon exposure in the tombs rises to 6,000 \(Bq/m^3\). Consequently, the ETA wants to conduct a test to determine if the true mean level of radon exposure in the tombs is less than 6,000 \(Bq/m^3\), using a Type I error probability of .10. An SPSS analysis of the data is shown (bottom, p. 384). Specify all the elements of the test: \(H_0\), \(H_a\), test statistic, p-value, \(\alpha\), and your conclusion.

Problem 55E Surface roughness of pipe. Refer to the Anti-corrosion Methods and Materials (Vol. 50, 2003) study of the surface roughness of coated interior pipe used in oil fields, Exercise 6.33 (p. 318). The data (in micrometers) for 20 sampled pipe sections are reproduced in the next table. a. Give the null and alternative hypotheses for testing whether the mean surface roughness of coated interior pipe, , differs from 2 micrometers. b. Find the test statistic for the hypothesis test. c. Give the rejection region for the hypothesis test, using = .05. d. State the appropriate conclusion for the hypothesis test. e. A Minitab printout giving the test results is shown at the top of the page. Find and interpret the p-value of the test. f. In Exercise 6.33 you found a 95% confidence interval for . Explain why the confidence interval and test lead to the same conclusion about .

Problem 58E Shopping vehicle and judgment. Refer to the Journal of Marketing Research (Dec. 2011) study of when grocery store shoppers’ judgments, Exercise 2.85 (p. 85). For one part of the study, 11 consumers were told to put their arm in a flex position (similar to a shopping basket) and then each consumer was offered several choices between a vice product and a virtue product (e.g., a movie ticket vs. a shopping coupon, pay later with a larger amount vs. pay now). Based on these choices, a vice choice score was determined on a scale of 0 to 100 (where higher scores indicate a greater preference for vice options). The data in the next table are (simulated) choice scores for the 11 consumers. Suppose that the average choice score for consumers with an extended arm position (similar to pushing a shopping cart) is known to be = 50. The researchers theorize that the mean choice score for consumers shopping with a flexed arm will be higher than 43 (reflecting their higher propensity to select a vice product. Test the theory at = .05.

Problem 57E Water distillation with solar energy. In countries with a water shortage, converting salt water to potable water is big business. The standard method of water distillation is with a single-slope solar still. Several enhanced solar energy water distillation systems were investigated in Applied Solar Energy (Vol. 46, 2010). One new system employs a sun-tracking meter and a step-wise basin. The new system was tested over 3 randomly selected days at a location in Amman, Jordan. The daily amounts of distilled water collected by the new system over the 3 days were 5.07, 5.45, and 5.21 liters per square meter (l/m2). Suppose it is known that the mean daily amount of distilled water collected by the standard method at the same location in Jordan is = 1.4 l/m2. a. Set up the null and alternative hypotheses for determining whether the mean daily amount of distilled water collected by the new system is greater than 1.4. b. For this test, give a practical interpretation of the value = .10. c. Find the mean and standard deviation of the distilled water amounts for the sample of 3 days. d. Use the information from part c to calculate the test statistic. e. Find the observed significance level (p-value) of the test. f. State, practically, the appropriate conclusion.

Problem 59E Minimizing tractor skidding distance. Refer to the Journal of Forest Engineering (July 1999) study of minimizing tractor skidding distances along a new road in a European forest, Exercise 6.35 (p. 319). The skidding distances (in meters) were measured at 20 randomly selected road sites. The data are repeated below. Recall that a logger working on the road claims the mean skidding distance is at least 425 meters. Is there sufficient evidence to refute this claim? Use = .10.

Arsenic in smelters. The Occupational Safety and Health Act (OSHA) allows issuance of engineering standards to ensure safe workplaces for all Americans. The maximum allowable mean level of arsenic in smelters, herbicide production facilities, and other places where arsenic is used is .004 milligrams per cubic meter of air. Suppose smelters at two plants are being investigated to determine whether they are meeting OSHA standards. Two analyses of the air are made at each plant, and the results (in milligrams per cubic meter of air) are shown in the table. A claim is made that the OSHA standard is violated at Plant 2 but not at Plant 1. Do you agree?

Problem 64E Suppose a random sample of 100 observations from a binomial population gives a value of = .63 and you wish to test the null hypothesis that the population parameter p is equal to .70 against the alternative hypothesis that p is less than .70. a. Noting that = .63, what does your intuition tell you? Does the value of appear to contradict the null hypothesis? b. Use the large-sample z-test to test H0: p = .70 against the alternative hypothesis, Ha: p < .70. Use = .05. How do the test results compare with your intuitive decision from part a? c. Find and interpret the observed significance level of the test you conducted in part b.

Suppose the sample in Exercise 7.64 has produced \(\hat{p}=.83\) and we wish to test \(H_0: p=.9\) against the alternative \(H_{\mathrm{a}}: p<.9\) a. Calculate the value of the z-statistic for this test. b. Note that the numerator of the z-statistic \(\left(\hat{p}-p_0=\right.\) \(.83-.90=-.07)\) is the same as for Exercise 7.64. Considering this, why is the absolute value of z for this exercise larger than that calculated in Exercise 7.64? c. Complete the test using \(\alpha=.05\) and interpret the result. d. Find the observed significance level for the test and interpret its value.

Active nuclear power plants. Refer to the U.S. Energy Information Administration’s list of active nuclear power plants operating in each of a sample of 20 states, Exercise 2.54 (p. 73). The data are reproduced in the next table. a. Is there sufficient evidence to claim that the mean number of active nuclear power plants operating in all states exceeds 3? Test using \(\alpha\) = .10. b. Are the conditions required for a valid small-sample test reasonably satisfied? Explain. c. Eliminate the lowest two values and the highest two values from the data set, then conduct the test of part a on the smaller data set. What impact does this have on the test results? d. Why is it dangerous to eliminate data points in order to satisfy an assumption for a test of hypothesis?

Problem 60E Crude oil biodegradation. Refer to the Journal of Petroleum Geology (April 2010) study of the environmental factors associated with biodegradation in crude oil reservoirs, Exercise 6.37 (p. 319). Recall that 16 water specimens were randomly selected from various locations in a reservoir on the floor of a mine and that the amount of dioxide (milligrams/liter)—a measure of biodegradation—as well as presence of oil were determined for each specimen. These data are reproduced in the accompanying table. a. Conduct a test to determine if the true mean amount of dioxide present in water specimens that contained oil was less than 3 milligrams/liter. Use = .10. b. Repeat part a for water specimens that did not contain oil.

Problem 61E Crack intensity of paved highways. The Mississippi Department of Transportation collected data on the number of cracks (called crack intensity) in an undivided two-lane highway using van-mounted, state-of-the-art video technology (Journal of Infrastructure Systems, Mar. 1995). The mean number of cracks found in a sample of eight 50-meter sections of the highway was = .210, with a variance of s2 = .011. Suppose the American Association of State Highway and Transportation Officials (AASHTO) recommends a maximum mean crack intensity of .100 for safety purposes. Is there evidence to say that the true mean crack intensity of the Mississippi highway exceeds the AASHTO recommended maximum? Use = .01 in the test.

