9.16 A bottled water distributor wants to determine whether the mean amount of water contained in 1-gallon bottles purchased from a nationally known water bottling company is actually 1 gallon. You know from the water bottling company specifications that the standard deviation of the amount of water per bottle is 0.02 gallon. You select a random sample of 50 bottles, and the mean amount of water per 1-gallon bottle is 0.995 gallon. a. Is there evidence that the mean amount is different from 1.0 gallon? (Use a = 0.01 ) b. Compute the p-value and interpret its meaning. c. Construct a 99% confidence interval estimate of the population mean amount of water per bottle. d. Compare the results of (a) and (c). What conclusions do you reach?9.51 The population mean waiting time to check out of a super- market has been 4 minutes. Recently, in an effort to reduce the waiting time, the supermarket has experimented with a system in which infrared cameras use body heat and in-store software to de- termine how many lanes should be opened. A sample of 100 cus- tomers was selected, and their mean waiting time to check out was 3.25 minutes, with a sample standard deviation of 2.7 minutes. a. At the 0.05 level of significance, using the critical value ap- proach to hypothesis testing, is there evidence that the popu- lation mean waiting time to check out is less than 4 minutes? b. At the 0.05 level of significance, using the p-value approach to hypothesis testing, is there evidence that the population mean waiting time to check out is less than 4 minutes? c. Interpret the meaning of the p-value in this problem. d. Compare your conclusions in (a) and (b).9.60 Actuation Consulting conducted a global survey of prod- uct teams with the goal of better understanding the dynamics of product team performance and uncovering the practices that make these teams successful. One of the survey findings was that 42% of organizations consider the project manager role vital to successful completion of product development activities (Source: The Study of Product Team Performance, 2014, bit.ly/1GHTTOM.) Suppose another study is conducted to check the validity of this result, with the goal of proving that the percentage is less than 42%. a. State the null and research hypotheses. b. A sample of 100 organizations is selected, and results indicate that 40 consider the project manager role vital to successful completion of product development activities. Use either the six-step critical value hypothesis testing approach or the five- step p-value approach to determine at the 0.05 level of signif- icance whether there is evidence that the percentage is less than 42%.9.69 In hypothesis testing, the common level of significance is a =0.05. Some might argue for a level of significance greater than 0.05. Suppose that web designers tested the proportion of potential web page visitors with a preference for a new web design over the existing web design. The null hypothesis was that the popula- tion proportion of web page visitors preferring the new design was 0.50, and the alternative hypothesis was that it was not equal to 0.50. The p-value for the test was 0.20. a. State, in statistical terms, the null and alternative hypotheses for this example. b. Explain the risks associated with Type I and Type II errors in this case. c. What would be the consequences if you rejected the null hypothesis for a p-value of 0.20? d. What might be an argument for raising the value of a? e. What would you do in this situation? f. What is your answer in (e) if the p-value equals 0.12? What if it equals 0.06?9.70 Financial institutions utilize prediction models to predict bankruptcy. One such model is the Altman Z-score model, which uses multiple corporate income and balance sheet values to mea- sure the financial health of a company. If the model predicts a low Z-score value, the firm is in financial stress and is predicted to go bankrupt within the next two years. If the model predicts a moderate or high Z-score value, the firm is financially healthy and is predicted to be a non-bankrupt firm (see pages.stern.nyu .edu/~ealtman/Zscores.pdf). This decision-making procedure can be expressed in the hypothesis-testing framework. The null hypothesis is that a firm is predicted to be a non-bankrupt firm. The alternative hypothesis is that the firm is predicted to be a bankrupt firm. a. Explain the risks associated with committing a Type I error in this case. b. Explain the risks associated with committing a Type II error in this case. c. Which type of error do you think executives want to avoid? Explain. d. How would changes in the model affect the probabilities of committing Type I and Type II errors?9.73 An auditor for a government agency was assigned the task of evaluating reimbursement for office visits to physicians paid by Medicare. The audit was conducted on a sample of 75 reimburse- ments, with the following results: . In 12 of the office visits, there was an incorrect amount of reimbursement. . The amount of reimbursement was X = $93.70, S = $34.55. a. At the 0.05 level of significance, is there