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An independent random sample is selected from an approximately normal population with an unknown standard deviation. Find the p-value for the given sample size and test statistic. Also determine if the null hypothesis would be rejected at \(\alpha=0.01\). (a) \(n=26, T=2.485\) (b) \(n=18, T=0.5\)

Georgianna claims that in a small city renowned for its music school, the average child takes at least 5 years of piano lessons. We have a random sample of 30 children from the city, with a mean of 4.6 years of piano lessons and a standard deviation of 2.2 years. (a) Use a hypothesis test to determine if there is sufficient evidence against Georgianna's claim. (b) Construct a \(95 \%\) confidence interval for the number of years students in this city take piano lessons, and interpret it in context of the data. (c) Do your results from the hypothesis test and the confidence interval agree? Explain your reasoning.

In each of the following scenarios, determine if the data are paired. (a) We would like to know if Intel's stock and Southwest Airlines' stock have similar rates of return. To find out, we take a random sample of 50 days, and record Intel's and Southwest's stock on those same days. (b) We randomly sample 50 items from Target stores and note the price for each. Then we visit Walmart and collect the price for each of those same 50 items. (c) A school board would like to determine whether there is a difference in average SAT scores for students at one high school versus another high school in the district. To check, they take a simple random sample of 100 students from each high school.

Age at first marriage, Part I. The National Survey of Family Growth conducted by the Centers for Disease Control gathers information on family life, marriage and divorce, pregnancy, infertility, use of contraception, and men's and women's health. One of the variables collected on this survey is the age at first marriage. The histogram below shows the distribution of ages at first marriage of 5,534 randomly sampled women between 2006 and \(2010 .\) The average age at first marriage among these women is 23.44 with a standard deviation of \(4.72 .^{27}\) Estimate the average age at first marriage of women using a \(95 \%\) confidence interval, and interpret this interval in context. Discuss any relevant assumptions.