Problem 66E A statistics student used a computer program to test the null hypothesis H0: p = .5 against the one-tailed alternative, Ha: p > .5. A sample of 500 observations are input into SPSS, which returns the following results: z = .44, two-tailed p-value = .33. a. The student concludes, based on the p-value, that there is a 33% chance that the alternative hypothesis is true. Do you agree? If not, correct the interpretation. b. How would the p-value change if the alternative hypothesis was two-tailed, Ha: p ? .5? Interpret this p-value.

Problem 67E Refer to Exercise 6.44 (p. 325), in which 50 consumers taste-tested a new snack food. Their responses (where 0 = do not like; 1 = like; 2 = indifferent) are reproduced below. a. Test H0: p = .5 against Ha: p > .5, where p is the proportion of customers who do not like the snack food. Use = .10. b. Find the observed significance level of your test.

Problem 68E For the binomial sample sizes and null hypothesized values of p in each part, determine whether the sample size is large enough to use the normal approximation methodology presented in this section to conduct a test of the null hypothesis H0: p = p0.

Problem 69E Paying for music downloads. If you use the Internet, have you ever paid to access or download music? This was one of the questions of interest in a Pew Internet & American Life Project Survey (October 2010). In a representative sample of 755 adults who use the Internet, 506 admitted that they have paid to download music. Let p represent the true proportion of all Internet-using adults who have paid to download music. a. Compute a point estimate of p. b. Set up the null and alternative hypotheses for testing whether the true proportion of all Internet-using adults who have paid to download music exceeds .7. c. Compute the test statistic for part b. d. Find the rejection region for the test if = .01. e. Find the p-value for the test. f. Make the appropriate conclusion using the rejection region. g. Make the appropriate conclusion using the p-value.

Problem 70E Satellite radio in cars. A spokesperson for the National Association of Broadcasters (NAB) claims that 80% of all satellite radio subscribers have a satellite radio receiver in their car. That in a June 2007 survey of 501 satellite radio subscribers, 396 had a satellite receiver in their car. Consider a test of the NAB spokesperson’s claim. a. Define the parameter of interest to the NAB spokesperson. ________________ b. Set up the null hypothesis for testing the claim. ________________ c. Specify the alternative hypothesis if you believe that the spokesperson’s claim is too high. ________________ d. Compute the value of the test statistic. ________________ e. Determine the rejection region for the test using ? =.10. ________________ f. Compute the p -value of the test. ________________ g. Make the appropriate conclusion. Show that the decision based on the rejection region agrees with the decision based on the p -value. Satellite radio in cars. A recent survey conducted for the National Association of Broadcasters investigated satellite radio subscriber service and usage. The June 2007 survey, conducted by Wilson Research Strategies, consisted of a random sample of 501 satellite radio subscribers. One of the questions of interest was, “Do you have a satellite radio receiver in your car?” The survey found that 396 subscribers did, in fact, have a satellite receiver in their car. a. Identify the population of interest to the National Association of Broadcasters. ________________ b. Based on the survey question, what is the variable of interest? ________________ c. Does the variable produce quantitative or qualitative data? ________________ d. Describe that sample of interest. ________________ e. What inference can be made from the survey results?

History of corporate acquisitions. Refer to the Academy of Management Journal (Aug. 2008) investigation of the performance and timing of corporate acquisitions, Exercise 6.53 (p. 326). Recall that the investigation discovered that in a random sample of 2,778 firms, 748 announced one or more acquisitions during the year 2000. Does the sample provide sufficient evidence to indicate that the true percentage of all firms that announced one or more acquisitions during the year 2000 is less than 30%? Use \(\alpha = .05\) to make your decision. Text Transcription: alpha = .05

Problem 72E Gummy bears: red or yellow? Chance (Winter 2010) presented a lesson in hypothesis testing carried out by medical students in a biostatistics class. Students were blind-folded and then given a red-colored or yellow-colored gummy bear to chew. (Half the students were randomly assigned to receive the red gummy bear and half to receive the yellow bear. The students could not see what color gummy bear they were given.) After chewing, the students were asked to guess the color of the candy based on the flavor. Of the 121 students who participated in the study, 97 correctly identified the color of the gummy bear. a. If there is no relationship between color and gummy bear flavor, what proportion of the population of students will correctly identify the color? ________________ b. Specify the null and alternative hypothesis for testing whether color and flavor are related. ________________ c. Carry out the test and give the appropriate conclusion at ? =.01. Use the p -value of the test to make your decision.

Problem 74E Vacation-home owners. The National Association of Realtors (NAR) reported the results of a March 2010 survey of home buyers. In a random sample of 1,982 residential properties purchased during the year, 198 were purchased as a vacation home. Five years ago, 14% of residential properties were vacation homes. a. Do the survey results allow the NAR to conclude (at = .01) that the percentage of all residential properties purchased for vacation homes is less that 14%. b. The NAR sent the survey questionnaire to a nationwide sample of 45,000 new home owners, of which 1,982 responded to the survey. How might this bias the results?

Toothpaste brands with the ADA seal. Consumer Reports evaluated and rated 46 brands of toothpaste. One attribute examined in the study was whether or not a toothpaste brand carries an American Dental Association (ADA) seal verifying effective decay prevention. The data for the 46 brands (coded 1 = ADA seal, 0 = no ADA seal) are listed here. a. Give the null and alternative hypotheses for testing whether the true proportion of toothpaste brands with the ADA seal verifying effective decay prevention is less than .5. b. Locate the p-value on the Minitab printout below. c. Make the appropriate conclusion using \(\alpha = .10\). Text Transcription: alpha = .10

Problem 76E Unemployment and a reduced workweek. In an effort to increase employment, France mandated in February 2000 that all companies with 20 or more employees reduce the workweek to 35 hours. The economic impact of the shortened workweek was analyzed in Economic Policy (July 2008). The researchers focused on several key variables such as hourly wages, dual-job holdings, and level of employment. Assume that in the year prior to the 35-hour-workweek law, unemployment in France was at 12%. Suppose that in a random sample of 500 French citizens (eligible workers) taken several years after the law was enacted, 53 were unemployed. Conduct a test of hypothesis to determine if the French unemployment rate dropped after the enactment of the 35-hour-workweek law. Test using = .05.

Problem 75E Organic-certified coffee. Coffee markets that conform to organic standards focus on the environmental aspects of coffee growing, such as the use of shade trees and a reduced reliance on chemical pesticides. A study of organic coffee growers was published in Food Policy (Vol. 36, 2010). In a representative sample of 845 coffee growers from southern Mexico, 417 growers were certified to sell to organic coffee markets while 77 growers were transitioning to become organic certified. In the United States, 60% of coffee growers are organic certified. Is there evidence to indicate that fewer than 60% of the coffee growers in southern Mexico are either organic certified or transitioning to become organic certified? State your conclusion so that there is only a 5% chance of making a Type I error.