evidence that the pop- ulation mean reimbursement was less than $100? b. At the 0.05 level of significance, is there evidence that the proportion of incorrect reimbursements in the population was greater than 0.10? c. Discuss the underlying assumptions of the test used in (a). d. What is your answer to (a) if the sample mean equals $90? e. What is your answer to (b) if 15 office visits had incorrect reimbursements?9.74 A bank branch located in a commercial district of a city has the business objective of improving the process for serving cus- tomers during the noon-to-1:00 p.m. lunch period. The waiting time (defined as the time the customer enters the line until he or she reaches the teller window) of a random sample of 15 custom- ers is collected, and the results are organized and stored in Bank1 These data are: 4.21 5.55 3.02 5.13 4.77 2.34 3.54 3.20 4.50 6.10 0.38 5.12 6.46 6.19 3.79 a. At the 0.05 level of significance, is there evidence that the pop- ulation mean waiting time is less than 5 minutes? b. What assumption about the population distribution is needed in order to conduct the t test in (a)? c. Construct a boxplot or a normal probability plot to evaluate the assumption made in (b). d. Do you think that the assumption needed in order to conduct the t test in (a) is valid? Explain. e. As a customer walks into the branch office during the lunch hour, she asks the branch manager how long she can expect to wait. The branch manager replies, "Almost certainly not longer than 5 minutes." On the basis of the results of (a), evaluate this statement.9.75 A manufacturing company produces electrical insulators. If the insulators break when in use, a short circuit is likely to occur. To test the strength of the insulators, destructive testing is carried out to determine how much force is required to break the insulators. Force is measured by observing the number of pounds of force applied to the insulator before it breaks. The following data (stored in Force ) are from 30 insulators sub- jected to this testing: 1,870 1,728 1,656 1,610 1,634 1,784 1,522 1,696 1,592 1,662 1,866 1,764 1,734 1,662 1,734 1,774 1,550 1,756 1,762 1,866 1,820 1,744 1,788 1,688 1,810 1,752 1,680 1,810 1,652 1,736 a. At the 0.05 level of significance, is there evidence that the pop- ulation mean force required to break the insulator is greater than 1,500 pounds? b. What assumption about the population distribution is needed in order to conduct the t test in (a)? c. Construct a histogram, boxplot, or normal probability plot to evaluate the assumption made in (b). d. Do you think that the assumption needed in order to conduct the t test in (a) is valid? Explain.9.77 Studies conducted by the manufacturer of Boston and Ver- mont asphalt shingles have shown product weight to be a major factor in the customer's perception of quality. Moreover, the weight represents the amount of raw materials being used and is therefore very important to the company from a cost standpoint. The last stage of the assembly line packages the shingles before the packages are placed on wooden pallets. Once a pallet is full (a pallet for most brands holds 16 squares of shingles), it is weighed, and the measurement is recorded. The file Pallet contains the weight (in pounds) from a sample of 368 pallets of Boston shin- gles and 330 pallets of Vermont shingles. a. For the Boston shingles, is there evidence at the 0.05 level of significance that the population mean weight is different from 3,150 pounds? b. Interpret the meaning of the p-value in (a). c. For the Vermont shingles, is there evidence at the 0.05 level of significance that the population mean weight is different from 3,700 pounds? d. Interpret the meaning of the p-value in (c). e. In (a) through (d), do you have to be concerned with the nor- mality assumption? Explain.9.32 A manufacturing company produces steel housings for electrical equipment. The main component part of the housing is a steel trough that is made out of a 14-gauge steel coil. It is pro- duced using a 250-ton progressive punch press with a wipe-down operation that puts two 90-degree forms in the flat steel to make the trough. The distance from one side of the form to the other is critical because of weatherproofing in outdoor applications. The company requires that the width of the trough be between 8.31 inches and 8.61 inches. The file Trough contains the widths of the troughs, in inches, for a sample of n = 49: 8.312 8.343 8.317 8.383 8.348 8.410 8.351 8.373 8.481 8.422 8.476 8.382 8.484 8.403 8.414 8.419 8.385 8.465 8.498 8.447 8.436 8.413 8.489 8.414 8.481 8.415 8.479 8.429 8.458 8.462 8.460 8.444 8.429 8.460 8.412 8.420 8.410 8.405 8.323 8.420 8.396 8.447 8.405 8.439 8.411 8.427 8.420 8.498 8.409 a. At the 0.05 level of significance, is there evidence that the mean width of the troughs is different from 8.46 inches? D. What assumption about the population distribution is needed in order to conduct the t test in (a)? c. Evaluate the assumption made in (b). d. Do you think that the assumption needed in order to conduct the t test in (a) is valid? Explain