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Chapter 9: Problem 3

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.

Short Answer

Expert verified
In a two-tailed test, the null hypothesis assumes the parameter equals a specific value while the alternative hypothesis states it does not equal that value. In a left-tailed test, the null hypothesis assumes the parameter is greater or equal to a specific value while the alternative hypothesis states it's less than that value. In a right-tailed test, the null hypothesis assumes the parameter is less or equal to a specific value while the alternative hypothesis states it's greater than that value.

Step by step solution

01

Two-Tailed Test

For a two-tailed test, the alternative hypothesis assumes a difference from a stated value, but it does not specify the direction of the difference. This means the null hypothesis (\(H_0\)) would be \(H_0: \mu = \mu_0\), implying that there is no change or the parameter equals the specific value. While the alternative hypothesis (\(H_a\) or \(H_1\)), would be \(H_1: \mu \neq \mu_0\), stating the parameter is not equal to the specific value.
02

Left-Tailed Test

In the left-tailed test, the alternative hypothesis assumes that the parameter is less than a specific value. This implies that the null hypothesis (\(H_0\)) would be \(H_0: \mu \geq \mu_0\), stating that the parameter is greater than or equal to the specific value, while the alternative hypothesis would be \(H_1: \mu < \mu_0\), stating that the parameter is less than the specific value.
03

Right-Tailed Test

In a right-tailed test, the alternative hypothesis assumes that the parameter is greater than a specific value. So the null hypothesis (\(H_0\)) would be \(H_0: \mu \leq \mu_0\), stating that the parameter is less than or equal to the specific value. Conversely, the alternative hypothesis would be \(H_1: \mu > \mu_0\), stating that the parameter is greater than the specific value.

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Most popular questions from this chapter

\(\quad\) 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 \(\mathrm{b}\).

Write the null and alternative hypotheses for each of the following examples. Determine if each is a case of a two-tailed, a left-tailed, or a right-tailed test. a. To test if the mean amount of time spent per week watching sports on television by all adult men is different from \(9.5\) hours b. To test if the mean amount of money spent by all customers at a supermarket is less than \(\$ 105\) c. To test whether the mean starting salary of college graduates is higher than \(\$ 47,000\) per year d. To test if the mean waiting time at the drive-through window at a fast food restaurant during rush hour differs from 10 minutes e. To test if the mean time spent per week on house chores by all housewives is less than 30 hours

A study claims that \(65 \%\) of students at all colleges and universities hold off-campus (part-time or full-time) jobs. You want to check if the percentage of students at your school who hold off-campus jobs is different from \(65 \%\). Briefly explain how you would conduct such a test. Collect data from 40 students at your school on whether or not they hold off-campus jobs. Then, calculate the proportion of students in this sample who hold off-campus jobs. Using this information, test the hypothesis. Select your own significance level.

Consider \(H_{0}: p=.45\) versus \(H_{1}: p<.45\). a. A random sample of 400 observations produced a sample proportion equal to .42. Using \(\alpha=.025\), would you reject the null hypothesis? b. Another random sample of 400 observations taken from the same population produced a sample proportion of .39. Using \(a=.025\), would you reject the null hypothesis? Comment on the results of parts a and \(\mathrm{b}\).

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 \(n=55\)
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