Problem 77E Effectiveness of skin cream. Pond’s Age-Defying Complex, a cream with alpha hydroxy acid, advertised that it could reduce wrinkles and improve the skin. In a study published in Archives of Dermatology (June 1996), 33 middle-aged women used a cream with alpha hydroxy acid for 22 weeks. At the end of the study period, a dermatologist judged whether each woman exhibited skin improvement. The results for the 33 women (where I = improved skin and N = no improvement) are listed in the next table. [Note: Pond’s recently discontinued the production of this cream product, replacing it with Age-Defying Towlettes.] a. Do the data provide sufficient evidence to conclude that the cream improved the skin of more than 60% of middle-aged women? Test using = .05. b. Find and interpret the p-value of the test.

TVs with DVRs. According to Nielsen’s Television Audience Report (2011), 41% of all households with televisions in the United States have a digital video recorder (DVR). Develop a sampling plan that will allow you to test this claim. Identify the target population, experimental units, variable to be measured, parameter of interest, null and alternative hypotheses, and the form of the test statistic.

Problem 78E Detection of motorcycles while driving. Motorcycle fatalities have increased dramatically over the past decade. As a result, manufacturers of powered two-wheelers (PTWs) are tweaking with their design in order to improve visibility by automobile drivers. The factors that impact the visibility of PTWs on the road were investigated in Accident Analysis and Prevention (Vol. 44, 2012). A visual search study was conducted in which viewers were presented with pictures of driving scenarios and asked to identify the presence or absence of a PTW. Of interest to the researchers is the detection rate, i.e., the proportion of pictures showing a PTW in which the viewer actually detected the presence of the PTW. Suppose that, in theory, the true detection rate for pictures of PTWs is .70. The study revealed that in a sample of 2,376 pictures that included a PTW, only 1,554 were detected by the viewers. Use this results to test the theory at = .10.

Problem 80E Choosing portable grill displays. Refer to the Journal of Consumer Research (Mar. 2003) experiment on influencing the choices of others by offering undesirable alternatives, Exercise 3.23 (p. 142). Recall that each of 124 college students selected three portable grills from five to display on the showroom floor. The students were instructed to include Grill #2 (a smaller-sized grill) and select the remaining two grills in the display to maximize purchases of Grill #2. If the six possible grill display combinations (1-2-3, 1-2-4, 1-2-5, 2-3-4, 2-3-5, and 2-4-5) were selected at random, then the proportion of students selecting any display was 1/6 = .167. One theory tested by the researcher was that the students would tend to choose the three-grill display so that Grill #2 was a compromise between a more desirable and a less desirable grill (i.e., display 1-2-3, 1-2-4, or 1-2-5). Of the 124 students, 85 selected a three-grill display that was consistent with this theory. Use this information to test the theory proposed by the researcher at = .05.

Let \(X_{0}^{2}\) be a particular value of \(X^2\). Find the value of \(X_{0}^{2}\) such that a. \(P\left(x^{2}\ >\ x_{0}^{2}\right)=.10 \text { for } n=12\) b. \(P\left(x^{2}\ >\ x_{0}^{2}\right)=.05 \text { for } n=9\) c. \(P\left(x^{2}\ >\ x_{0}^{2}\right)=.025 \text { for } n=5\)

The Pepsi Challenge. “Take the Pepsi Challenge” was a famous marketing campaign used by the Pepsi-Cola Company. Coca-Cola drinkers participated in a blind taste test where they were asked to taste unmarked cups of Pepsi and Coke and were asked to select their favorite. In one Pepsi television commercial, an announcer stated that “in recent blind taste tests, more than half the Diet Coke drinkers surveyed said they preferred the taste of Diet Pepsi.” Suppose 100 Diet Coke drinkers took the Pepsi Challenge and 56 preferred the taste of Diet Pepsi. Determine if more than half of all Diet Coke drinkers selected Diet Pepsi in the blind taste test. Select \(\alpha\) to minimize the probability of a Type I error. What were the consequences of the test results from Coca-Cola’s perspective?

A random sample of n observations is selected from a normal population to test the null hypothesis that \(\sigma^2=25\). Specify the rejection region for each of the following combinations of \(H_{\mathrm{a}}, \alpha\), and n: a. \(H_{\mathrm{a}}: \sigma^2 \neq 25 ; \alpha=.05 ; n=16\) b. \(H_{\mathrm{a}}: \sigma^2>25 ; \alpha=.01 ; n=23\) c. \(H_{\mathrm{a}}: \sigma^2>25 ; \alpha=.10 ; n=15\) d. \(H_{\mathrm{a}}: \sigma^2<25 ; \alpha=.01 ; n=13\) e. \(H_{\mathrm{a}}: \sigma^2 \neq 25 ; \alpha=.10 ; n=7\) f. \(H_{\mathrm{a}}: \sigma^2<25 ; \alpha=.05 ; n=25\)

Refer to Exercise 7.84. Suppose we had n = 100, \(\bar{x}=9.4\), and \(s^2=4.84\).. a. Test the null hypothesis, \(H_0:\sigma^2\ >\ 1\), against the alternative hypothesis, \(H_0:\sigma^2\ >\ 1\). b. Compare your test result with that of Exercise 7.84.

Problem 84E A random sample of seven measurements gave = 9.4 and s2 = 4.84. a. What assumptions must you make concerning the population in order to test a hypothesis about ? b. Suppose the assumptions in part a are satisfied. Test the null hypothesis, = 1, against the alternative hypothesis, > 1. Use = .05. c. Test the null hypothesis that = 1 against the alternative hypothesis that ? 1. Use = .05.

Problem 86E A random sample of n = 7 observations from a normal population produced the following measurements: 4, 0, 6, 3, 3, 5, 9. Do the data provide sufficient evidence to indicate that < 1? Test using = .05.

Problem 88E Lobster trap placement. Refer to the Bulletin of Marine Science (April 2010) observational study of lobster trap placement by teams fishing for the red spiny lobster in Baja California Sur, Mexico, Exercise 7.52 (p. 384). Trapspacing measurements (in meters) for a sample of seven teams of red spiny lobster fishermen are repeated in the table. (These measurements are for the BT cooperative in the accompanying data file.) The researchers want to know whether = 10. a. Specify the null and alternative hypotheses for this test. b. Find the variance of the sample data, s2. c. Note that s2 > 10. Consequently, a fisherman wants to reject the null hypothesis. What are the problems with using such a decision rule? d. Compute the value of the test statistic. e. Use statistical software to find the p-value of the test. f. Give the appropriate conclusion. g. What conditions must be satisfied for the test results to be valid?

Problem 87E Trading skills of institutional investors. Refer to The Journal of Finance (April 2011) analysis of trading skills of institutional investors, Exercise 7.36 (p. 377). Recall that the study focused on “round-trip” trades, i.e., trades in which the same stock was both bought and sold in the same quarter. In a random sample of 200 round-trip trades made by institutional investors, the sample standard deviation of the rates of return was 8.82%. One property of a consistent performance of institutional investors is a small variance in the rates of return of round-trip trades, say, a standard deviation of less than 10%. a. Specify the null and alternative hypotheses for determining whether the population of institutional investors performs consistently. b. Find the rejection region for the test using = .05. c. Interpret the value of a in the words of the problem. d. A Minitab printout of the analysis is shown below. Locate the test statistic and p-value on the printout. e. Give the appropriate conclusion in the words of the problem. f. What assumptions about the data are required for the inference to be valid?

