91Ó°ÊÓ

Make the following tests of hypotheses. a. \(H_{0}: \mu=80, \quad H_{1}: \mu \neq 80, \quad n=33, \quad \bar{x}=76.5, \quad \sigma=15, \quad \alpha=.10\) b. \(H_{0}=\mu=32, \quad H_{1}: \mu<32, \quad n=75, \quad \bar{x}=26.5, \quad \sigma=7.4, \quad \alpha=.01\) c. \(H_{0}=\mu=55, \quad H_{1}: \mu>55, \quad n=40, \bar{x}=60.5, \quad \sigma=4, \quad \alpha=.05\)

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 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.

Consider the null hypothesis \(H_{0}: \mu=12.80 .\) A random sample of 58 observations is taken from this population to perform this test. Using \(\alpha=.05\), show the rejection and nonrejection regions on the sampling distribution curve of the sample mean and find the critical value(s) of \(t\) for the following. a. a right-tailed test \(\underline{\text { b. a left-tailed test }}\) c. a two-tailed test

Consider \(H_{0^{\circ}} \mu=80\) versus \(H_{1}: \mu \neq 80\) for a population that is normally distributed. a. A random sample of 25 observations taken from this population produced a sample mean of 77 and a standard deviation of 8 . Using \(\alpha=.01\), would you reject the null hypothesis? b. Another random sample of 25 observations taken from the same population produced a sample mean of 86 and a standard deviation of \(6 .\) Using \(\alpha=.01\), would you reject the null hypothesis?

Consider \(H_{0}=\mu=40\) versus \(H_{1}: \mu>40 .\) a. A random sample of 64 observations taken from this population produced a sample mean of 43 and a standard deviation of \(5 .\) Using \(\alpha=.025\), would you reject the null hypothesis? b. Another random sample of 64 observations taken from the same population produced a sample mean of 41 and a standard deviation of \(7 .\) Using \(\alpha=.025\), would you reject the null hypothesis? Comment on the results of parts a and \(\mathrm{b}\).

Perform the following tests of hypothesis. a. \(H_{0}: \mu=285, \quad H_{1}: \mu<285\) \(n=55, \quad \bar{x}=267.80, \quad s=42.90, \quad \alpha=.05\) b. \(H_{0-\mu}=10.70, \quad H_{1}: \mu \neq 10.70, \quad n=47, \bar{x}=12.025, \quad s=4.90, \quad \alpha=.01\) c. \(H_{0}=\mu=147,500, \quad H_{1}: \mu>147,500, n=41, \bar{x}=149,812, s=22,972, \alpha=.10\)

The police that patrol a heavily traveled highway claim that the average driver exceeds the 65 miles per hour speed limit by more than 10 miles per hour. Seventy-two randomly selected cars were clocked by airplane radar. The average speed was \(77.40\) miles per hour, and the standard deviation of the speeds was \(5.90\) miles per hour. Find the range for the \(p\) -value for this test. What will your conclusion be using this \(p\) -value range and \(\alpha=.02\) ?

A soft-drink manufacturer claims that its 12 -ounce cans do not contain, on average, more than 30 calories. A random sample of 64 cans of this soft drink, which were checked for calories, contained a mean of 32 calories with a standard deviation of 3 calories. Does the sample information support the altemative hypothesis that the manufacturer's claim is false? Use a significance level of \(5 \%\). Find the range for the \(p\) -value for this test. What will your conclusion be using this \(p\) -value and \(\alpha=.05 ?\)

A paint manufacturing company claims that the mean drying time for its paints is not longer than 45 minutes. A random sample of 20 gallons of paints selected from the production line of this company showed that the mean drying time for this sample is \(49.50\) minutes with a standard deviation of 3 minutes. Assume that the drying times for these paints have a normal distribution. a. Using a \(1 \%\) significance level, would you conclude that the company's claim is true? b. What is the Type I error in this exercise? Explain in words. What is the probability of making such an error?