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Chapter 9: Problem 82

Consider the null hypothesis \(H_{0}: p=.25 .\) Suppose a random sample of 400 observations is taken to perform this test about the population proportion. Using \(\alpha=.01\), show the rejection and nonrejection regions and find the critical value(s) of \(z\) for a a. left-tailed test b. two-tailed test c. right-tailed test

Short Answer

Expert verified
For the given \(\alpha = .01\) and random sample of size 400, the critical value of z is -2.33 for a left-tailed test, -2.58 and 2.58 for a two-tailed test, and 2.33 for a right-tailed test. The null hypothesis is rejected if the value of z lies in the rejection region (less than -2.33 for left-tailed; less than -2.58 or greater than 2.58 for two-tailed; greater than 2.33 for right-tailed), and not rejected if the value of z lies in the nonrejection region.

Step by step solution

01

Identify and compute the critical values

The critical value of z is the threshold for the rejection region. For a left-tailed test, it is negative; for a two-tailed test, it has both positive and negative values; for a right-tailed test, it is positive. So,a. left-tailed test: Find the critical value corresponding to \(\alpha = .01\). You can use the z-table or online calculator to get \(z = -2.33\). b. two-tailed test: Here, \(\alpha\) is divided into two, that is \(.01/2 = .005\) on each side. So, the critical values are \(-2.58\) and \(2.58\).c. right-tailed test: The critical value is the value which the test statistic must exceed for \(H_{0}\) to be rejected. For a right-tailed test, it's found the same way as a left-tailed test but the sign is positive, hence \(z = 2.33\).
02

Determine the rejection region

The rejection region depends on the location of the critical values.a. left-tailed test: The rejection region is \(z < -2.33\)b. two-tailed test: There are two rejection regions: \(z < -2.58\) or \(z > 2.58\)c. right-tailed test: The rejection region is \(z > 2.33\)
03

Determine the nonrejection region

The nonrejection (or acceptance) region is the range of values which supports the null hypothesis.a. left-tailed test: The nonrejection region is \(z \geq -2.33\)b. two-tailed test: The nonrejection region is \(-2.579 \leq z \leq 2.58\)c. right-tailed test: The nonrejection region is \(z \leq 2.33\)

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Most popular questions from this chapter

A 2008 AARP survey reported that \(85 \%\) of U.S. workers aged 50 years and older with at least one 4-year college degree had taken employer-based training within the previous 2 years, compared to only \(50 \%\) of workers aged 50 years and older with a high school degree or less. In a current survey of \(640 \mathrm{U.S}\). workers aged 50 years and older with a high school degree or less, 341 had taken employer-based training within the previous 2 years. a. Using the critical-value approach and \(\alpha=.05\), test whether the current percentage of all U.S. workers aged 50 years and older with a high school degree or less who have taken employerbased training within the previous 2 years is different from \(50 \%\). b. How do you explain the Type I error in part a? What is the probability of making this error in part a? c. Calculate the \(p\) -value for the test of part a. What is your conclusion if \(\alpha=.05 ?\)

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