Problem 89E Golf tees produced from an injection mold. Refer to Exercise 7.37 (p. 378) and the weights of tees produced by an injection mold process. If operating correctly, the process will produce tees with a weight variance of .000004 (ounces)2. If the weight variance differs from .000004, the injection molder is out of control. a. Set up the null and alternative hypotheses for testing whether the injection mold process is out of control. b. Use the data saved in the accompanying file to conduct the test, part a. Use = .01. c. What conditions are required for inferences derived from the test to be valid? Are they reasonably satisfied?

Jitter in a water power system. Refer to the Journal of Applied Physics investigation of throughput jitter in the opening switch of a prototype water power system, Exercise 6.100 (p. 340). Recall that low throughput jitter is critical to successful waterline technology. An analysis of conduction time for a sample of 18 trials of the prototype system yielded \(\bar{x}\) = 334.8 nanoseconds and s = 6.3 nanoseconds. (Conduction time is defined as the length of time required for the downstream current to equal 10% of the upstream current.) A system is considered to have low throughput jitter if the true conduction time standard deviation is less than 7 nanoseconds. Does the prototype system satisfy this requirement? Test using \(\alpha\) = .01.

A new dental bonding agent. Refer to the Trends in Biomaterials & Artificial Organs (Jan. 2003) study of a new dental bonding adhesive (called Smartbond), Exercise 7.54 (p. 385). Recall that tests on a sample of 10 extracted teeth bonded with the new adhesive resulted in a mean breaking strength (after 24 hours) of \(\bar{x}=5.07\) Mpa and a standard deviation of s = .46 Mpa. The manufacturer must demonstrate that the breaking strength variance of the new adhesive is less than the variance of the standard composite adhesive, \(\sigma^2=.25\). a. Set up the null and alternative hypotheses for the test. b. Find the rejection region for the test using \(\alpha=.01\). c. Compute the test statistic. d. Give the appropriate conclusion for the test. e. What conditions are required for the test results to be valid?

Problem 92E Drug content assessment. Refer to the Analytical Chemistry (Dec. 15, 2009) study of a new method used by GlaxoSmithKline Medicines Research Center to determine the amount of drug in a tablet, Exercise 6.101 (p. 340). Drug concentrations (measured as a percentage) for 50 randomly selected tablets are repeated in the accompanying table. The standard method of assessing drug content yields a concentration variance of 9. Can the scientists at GlaxoSmithKline conclude that the new method of determining drug concentration is less variable than the standard method? Test using = .01.

Problem 93E Do ball bearings conform to specifications? It is essential in the manufacture of machinery to use parts that conform to specifications. In the past, diameters of the ball bearings produced by a certain manufacturer had a variance of .00156. To cut costs, the manufacturer instituted a less expensive production method. The variance of the diameters of 100 randomly sampled bearings produced by the new process was .00211. Do the data provide sufficient evidence to indicate that diameters of ball bearings produced by the new process are more variable than those produced by the old process?

Problem 95E Why do small firms export? Refer to the Journal of Small Business Management (Vol. 40, 2002) study of what motivates small firms to export, Exercise 7.45 (p. 379). Recall that in a survey of 137 exporting firms, each CEO was asked to respond to the statement “Management believes that the firm can achieve economies of scale by exporting” on a scale of 1 (strongly disagree) to 5 (strongly agree). Summary statistics for the n = 137 scale scores were reported as = 3.85 and s = 1.5. a. Explain why the researcher will be unable to conclude that the true mean scale score exceeds 3.5 (as in Exercise 7.45) if the standard deviation of the scale scores is too large. b. Give the largest value of the true standard deviation, , for which you will reject the null hypothesis H0: = 3.5 in favor of the alternative hypothesis Ha: > 3.5 using = .01. c. Based on the study results, is there evidence (at = .01) to indicate that is smaller than the value you determined in part b?

Cooling method for gas turbines. Refer to the Journal of Engineering for Gas Turbines and Power (Jan. 2005) study of the performance of augmented gas turbine engines, Exercise 7.40 (p. 378). Recall that performance for each in a sample of 67 gas turbines was measured by heat rate (kilojoules per kilowatt per hour). The data are saved in the accompanying file. Suppose that standard gas turbines have heat rates with a standard deviation of 1,500 kJ/kWh. Is there sufficient evidence to indicate that the heat rates of the augmented gas turbine engine are more variable than the heat rates of the standard gas turbine engine? Test using \(\alpha\) = .05.

a. List three factors that will increase the power of a test. b. What is the relationship between b, the probability of committing a Type II error, and the power of a test?

Suppose you want to test \(H_0: \mu\ =\ 500\) against \(H_a: \mu\ >\ 500\) using \(\alpha\ =\ .05\). The population in question is normally distributed with standard deviation 100. A random sample of size n = 25 will be used. a. Sketch the sampling distribution of \(\bar{x}\) assuming that \(H_0\) is true. b. Find the value of \(\bar{x}_0\), that value of \(\bar{x}\) above which the null hypothesis will be rejected. Indicate the rejection region on your graph of part a. Shade the area above the rejection region and label it \(\alpha\). c. On your graph of part a, sketch the sampling distribution of \(\bar{x}\) if \(\mu\ =\ 550\). Shade the area under this distribution that corresponds to the probability that \(\bar{x}\) falls in the nonrejection region when \(\mu\ =\ 550\). Label this area \(\beta\). d. Find \(\beta\). e. Compute the power of this test for detecting the alternative \(H_a:\mu\ =\ 550\).

Problem 98E Refer to Exercise 7.97. a. If = 575 instead of 550, what is the probability that the hypothesis test will incorrectly fail to reject H0? That is, what is b? b. If = 575, what is the probability that the test will correctly reject the null hypothesis? That is, what is the power of the test? c. Compare b and the power of the test when = 575 to the values you obtained in Exercise 7.97 for = 550. Explain the differences.

It is desired to test \(H_0: \mu=75\) against \(H_{\mathrm{a}}: \mu<75\) using \(\alpha=.10\). The population in question is uniformly distributed with standard deviation 15 . A random sample of size 49 will be drawn from the population. a. Describe the (approximate) sampling distribution of \(\bar{x}\) under the assumption that \(H_0\) is true. b. Describe the (approximate) sampling distribution of \(\bar{x}\) under the assumption that the population mean is 70 . c. If \(\mu\) were really equal to 70 , what is the probability that the hypothesis test would lead the investigator to commit a Type II error? d. What is the power of this test for detecting the alternative \(H_{\mathrm{a}}: \mu=70\)?

Problem 100E Refer to Exercise 7.99. a. Find for each of the following values of the population mean: 74, 72, 70, 68, and 66. b. Plot each value of you obtained in part a against its associated population mean. Show on the vertical axis and on the horizontal axis. Draw a curve through the five points on your graph. c. Use your graph of part b to find the approximate probability that the hypothesis test will lead to a Type II error when = 73. d. Convert each of the values you calculated in part a to the power of the test at the specified value of . Plot the power on the vertical axis against on the horizontal axis. Compare the graph of part b to the power curve of this part. e. Examine the graphs of parts b and d. Explain what they reveal about the relationships among the distance between the true mean and the null hypothesized mean 0, the value of , and the power.

