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Chapter 7: Q20E (page 403)
If a hypothesis test were conducted using 伪= 0.05, 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
The null hypothesis would be rejected for every case because all of the given p-values are less than the 伪 value.
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For each of the following rejection regions, sketch the sampling distribution for z and indicate the location of the rejection region.
a. \({H_0}:\mu \le {\mu _0}\) and \({H_a}:\mu > {\mu _0};\alpha = 0.1\)
b. \({H_0}:\mu \le {\mu _0}\) and \({H_a}:\mu > {\mu _0};\alpha = 0.05\)
c. \({H_0}:\mu \ge {\mu _0}\) and \({H_a}:\mu < {\mu _0};\alpha = 0.01\)
d. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.05\)
e. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.1\)
f. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.01\)
g. For each rejection region specified in parts a鈥揻, state the probability notation in z and its respective Type I error value.
Trading skills of institutional investors. The trading skills of institutional stock investors were quantified and analyzed in The Journal of Finance (April 2011). The study focused on 鈥渞ound-trip鈥� 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\(\alpha = 0.05\).
c. Interpret the value of\(\alpha \)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 鈮� t.)
e. Give the appropriate conclusion in the words of the problem.
Jury trial outcomes. Sometimes, the outcome of a jury trial defies the 鈥渃ommon sense鈥� 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 鈥渘ot guilty鈥� verdict. The court never accepts the null hypothesis; that is, the court never declares the accused 鈥渋nnocent.鈥� A 鈥渘ot guilty鈥� 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 伪 and 尾 ; 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伪 increase or decrease? Explain.
f. For a jury prejudiced against a guilty verdict as the trial begins, will the value of 尾 increase or decrease? Explain
Refer to Exercise 6.44 (p. 356), in which 50 consumers taste-tested a new snack food. Their responses (where 0 = do not like; 1 = like; 2 = indifferent) are reproduced below
鈥淪treaming鈥� of television programs is trending upward. According to The Harris Poll (August 26, 2013), over one-third of American鈥檚 qualify as 鈥渟ubscription streamers,鈥� i.e., those who watch streamed TV programs through a subscription service such as Netflix, Hulu Plus, or Amazon Prime. The poll included 2,242 adult TV viewers, of which 785 are subscription streamers. On the basis of this result, can you conclude that the true fraction of adult TV viewers who are subscription streamers differs from one-third? Carry out the test using a Type I error rate of . Be sure to give the null and alternative hypotheses tested, test statistic value, rejection region or p-value, and conclusion.
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