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Chapter 9: Problem 16

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 regions are as follows: For the two-tailed test, it's z-scores less than -1.96 or greater than 1.96. For the left-tailed test, it's any z-score less than -2.33. For the right-tailed test, it's any z-score greater than 2.05.

Step by step solution

01

Two-tailed Test

Essentially, a two-tailed test is applied when the null hypothesis is rejected if the test statistic is either greater-than or less-than a critical value. Here, the level of significance is \(\alpha=0.05\). Since this is a two-tailed test, split the \(\alpha\) in half so it equals 0.025 on each side (or tail) of the distribution. Look this value up in the z-table to find the corresponding z-scores. The rejection regions are then z-scores smaller than -1.96 or larger than 1.96.
02

Left-tailed Test

In this situation, a left-tailed test is used when the null hypothesis can be rejected if the test statistic is less than the critical value. Therefore, the entire significance level of \(\alpha=0.01\) is located to the left (or below) the critical value. When looking up this value in the z-table, it indicates a z-score of -2.33. Therefore, the rejection region is any z-score smaller than -2.33.
03

Right-tailed Test

For a right-tailed test, the null hypothesis is rejected if the test statistic is greater than the critical value. Hence the whole significance level of \(\alpha=0.02\) is placed to the right (or above) the critical value. After looking up this value in the z-table, it corresponds to a z-score of 2.05. Therefore, any z-score greater than 2.05 would fall into the rejection region.

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Most popular questions from this chapter

A random sample of 8 observations taken from a population that is normally distributed produced a sample mean of \(44.98\) and a standard deviation of \(6.77\). Find the critical and observed values of \(t\) and the range for the \(p\) -value for each of the following tests of hypotheses, using \(\alpha=.05\). a. \(H_{0}: \mu=50\) versus \(H_{1}: \mu \neq 50\) b. \(H_{0}: \mu=50\) versus \(H_{1}: \mu<50\)

According to a survey by the College Board, undergraduate students at private nonprofit four-year colleges spent an average of \(\$ 1244\) on books and supplies in \(2014-2015\) (www.collegeboard.org). A recent random sample of 200 undergraduate college students from a large private nonprofit four-year college showed that they spent an average of \(\$ 1204\) on books and supplies during the last academic year. Assume that the standard deviation of annual expenditures on books and supplies by all such students at this college is \(\$ 200\). a. Find the \(p\) -value for the test of hypothesis with the alternative hypothesis that the annual mean expenditure by all such students at this college is less than \(\$ 1244\). Based on this \(p\) -value, would you reject the null hypothesis if the significance level is \(.025\) ? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.025 .\) Would you reject the null hypothesis? Explain.

\(\quad\) Consider \(H_{0}: \mu=100\) versus \(H_{1}: \mu \neq 100\). a. A random sample of 64 observations produced a sample mean of \(98 .\) Using \(\alpha=.01\), would you reject the null hypothesis? The population standard deviation is known to be \(12 .\) b. Another random sample of 64 observations taken from the same population produced a sample mean of 104 . Using \(\alpha=.01\), would you reject the null hypothesis? The population standard deviation is known to be \(12 .\) Comment on the results of parts a and \(\mathrm{b}\).

You read an article that states "50 hypothesis tests of \(H_{0}\) - \(\mu=35\) versus \(H_{1}: \mu \neq 35\) were performed using \(\alpha=.05\) on 50 different samples taken from the same population with a mean of \(35 .\) Of these, 47 tests failed to reject the null hypothesis." Explain why this type of result is not surprising.

Explain which of the following is a two-tailed test, a left-tailed test, or a right-tailed test. a. \(H_{0}: \mu=12, H_{1}: \mu<12\) b. \(H_{0}: \mu \leq 85, H_{1}: \mu>85\) c. \(H_{0}: \mu=33, H_{1}: \mu \neq 33\) Show the rejection and nonrejection regions for each of these cases by drawing a sampling distribution curve for the sample mean, assuming that it is normally distributed.
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