91Ó°ÊÓ

Describe the sampling distribution of \(\bar{x}_{1}-\bar{x}_{2}\) for two independent samples when \(\sigma_{1}\) and \(\sigma_{2}\) are known and either both sample sizes are large or both populations are normally distributed. What are the mean and standard deviation of this sampling distribution?

The following information is obtained from two independent samples selected from two populations. $$ \begin{array}{lll} n_{1}=650 & \bar{x}_{1}=1.05 & \sigma_{1}=5.22 \\ n_{2}=675 & \bar{x}_{2}=1.54 & \sigma_{2}=6.80 \end{array} $$ a. What is the point estimate of \(\mu_{1}-\mu_{2} ?\) b. Construct a \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). Find the margin of error for this estimate.

Using data from the U.S. Census Bureau and other sources, www.nerdwallet.com estimated that considering only the households with credit card debts, the average credit card debt for U.S. households was \(\$ 15,523\) in 2014 and \(\$ 15,242\) in 2013 . Suppose that these estimates were based on random samples of 600 households with credit card debts in 2014 and 700 households with credit card debts in 2013 . Suppose that the sample standard deviations for these two samples were \(\$ 3870\) and \(\$ 3764\), respectively. Assume that the standard deviations for the two populations are unknown but equal. a. Let \(\mu_{1}\) and \(\mu_{2}\) be the average credit card debts for all such households for the years 2014 and 2013 , respectively. What is the point estimate of \(\mu_{1}-\mu_{2}\) ? b. Construct a \(98 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). c. Using a \(1 \%\) significance level, can you conclude that the average credit card debt for such households was higher in 2014 than in \(2013 ?\) Use both the \(p\) -value and the critical-value approaches to make this test.

Assuming that the two populations are normally distributed with unequal and unknown population standard deviations, construct a \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) for the following. $$ \begin{array}{lll} n_{1}=14 & \bar{x}_{1}=109.43 & s_{1}=2.26 \\ n_{2}=15 & \bar{x}_{2}=113.88 & s_{2}=5.84 \end{array} $$

A company that has many department stores in the southern states wanted to find at two such stores the percentage of sales for which at least one of the items was returned. A sample of 800 sales randomly selected from Store A showed that for 280 of them at least one item was returned. Another sample of 900 sales randomly selected from Store B showed that for 279 of them at least one item was returned. a. Construct a 98\% confidence interval for the difference between the proportions of all sales at the two stores for which at least one item is returned. b. Using a \(1 \%\) significance level, can you conclude that the proportions of all sales for which at least one item is returned is higher for Store A than for Store \(\mathrm{B}\) ?