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Consider a hypothesis test of difference of means for two independent populations \(x_{1}\) and \(x_{2} .\) What are two ways of expressing the null hypothesis?

A random sample has 49 values. The sample mean is \(8.5\) and the sample standard deviation is \(1.5 .\) Use a level of significance of \(0.01\) to conduct a left-tailed test of the claim that the population mean is \(9.2\). (a) Is it appropriate to use a Student's \(t\) distribution? Explain. How many degrees of freedom do we use? (b) What are the hypotheses? (c) Compute the sample test statistic \(t\). (d) Estimate the \(P\) -value for the test. (e) Do we reject or fail to reject \(H_{0}\) ? (f) The results.

REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults (Reference: Secrets of Sleep by Dr. A. Borbely). Assume that REM sleep time is normally distributed for both children and adults. A random sample of \(n_{1}=10\) children (9 years old) showed that they had an average REM sleep time of \(\bar{x}_{1}=2.8\) hours per night. From previous studies, it is known that \(\sigma_{1}=0.5\) hour. Another random sample of \(n_{2}=10\) adults showed that they had an average REM sleep time of \(\bar{x}_{2}=2.1\) hours per night. Previous studies show that \(\sigma_{2}=0.7\) hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a \(1 \%\) level of significance.

Weatherwise magazine is published in association with the American Meteorological Society. Volume 46 , Number 6 has a rating system to classify Nor'easter storms that frequently hit New England states and can cause much damage near the ocean coast. A severe storm has an average peak wave height of \(16.4\) feet for waves hitting the shore. Suppose that a Nor'easter is in progress at the severe storm class rating. (a) Let us say that we want to set up a statistical test to see if the wave action (i.e., height) is dying down or getting worse. What would be the null hypothesis regarding average wave height? (b) If you wanted to test the hypothesis that the storm is getting worse, what would you use for the alternate hypothesis? (c) If you wanted to test the hypothesis that the waves are dying down, what would you use for the alternate hypothesis? (d) Suppose you do not know whether the storm is getting worse or dying out. You just want to test the hypothesis that the average wave height is different (either higher or lower) from the severe storm class rating. What would you use for the alternate hypothesis? (e) For each of the tests in parts (b), (c), and (d), would the area corresponding to the \(P\) -value be on the left, on the right, or on both sides of the mean? Explain your answer in each case.