Problem 101E Suppose you want to conduct the two-tailed test of H0: p = .7 against Ha: p ? .7 using = .05. A random sample of size 100 will be drawn from the population in question. a. Describe the sampling distribution of under the assumption that H0 is true. b. Describe the sampling distribution of under the assumption that p = .65. c. If p were really equal to .65, find the value of b associated with the test. d. Find the value of b for the alternative Ha: p = .71.

Problem 102E Square footage of new California homes. The average size of single-family homes built in the United States is 2,390 square feet (Statistical Abstract of the United States, 2011). A random sample of 100 new homes sold in California yielded the following size information: = 2,507 square feet and s = 257 square feet. a. Assume the average size of U.S. homes is known with certainty. Do the sample data provide sufficient evidence to conclude that the mean size of California homes built exceeds the national average? Test using = .01. b. Suppose the actual mean size of new California homes was 2,490 square feet. What is the power of the test in part a to detect this 100-square-foot difference? c. If the California mean were actually 2,440 square feet, what is the power of the test in part a to detect this 50-square-foot difference?

Problem 103E Manufacturers that practice sole sourcing. If a manufacturer (the vendee) buys all items of a particular type from a particular vendor, the manufacturer is practicing sole sourcing (Schonberger and Knod, Operations Management, 2001). As part of a sole-sourcing arrangement, a vendor agrees to periodically supply its vendee with sample data from its production process. The vendee uses the data to investigate whether the mean length of rods produced by the vendor’s production process is truly 5.0 millimeters (mm) or more, as claimed by the vendor and desired by the vendee. a. If the production process has a standard deviation of .01 mm, the vendor supplies n = 100 items to the vendee, and the vendee uses = .05 in testing H0: = 5.0 mm against Ha: < 5.0 mm, what is the probability that the vendee’s test will fail to reject the null hypothesis when in fact = 4.9975 mm? What is the name given to this type of error? b. Refer to part a. What is the probability that the vendee’s test will reject the null hypothesis when in fact = 5.0? What is the name given to this type of error? c. What is the power of the test to detect a departure of .0025 mm below the specified mean rod length of 5.0 mm?

Problem 104E Satellite radio in cars. Refer to the National Association of Broadcasters (NAB) survey of 501 satellite radio subscribers, Exercise 7.70 (p. 392). Recall that an NAB spokesperson claims that 80% of all satellite radio subscribers have a satellite radio receiver in their car. You conducted a test to determine if the claimed value is too high using = .10. What is the probability that the test will conclude that the claim is too high, if in fact the true percentage of all satellite radio subscribers who have a satellite radio receiver in their car is 82%?

Problem 105E Gummi Bears: Red or yellow? Refer to the Chance (Winter 2010) experiment to determine if color of a Gummi Bear is related to its flavor, Exercise 7.72 (p. 392). You tested the null hypothesis of p = .5 against the two-tailed alternative hypothesis of p ? .5 using = .01, where p represents the true proportion of blind folded students who correctly identified the color of the Gummi Bear. Recall that of the 121 students who participated in the study, 97 correctly identified the color. Find the power of the test if the true proportion is p = .65.

Problem 106E Fuel economy of the Honda Civic. According to the Environmental Protection Agency (EPA) Fuel Economy Guide, the 2011 Honda Civic automobile obtains a mean of 36 miles per gallon (mpg) on the highway. Suppose Honda claims that the EPA has underestimated the Civic’s mileage. To support its assertion, the company selects n = 50 model 2011 Civic cars and records the mileage obtained for each car over a driving course similar to the one used by the EPA. The following data resulted: = 38.3 mpg, s = 6.4 mpg. a. If Honda wishes to show that the mean mpg for 2011 Civic autos is greater than 36 mpg, what should the alternative hypothesis be? The null hypothesis? b. Do the data provide sufficient evidence to support the auto manufacturer’s claim? Test using = .05. List any assumptions you make in conducting the test. c. Calculate the power of the test for the mean values of 36.5, 37.0, 37.5, 38.0, and 38.5, assuming s = 6.4 is a good estimate of d. Plot the power of the test on the vertical axis against the mean on the horizontal axis. Draw a curve through the points. e. Use the power curve of part d to estimate the power for the mean value = 37.75. Calculate the power for this value of and compare it to your approximation. f. Use the power curve to approximate the power of the test when = 41. If the true value of the mean mpg for this model is really 41, what (approximately) are the chances that the test will fail to reject the null hypothesis that the mean is 36?

Problem 107E Solder-joint inspections. Refer to Exercise 7.44 (p. 379), in which the performance of a particular type of laserbased inspection equipment was investigated. Assume that the standard deviation of the number of solder joints inspected on each run is 1.2. If = .05 is used in conducting the hypothesis test of interest using a sample of 48 circuit boards, and if the true mean number of solder joints that can be inspected is really equal to 9.5, what is the probability that the test will result in a Type II error?

Problem 108SE Specify the differences between a large-sample and smallsample test of hypothesis about a population mean Focus on the assumptions and test statistics.

Complete the following statement: The smaller the p-value associated with a test of hypothesis, the stronger the support for the _____ hypothesis. Explain your answer.

Which of the elements of a test of hypothesis can and should be specified prior to analyzing the data that are to be used to conduct the test?

If you select a very small value for \(\alpha\) when conducting a hypothesis test, will \(\beta\) tend to be big or small? Explain.

If the rejection of the null hypothesis of a particular test would cause your firm to go out of business, would you want a to be small or large? Explain.

Problem 113SE A random sample of 20 observations selected from a normal population produced = 72.6 and s2 = 19.4.

A random sample of 175 measurements possessed a mean \(\bar{x}=8.2\) and a standard deviation s = .79. a. Test \(H_{0}: \mu=8.3\) against \(H_{\mathrm{a}}: \mu \neq 8.3\). Use \(\alpha=.05\). b. Test \(H_{0}: \mu=8.4\) against \(H_{\mathrm{a}}: \mu \neq 8.4\). Use \(\alpha=.05\). c. Test \(H_{0}: \sigma=1\) against \(H_{\mathrm{a}}: \sigma \neq 1\). Use \(\alpha=.05\). d. Find the power of the test, part a, if \(\mu_{\mathrm{a}}=8.5\). Text Transcription: bar{x} = 8.2 H_0: mu = 8.3 H_a: mu neq 8.3 alpha = .05 H_0: mu = 8.4 H_a: mu neq 8.4 H_0: sigma = 1 H_a: sigma neq 1 mu_a = 8.5

A random sample of n = 200 observations from a binomial population yields \(\hat{p}=.29\). a. \(\text { Test } H_{0}: p=.35 \text { against } H_{\mathrm{a}}: p<.35 \text {. Use } \alpha=.05 \text {. }\) b. \(\text { Test } H_{0}: p=.35 \text { against } H_{a}: p \neq .35 \text {. Use } \alpha=.05\).

A t-test is conducted for the null hypothesis \(H_0:\mu\ =\ 10\) versus the alternative \(H_a: \mu\ >\ 10\) for a random sample of n = 17 observations. The test results are t = 1.174, p-value = .1288. a. Interpret the p-value. b. What assumptions are necessary for the validity of this test? c. Calculate and interpret the p-value assuming the alternative hypothesis was instead \(H_a: \mu\ \neq\ 10\).

