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What are the four possible outcomes for a test of hypothesis? Show these outcomes by writing a table. Briefly describe the Type I and Type II errors.

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.

Explain which of the following is a two-tailed test, a left-tailed test, or a right-tailed test. a. \(H_{0}: \mu=45, \quad H_{1}: \mu>45\) b. \(H_{0}: \mu=23, \quad H_{1}: \mu \neq 23\) c. \(H_{0}: \mu \geq 75, \quad H_{1}: \mu<75\)

A random sample of 80 observations produced a sample mean of \(86.50\). Find the critical and observed values of \(z\) for each of the following tests of hypothesis using \(\alpha=.10 .\) The population standard deviation is known to be \(7.20\). a. \(H_{0}: \mu=91\) versus \(H_{1}: \mu \neq 91\) b. \(H_{0}: \mu=91 \quad\) versus \(\quad H_{1}: \mu<91\)

A study claims that all adults spend an average of 14 hours or more on chores during a weekend. A researcher wanted to check if this claim is true. A random sample of 200 adults taken by this researcher showed that these adults spend an average of \(14.65\) hours on chores during a weekend. The population standard deviation is known to be \(3.0\) hours. a. Find the \(p\) -value for the hypothesis test with the alternative hypothesis that all adults spend more than 14 hours on chores during a weekend. Will you reject the null hypothesis at \(\alpha=.01\) ? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.01\).

A telephone company claims that the mean duration of all long-distance phone calls made by its residential customers is 10 minutes. A random sample of 100 long-distance calls made by its residential customers taken from the records of this company showed that the mean duration of calls for this sample is \(9.20\) minutes. The population standard deviation is known to be \(3.80\) minutes. a. Find the \(p\) -value for the test that the mean duration of all long- distance calls made by residential customers is different from 10 minutes. If \(\alpha=.02\), based on this \(p\) -value, would you reject the null hypothesis? Explain. What if \(\alpha=.05\) ? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.02\). Does your conclusion change if \(\alpha=.05 ?\)

Records in a three-county area show that in the last few years, Girl Scouts sell an average of \(47.93\) boxes of cookies per year, with a population standard deviation of \(8.45\) boxes per year. Fifty randomly selected Girl Scouts from the region sold an average of \(46.54\) boxes this year. Scout leaders are concerned that the demand for Girl Scout cookies may have decreased. a. Test at the \(10 \%\) significance level whether the average number of boxes of cookies sold by all Girl Scouts in the three-county area is lower than the historical average. b. What will your decision be in part a if the probability of a Type I error is zero? Explain.

A study claims that all homeowners in a town spend an average of 8 hours or more on house cleaning and gardening during a weekend. A researcher wanted to check if this claim is true. A random sample of 20 homeowners taken by this researcher showed that they spend an average of \(7.68\) hours on such chores during a weekend. The population of such times for all homeowners in this town is normally distributed with the population standard deviation of \(2.1\) hours. a. Using the \(1 \%\) significance level, can you conclude that the claim that all homeowners spend an average of 8 hours or more on such chores during a weekend is false? Use both approaches. b. Make the test of part a using a \(2.5 \%\) significance level. Is your decision different from the one in part a? Comment on the results of parts a and \(b\).

A company claims that the mean net weight of the contents of its All Taste cereal boxes is at least 18 ounces. Suppose you want to test whether or not the claim of the company is true. Explain briefly how you would conduct this test using a large sample. Assume that \(\sigma=.25\) ounce.

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 \(n=22\)