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Chapter 7: Q23E (page 403)
In a test of against, the sample data yielded the test statistic z = 2.17. Find and interpret the p-value for the test.
The p-value is 0.0150, representing the probability of the test statistic being more than 2.17 when the claim is true.
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FDA certification of new drugs. Pharmaceutical companies spend billions of dollars 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?
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. 339). 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 shown above gives the results of the test. Locate the test statistic and p-value on the printout.
d. At α = .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.
Jury trial outcomes. Sometimes, the outcome of a jury trial defies the “common sense” expectations of the general public (e.g., the 1995 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 case) 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 α and β ; 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 ofα increase or decrease? Explain.
f. For a jury prejudiced against a guilty verdict as the trial begins, will the value of β increase or decrease? Explain
Customers who participate in a store’s free loyalty card program save money on their purchases but allow the store to keep track of the customer’s shopping habits and potentially sell these data to third parties. A Pew Internet & American Life Project Survey (January 2016) revealed that 225 of a random sample of 250 U.S. adults would agree to participate in a store loyalty card program, despite the potential for information sharing. Let represent the true proportion of all customers who would participate in a store loyalty card program.
a. Compute a point estimate of
b. Consider a store owner who claims that more than 80% of all customers would participate in a loyalty card program. Set up the null and alternative hypotheses for testing whether the true proportion of all customers who would participate in a store loyalty card program exceeds .8
c. Compute the test statistic for part b.
d. Find the rejection region for the test if .
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.
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 was 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 \(Sx\) = \)2,185), while females with a postgraduate degree have a mean of \(32,227 (with standard error \(Sx\) = \)932).
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