Problem 117SE A random sample of 41 observations from a normal population possessed a mean = 88 and a standard deviation s = 6.9.

Problem 118SE Latex allergy in health care workers. Refer to the Current Allergy & Clinical Immunology (Mar. 2004) study of n = 46 hospital employees who were diagnosed with a latex allergy from exposure to the powder on latex gloves, Exercise 6.112 (p. 343). The number of latex gloves used per week by the sampled workers is summarized as follows: = 19.3 and s = 11.9. Let represent the mean number of latex gloves used per week by all hospital employees. Consider testing H0 : = 20 against Ha: < 20. a. Give the rejection region for the test at a significance level of = .01. b. Calculate the value of the test statistic. c. Use the results, parts a and b, to make the appropriate conclusion.

Latex allergy in health care workers (cont'd). Refer to Exercise 7.118. Let \(\sigma^2\) represent the variance in the number of latex gloves used per week by all hospital employees. Consider testing \(H_0: \sigma^2=100\) against \(H_{\mathrm{a}}: \sigma^2 \neq 100\). a. Give the rejection region for the test at a significance level of \(\alpha=.01\). b. Calculate the value of the test statistic. c. Use the results, parts a and b, to make the appropriate conclusion.

Problem 120SE “Made in the USA” survey. Refer to the Journal of Global Business (Spring 2002) study of what “Made in the USA” means to consumers, Exercise 2.152 (p. 116). Recall that 64 of 106 randomly selected shoppers believed “Made in the USA” means 100% of labor and materials are from the United States. Let p represent the true proportion of consumers who believe “Made in the USA” means 100% of labor and materials are from the United States. a. Calculate a point estimate for p. b. A claim is made that p = .70. Set up the null and alternative hypotheses to test this claim. c. Calculate the test statistic for the test, part b. d. Find the rejection region for the test if = .01. e. Use the results, parts c and d, to make the appropriate conclusion.

Problem 121SE Beta value of a stock. The “beta coefficient” of a stock is a measure of the stock’s volatility (or risk) relative to the market as a whole. Stocks with beta coefficients greater than 1 generally bear greater risk (more volatility) than the market, whereas stocks with beta coefficients less than 1 are less risky (less volatile) than the overall market (Alexander, Sharpe, and Bailey, Fundamentals of Investments, 2000). A random sample of 15 high-technology stocks was selected at the end of 2009, and the mean and standard deviation of the beta coefficients were calculated: = 1.23, s = .37. a. Set up the appropriate null and alternative hypotheses to test whether the average high-technology stock is riskier than the market as a whole. b. Establish the appropriate test statistic and rejection region for the test. Use = .10. c. What assumptions are necessary to ensure the validity of the test? d. Calculate the test statistic and state your conclusion. e. What is the approximate p-value associated with this test? Interpret it. f. Conduct a test to determine if the variance of the stock beta values differs from .15. Use = .05.

Accuracy of price scanners at Wal-Mart. Refer to Exercise and the study of the accuracy of checkout scanners at Wal-Mart Stores in California. Recall that the National Institute for Standards and Technology (NIST) mandates that, for every 100 items scanned through the electronic checkout scanner at a retail store, no more than 2 should have an inaccurate price. A study of random items purchased at California Wal-Mart stores found that 8.3% had the wrong price ( Tampa Tribune , Nov. 22, 2005). Assume that the study included 1,000 randomly selected items. a. Identify the population parameter of interest in the study. b. Set up \(H_0\) and \(H_a\) for a test to determine whether the true proportion of items scanned at California Wal-Mart stores exceeds the 2% NIST standard. c. Find the test statistic and rejection region (at \(\alpha=.05\)) for the test. d. Give a practical interpretation of the test. e. What conditions are required for the inference made in part d to be valid? Are these conditions met?

A camera that detects liars. According to New Scientist (Jan. 2, 2002), a new thermal imaging camera that detects small temperature changes is now being used as a polygraph device. The U.S. Department of Defense Polygraph Institute (DDPI) claims the camera can correctly detect liars 75% of the time by monitoring the temperatures of their faces. a. Give the null hypothesis for testing the claim made by the DDPI. b. What is a Type I error for this problem? A Type II error?

Size of diamonds sold at retail. Refer to the Journal of Statistics Education data on diamonds saved in the file. In Exercise 6.120 (p. 344) you selected a random sample of 30 diamonds from the 308 diamonds and found the mean and standard deviation of the number of carats per diamond for the sample. Let \(\mu\) represent the mean number of carats in the population of 308 diamonds. Suppose you want to test \(H_{0}: \mu=.6\) against \(H_{\mathrm{a}}: \mu \neq .6\). a. In the words of the problem, define a Type I error and a Type II error. b. Use the sample information to conduct the test at a significance level of \(\alpha=.05\). c. Conduct the test, part b, using \(\alpha=.10\). d. What do the results suggest about the choice of \(\alpha\) in a test of hypothesis? Text Transcription: mu H_0: mu = .6 H_a: mu neq .6 alpha = .05 alpha = .10 alpha

Consumers’ use of discount coupons. In 1894, druggist Asa Candler began distributing handwritten tickets to his customers for free glasses of Coca-Cola at his soda fountain. That was the genesis of the discount coupon. In 1975, it was estimated that 65% of U.S. consumers regularly used discount coupons when shopping. In a more recent consumer survey, 72% said they regularly redeem coupons (Prospectiv 2008 Consumer Coupon Poll). Assume the recent survey consisted of a random sample of 1,000 shoppers. a. Does the survey provide sufficient evidence that the percentage of shoppers using cents-off coupons exceeds 65%? Test using \(\alpha\ =\ .05\). b. Is the sample size large enough to use the inferential procedures presented in this section? Explain. c. Find the observed significance level for the test you conducted in part a and interpret its value.

Errors in medical tests. Medical tests have been developed to detect many serious diseases. A medical test is designed to minimize the probability that it will produce a “false positive” or a “false negative.” A false positive refers to a positive test result for an individual who does not have the disease, whereas a false negative is a negative test result for an individual who does have the disease. a. If we treat a medical test for a disease as a statistical test of hypothesis, what are the null and alternative hypotheses for the medical test? b. What are the Type I and Type II errors for the test? Relate each to false positives and false negatives. c. Which of the errors has graver consequences? Considering this error, is it more important to minimize \(\alpha\) or \(\beta\) Explain.

Problem 129SE Point spreads of NFL games. Refer to the Chance (Fall 1998) study of point-spread errors in NFL games, Exercise 7.41 (p. 379). Recall that the difference between the actual game outcome and the point spread established by odds makers—the point-spread error—was calculated for 240 NFL games. The results are summarized as follows: = -1.6, s = 13.3. Suppose the researcher wants to know whether the true standard deviation of the pointspread errors exceeds 15. Conduct the analysis using = .10.

