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Chapter 7: Q2E (page 397)
Which element of a test of hypothesis is used to decide whether to reject the null hypothesis in favor of the alternative hypothesis?
Rejection region is the element of a test of hypothesis used to decide about the rejection of null hypothesis.
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In a test of \({H_0}:\mu = 100\) against \({H_a}:\mu \ne 100\), the sample data yielded the test statistic z = 2.17. Find the p-value for the test.
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
Americans鈥� favorite sport. The Harris Poll (December 2013) conducted an online survey of American adults to determine their favorite sport. Your friend believes professional (National Football League [NFL]) football鈥攚ith revenue of about $13 billion per year鈥攊s the favorite sport for 40% of American adults. Specify the null and alternative hypotheses for testing this belief. Be sure to identify the parameter of interest.
A simple random sample of 25 observations was selected from a normal population. The mean and standard deviation of this sample are 20 and 5, respectively.
a. Test against at the 10% significance level.
b. Test against at the 1% significance level.
Specify the differences between a large-sample and a small-sample test of a hypothesis about a population mean m. Focus on the assumptions and test statistics.
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