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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

Consider the following null and alternative hypotheses: $$ H_{0}: \mu=40 \quad \text { versus } \quad H_{1}: \mu \neq 40 $$ A random sample of 64 observations taken from this population produced a sample mean of \(38.4\). The population standard deviation is known to be \(6 .\) a. If this test is made at the \(2 \%\) significance level, would you reject the null hypothesis? Use the critical-value approach. b. What is the probability of making a Type I error in part a? c. Calculate the \(p\) -value for the test. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=01\) ? What if \(\alpha=05\) ?

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