Problem 128SE Drivers’ use of the Lincoln Tunnel. The Lincoln Tunnel (under the Hudson River) connects suburban New Jersey to midtown Manhattan. On Mondays at 8:30 a.m., the mean number of cars waiting in line to pay the Lincoln Tunnel toll is 1,220. Because of the substantial wait during rush hour, the Port Authority of New York and New Jersey is considering raising the amount of the toll between 7:30 and 8:30 a.m. to encourage more drivers to use the tunnel at an earlier or later time. Suppose the Port Authority experiments with peak-hour pricing for 6 months, increasing the toll from $4 to $7 during the rush hour peak. On 10 different workdays at 8:30 a.m. aerial photographs of the tunnel queues are taken and the number of vehicles counted. The results follow: Analyze the data for the purpose of determining whether peak-hour pricing succeeded in reducing the average number of vehicles attempting to use the Lincoln Tunnel during the peak rush hour.

Cost-of-living index. Each year, Kiplinger’s complies its list of Best Value Cities. One of the statistics used in the rankings is the cost-of-living index, complied by the U.S. Bureau of Labor Statistics. The index measures the cost of living in a city relative to the national average of 100. In 2011, the New York metropolitan area had a cost-of-living index of 218, the highest in the nation. This means that the cost of living in New York City is 118% higher than the average cost of living for the nation. In contrast, Pueblo, Colorado, had the lowest cost-of-living index (84). The table lists the cost-of-living index for each in a sample of seven Southeastern cities. a. Specify the null and alternative hypotheses for testing whether the true mean cost-of-living index for Southeastern cities differs from the national cost-of-living index of 100. b. What assumptions about the sample and population must hold in order for the test, part a, to yield valid results? c. Conduct the hypothesis test using \(\alpha\ =\ .05\). d. Is the observed significance level of the test greater or less than .05? Justify your answer.

Are manufacturers satisfied with trade promotions? Sales promotions that are used by manufacturers to entice retailers to carry, feature, or push the manufacturer’s products are called trade promotions. A survey of 132 manufacturers conducted by Nielsen found that 36% of the manufacturers were satisfied with their spending for trade promotions (Survey of Trade Promotion Practices, 2004). Is this sufficient evidence to reject a previous claim by the American Marketing Association that no more than half of all manufacturers are dissatisfied with their trade promotion spending? a. Conduct the appropriate hypothesis test at \(\alpha\ =\ .02\). Begin your analysis by determining whether the sample size is large enough to apply the testing methodology presented in this chapter. b. Report the observed significance level of the test and interpret its meaning in the context of the problem. c. Calculate \(\beta\), the probability of a Type II error, if in fact 55% of all manufacturers are dissatisfied with their trade promotion spending.

Improving the productivity of chickens. Refer to the Applied Animal Behaviour Science (Oct. 2000) study of the color of string preferred by pecking domestic chickens, Exercise 6.122 (p. 344). Recall that n = 72 chickens were exposed to blue string and the number of pecks each chicken took at the string over a specified time interval had a mean of \(\bar{x}=1.13\) pecks and a standard deviation of s = 2.21 pecks. Also recall that previous research had shown that \(\mu=7.5\) pecks if chickens are exposed to white string. a. Conduct a test (at \(\alpha\) = .01) to determine if the true mean number of pecks at blue string is less than \(\mu=7.5\) pecks. b. In Exercise 6.122, you used a 99% confidence interval as evidence that chickens are more apt to peck at white string than blue string. Do the test results, part a, support this conclusion? Explain.

Frequency marketing programs by restaurants. To instill customer loyalty, airlines, hotels, rental car companies, and credit card companies (among others) have initiated frequency marketing programs that reward their regular customers. More than 80 million people are members of the frequent flier programs of the airline industry (www.frequentflier.com). A large fast-food restaurant chain wished to explore the profitability of such a program. They randomly selected 12 of their 1,200 restaurants nationwide and instituted a frequency program that rewarded customers with a $5.00 gift certificate after every 10 meals purchased at full price. They ran the trial program for 3 months. The restaurants not in the sample had an average increase in profits of $1,050 over the previous 3 months, whereas the restaurants in the sample had the following changes in profit. Note that the last number is negative, representing a decrease in profits. a. Specify the appropriate null and alternative hypotheses for determining whether the mean profit change for restaurants with frequency programs was significantly greater (in a statistical sense) than $1,050. b. Conduct the test of part b using \(\alpha = .05\). Does it appear that the frequency program would be profitable for the company if adopted nationwide? Text Transcription: alpha = .05

Problem 133SE Arresting shoplifters (cont’d). Refer to Exercise 7.132. a. Describe a Type II error in terms of this application. b. Calculate the probability of a Type II error for this test assuming that the true fraction of shoplifters turned over to the police is p = .55. c. Suppose the number of retailers sampled is increased from 40 to 100. How does this affect the probability of a Type II error for p = .55?

Arresting shoplifters. Shoplifting in the United States costs retailers about $35 million a day. Despite the seriousness of the problem, the National Association of shoplifting Prevention (NASP) claims that only 50% of all shoplifters are turned over to police (www.shopliftingprevention.org). A random sample of 40 U.S. retailers were questioned concerning the disposition of the most recent shoplifter they apprehended. A total of 24 were turned over to police. Do these data provide sufficient evidence to contradict the NASP? a. Conduct a hypothesis test to answer the question of interest. Use \(\alpha\ =\ .05\). b. Is the sample size large enough to use the inferential procedure of part a? c. Find the observed significance level of the hypothesis test in part a. Interpret the value. d. For what values of \(\alpha\) would the observed significance level be sufficient to reject the null hypothesis of the test you conducted in part b?

EPA limits on vinyl chloride. The EPA sets an airborne limit of 5 parts per million (ppm) on vinyl chloride, a colorless gas used to make plastics, adhesives, and other chemicals. It is both a carcinogen and a mutagen (New Jersey Department of Health, Hazardous Substance Fact Sheet, 2010). A major plastics manufacturer, attempting to control the amount of vinyl chloride its workers are exposed to, has given instructions to halt production if the mean amount of vinyl chloride in the air exceeds 3.0 ppm. A random sample of 50 air specimens produced the following statistics: \(\bar{x}=3.1\ ppm\), s = .5 ppm. a. Do these statistics provide sufficient evidence to halt the production process? Use \(\alpha\ =\ .01.\) b. If you were the plant manager, would you want to use a large or a small value for \(\alpha\) for the test in part a? Explain. c. Find the p-value for the test and interpret its value.

Problem 136SE EPA limits vinyl chloride (cont’d). Refer to Exercise 7.135. a. In the context of the problem, define a Type II error. b. Calculate for the test described in part a of Exercise 7.135, assuming that the true mean is = 3.1 ppm. c. What is the power of the test to detect a departure from the manufacturer’s 3.0 ppm limit when the mean is 3.1 ppm? d. Repeat parts b and c assuming that the true mean is 3.2 ppm. What happens to the power of the test as the plant’s mean vinyl chloride level departs further from the limit?

