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Chapter 9: Problem 49

For each of the following examples of tests of hypothesis about \(\mu\), show the rejection and nonrejection regions on the \(t\) distribution curve. a. A two-tailed test with \(\alpha=.01\) and \(n=15\) b. A left-tailed test with \(\alpha=.005\) and \(n=25\) c. A right-tailed test with \(\alpha=.025\) and \(n=22\)

Short Answer

Expert verified
Rejection regions for the given tests are: a) \(t< -2.977\) and \(t > 2.977\), b) \(t < -2.492\), and c) \(t > 2.080\). Beyond these t-values, the null hypothesis would be rejected.

Step by step solution

01

Hypothesis Testing Overview

Hypothesis testing is a statistical method that is used in making statistical decisions using experimental data. It involves the statement of a null hypothesis, and the selection of a level of significance. The hypothesis is assumed to be true until statistical evidence in the form of a hypothesis test indicates otherwise.
02

Two-Tailed Test

A two-tailed test with an alpha level of .01 and sample size of 15 means we're checking for an extreme value in either tail of the distribution. We reject the null hypothesis if our test statistic is in the top 0.005 (i.e., \(\alpha/2\)) or the bottom 0.005. To determine the critical value, we also need to calculate the degrees of freedom, which is \(n-1\) or 14. Looking up these values in a t-table, we have a critical value of about \(\pm2.977\). Any t-score beyond these values falls in the rejection region.
03

Left-Tailed Test

A left-tailed test with an alpha level of .005 and n = 25 implies we're checking for an extreme value only in the left tail of the distribution. We reject the null hypothesis if our test statistic is in the bottom 0.005. Degrees of freedom are 24. From a t-table, the critical value is about -2.492, so any t-score below this is in the rejection region.
04

Right-Tailed Test

Right-tailed test with an alpha level of .025 and n = 22 implies we're checking for an extreme value only in the right tail of the distribution. We reject the null hypothesis if our test statistic is in the top 0.025. Degrees of Freedom this time are 21. From the t-table, the critical value is roughly 2.080, hence any t-score greater than this falls in the rejection region.

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Most popular questions from this chapter

According to the American Diabetes Association (www.diabetes.org), \(23.1 \%\) of Americans aged 60 years or older had diabetes in 2007. A recent random sample of 200 Americans aged 60 years or older showed that 52 of them have diabetes. Using a \(5 \%\) significance level, perform a test of hypothesis to determine if the current percentage of Americans aged 60 years or older who have diabetes is higher than that in 2007 . Use both the \(p\) -value and the critical-value approaches.

Thirty percent of all people who are inoculated with the current vaccine used to prevent a disease contract the disease within a year. The developer of a new vaccine that is intended to prevent this disease wishes to test for significant evidence that the new vaccine is more effective. a. Determine the appropriate null and alternative hypotheses. b. The developer decides to study 100 randomly selected people by inoculating them with the new vaccine. If 84 or more of them do not contract the disease within a year, the developer will conclude that the new vaccine is superior to the old one. What significance level is the developer using for the test? c. Suppose 20 people inoculated with the new vaccine are studied and the new vaccine is concluded to be better than the old one if fewer than 3 people contract the disease within a year. What is the significance level of the test?

Explain which of the following is a two-tailed test, a left-tailed test, or a right-tailed test. a. \(H_{0}: \mu=45, \quad H_{1}: \mu>45\) b. \(H_{0}: \mu=23, \quad H_{1}: \mu \neq 23\) c. \(H_{0}: \mu \geq 75, \quad H_{1}: \mu<75\)

Explain how the tails of a test depend on the sign in the alternative hypothesis. Describe the signs in the null and alternative hypotheses for a two-tailed, a left-tailed, and a right-tailed test, respectively.

In a 2009 nonscientific poll on the Web site of the Daily Gazette of Schenectady, New York, readers were asked the following question: "Are you less inclined to buy a General Motors or Chrysler vehicle now that they have filed for bankruptcy?" Of the respondents, \(56.1 \%\) answered "Yes" (http://www. dailygazette.com/polls/2009/jun/Bankruptcy/). In a recent survey of 1200 adult Americans who were asked the same question, 615 answered "Yes." Can you reject the null hypothesis at the \(1 \%\) significance level in favor of the alternative that the percentage of all adult Americans who are less inclined to buy a General Motors or Chrysler vehicle since the companies filed for bankruptcy is different from \(56.1 \%\) ? Use both the \(p\) -value and the critical-value approaches.
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