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Chapter 9: Problem 20

For each of the following examples of tests of hypotheses about \(\mu\), show the rejection and nonrejection regions on the sampling distribution of the sample mean assuming that it is normal. a. A two-tailed test with \(\alpha=.05\) and \(n=40\) b. A left-tailed test with \(\alpha=.01\) and \(n=20\) c. A right-tailed test with \(\alpha=.02\) and \(n=55\)

Short Answer

Expert verified
The rejection and non-rejection regions for the three tests are split around the critical values as follows: For test a, the rejection region is found at Z-scores above 1.96 or below -1.96. The non-rejection region lies between -1.96 and 1.96. For test b, the rejection region is found at Z-scores below -2.33 and the non-rejection region lies above this score. For test c, the rejection region is found at Z-scores above 2.05, and the non-rejection region lies below this score.

Step by step solution

01

Identify Test Type for a

Test a requires a two-tailed test, meaning we split the significance level (\(\alpha = .05\)) equally between the two tails of the distribution, giving us \(.025\) in each tail.
02

Calculate Critical Values for a

Using a standard normal distribution table, determine Z-scores for \(.025\) in each tail. You should find the Z-scores of approximately \(\pm 1.96\). These are the critical values that separate the rejection area from the non-rejection area.
03

Identify Test Type for b and for c

For test b, we use a left-tailed test at \(\alpha = .01\) and for test c, we use a right-tailed test at \(\alpha = .02\). This means that the entire significance level is in one tail of the distribution.
04

Calculate Critical Values for b and for c

As we did before, use the standard normal distribution table to find the Z-scores. For test b, we find the Z-score for .01 in the left tail, which is approximately \(-2.33\). For test c, we find the Z-score for .02 in the right tail, which is approximately \(2.05\). These are the critical values.

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Most popular questions from this chapter

Make the following tests of hypotheses. a. \(H_{0}: \mu=25, \quad H_{1}: \mu \neq 25, \quad n=81, \quad \bar{x}=28.5, \quad \sigma=3, \quad \alpha=.01\) b. \(H_{0}: \mu=12, \quad H_{1}: \mu<12, \quad n=45, \quad \bar{x}=11.25, \quad \sigma=4.5, \quad \alpha=.05\) c. \(H_{0}: \mu=40, \quad H_{1}: \mu>40, \quad n=100, \quad \bar{x}=47, \quad \sigma=7, \quad \alpha=.10\)

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