Problem 137SE EPA limits on vinyl chloride (cont’d). Refer to Exercises 7.135 and 7.136. a. Suppose an ? value of .05 is used to conduct the test. Does this change favor halting production? Explain. b. Determine the value of b and the power for the test when ? = .05 and µ = 3.1. c. What happens to the power of the test when ? is increased? 7.135 EPA limits on vinyl chloride. The EPA sets an airborne limit of 5 parts per million (ppm) on vinyl chloride, a colorless gas used to make plastics, adhesives, and other chemicals. It is both a carcinogen and a mutagen (New Jersey Department of Health, Hazardous Substance Fact Sheet, 2010). A major plastics manufacturer, attempting to control the amount of vinyl chloride its workers are exposed to, has given instructions to halt production if the mean amount of vinyl chloride in the air exceeds 3.0 ppm. A random sample of 50 air specimens produced the following statistics: = 3.1 ppm, s = .5 ppm. a. Do these statistics provide sufficient evidence to halt the production process? Use ? = .01. b. If you were the plant manager, would you want to use a large or a small value for ? for the test in part a? Explain. c. Find the p-value for the test and interpret its value. * 7.136 EPA limits vinyl chloride (cont’d). Refer to Exercise 7.135. a. In the context of the problem, define a Type II error. b. Calculate b for the test described in part a of Exercise 7.135, assuming that the true mean is µ = 3.1 ppm. c. What is the power of the test to detect a departure from the manufacturer’s 3.0 ppm limit when the mean is 3.1 ppm? d. Repeat parts b and c assuming that the true mean is 3.2 ppm. What happens to the power of the test as the plant’s mean vinyl chloride level departs further from the limit?

Problem 138SE Evaluating a measuring instrument. One way of evaluating a measuring instrument is to repeatedly measure the same item and compare the average of these measurements to the item’s known measured value. The difference is used to assess the instrument’s accuracy (American Society for Quality). To evaluate a particular Metlar scale, an item whose weight is known to be 16.01 ounces is weighed five times by the same operator. The measurements (in ounces) follow: a. In a statistical sense, does the average measurement differ from 16.01? Conduct the appropriate hypothesis test at = .05. What does your analysis suggest about the accuracy of the instrument? b. List any assumptions you make in conducting the hypothesis test, part a. c. Evaluate the instrument’s precision by testing whether the standard deviation of the weight measurements is greater than .01. Use = .05.

Graduation rates of student-athletes. Are student-athletes at Division I universities poorer students than nonathletes? The National Collegiate Athletic Association (NCAA) measures the academic outcomes of student-athletes with the Graduation Success Rate (GSR). The GSR is measured as the percentage of eligible athletes who graduate within 6 years of entering college. It is well known that the GSR for all students at Division I colleges is 60%. a. Suppose the NCAA reports that in a sample of 500 student-athletes, 315 graduated within 6 years. Is this sufficient information to conclude that the GSR for all student-athletes at Division I institutions differs from 60%? Test using \(\alpha\)=.01 b. The GSR statistics are also broken down by gender and sport. It is known that the GSR for all male college students is 58%. In a sample of 200 male basketball players at Division I institutions, 84 graduated within 6 years. Is this sufficient information to conclude that the GSR for all male basketball players at Division I institutions differs from 58%? Test using \(\alpha\)=.01

Problem 141SE Identifying type of urban land cover. For planning purposes, urban land cover must be identified as either grassland, commercial, or residential. This is typically done using remote sensing data from satellite pictures. In Geographical Analysis (Oct. 2006), researchers from Arizona State, Florida State, and Louisiana State universities collaborated on a new method for analyzing remote sensing data. A satellite photograph of an urban area was divided into 4 x 4 meter areas (called pixels). Of interest is a numerical measure of the distribution of gaps or hole sizes in the pixel, called lacunarity. The mean and standard deviation of the lacunarity measurements for a sample of 100 pixels randomly selected from a specific urban area are 225 and 20, respectively. It is known that the mean lacunarity measurement for all grassland pixels is 220. Do the data suggest that the area sampled is grassland? Test at = .01.

Problem 142SE Ages of cable TV shoppers. In a paper presented at the 2000 Conference of the International Association for Time Use Research, professor Margaret Sanik of Ohio State University reported the results of her study on American cable TV viewers who purchase items from one of the home shopping channels. She found that the average age of these cable TV shoppers was 51 years. Suppose you want to test the null hypothesis, H0: = 51, using a sample of n = 50 cable TV shoppers. a. Find the p-value of a two-tailed test if = 52.3 and s = 7.1. b. Find the p-value of an upper-tailed test if = 52.3 and s = 7.1. c. Find the p-value of a two-tailed test if = 52.3 and s = 10.4. d. For each of the tests, parts a–c, give a value of that will lead to a rejection of the null hypothesis. e. If = 52.3, give a value of s that will yield a two-tailed p-value of .01 or less.

Feminized faces in TV commercials. Television commercials most often employ females or “feminized” males to pitch a company’s product. Research published in Nature (Aug. 27, 1998) revealed that people are, in fact, more attracted to “feminized” faces, regardless of gender. In one experiment, 50 human subjects viewed both a Japanese female face and a Caucasian male face on a computer. Using special computer graphics, each subject could morph the faces (by making them more feminine or more masculine) until they attained the “most attractive” face. The level of feminization x (measured as a percentage) was measured. a. For the Japanese female face, \(\bar{x}\ =\ 10.2%\) and s = 31.3,. The researchers used this sample information to test the null hypothesis of a mean level of feminization equal to 0%. Verify that the test statistic is equal to 2.3. b. Refer to part a. The researchers reported the p-value of the test as p = .021. Verify and interpret this result. c. For the Caucasian male face, \(\bar{x}\ =\ 15.0%\) and s = 25.1%. The researchers reported the test statistic (for the test of the null hypothesis stated in part a) as 4.23 with an associated p-value of approximately 0. Verify and interpret these results.

Problem 144SE Testing the placebo effect. The placebo effect describes the phenomenon of improvement in the condition of a patient taking a placebo—a pill that looks and tastes real but contains no medically active chemicals. Physicians at a clinic in La Jolla, California, gave what they thought were drugs to 7,000 asthma, ulcer, and herpes patients. Although the doctors later learned that the drugs were really placebos, 70% of the patients reported an improved condition. Use this information to test (at = .05) the placebo effect at the clinic. Assume that if the placebo is ineffective, the probability of a patient’s condition improving is .5.

Factors that inhibit learning in marketing. What factors inhibit the learning process in the classroom? To answer this question, researchers at Murray State University surveyed 40 students from a senior-level marketing class (Marketing Education Review). Each student was given a list of factors and asked to rate the extent to which each factor inhibited the learning process in courses offered in their department. A 7-point rating scale was used, where 1 = “not at all” and 7 = “to a great extent.” The factor with the highest rating was instructor related: “Professors who place too much emphasis on a single right answer rather than overall thinking and creative ideas.” Summary statistics for the student ratings of this factor are \(\bar{x}\ =\ 4.70\), s = 1.62. a. Conduct a test to determine if the true mean rating for this instructor-related factor exceeds 4. Use \(\alpha\ =\ .05\). Interpret the test results. b. Examine the results of the study from a practical view, then discuss why “statistically significant” does not always imply “practically significant.” c. Because the variable of interest, rating, is measured on a 7-point scale, it is unlikely that the population of ratings will be normally distributed. Consequently, some analysts may perceive the test, part a, to be invalid and search for alternative methods of analysis. Defend or refute this argument.