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The manufacturer of a gasoline additive claims that the use of this additive increases gasoline mileage. A random sample of six cars was selected, and these cars were driven for 1 week without the gasoline additive and then for 1 week with the gasoline additive. The following table gives the miles per gallon for these cars without and with the gasoline additive. \begin{tabular}{l|llllll} \hline Without & \(24.6\) & \(28.3\) & \(18.9\) & \(23.7\) & \(15.4\) & \(29.5\) \\ \hline With & \(26.3\) & \(31.7\) & \(18.2\) & \(25.3\) & \(18.3\) & \(30.9\) \\ \hline \end{tabular} a. Construct a \(99 \%\) confidence interval for the mean \(\mu_{d}\) of the population paired differences, where a paired difference is equal to the miles per gallon without the gasoline additive minus the miles per gallon with the gasoline additive. b. Using a \(2.5 \%\) significance level, can you conclude that the use of the gasoline additive increases the gasoline mileage?

What is the shape of the sampling distribution of \(\hat{p}_{1}-\hat{p}_{2}\) for two large samples? What are the mean and standard deviation of this sampling distribution?

When are the samples considered large enough for the sampling distribution of the difference between two sample proportions to be (approximately) normal?

The lottery commissioner's office in a state wanted to find if the percentages of men and women who play the lottery often are different. A sample of 500 men taken by the commissioner's office showed that 160 of them play the lottery often. Another sample of 300 women showed that 66 of them play the lottery often. a. What is the point estimate of the difference between the two population proportions? b. Construct a \(99 \%\) confidence interval for the difference between the proportions of all men and all women who play the lottery often. c. Testing at a \(1 \%\) significance level, can you conclude that the proportions of all men and all women who play the lottery often are different?

The owner of a mosquito-infested fishing camp in Alaska wants to test the effectiveness of two rival brands of mosquito repellents, \(\mathrm{X}\) and \(\mathrm{Y}\). During the first month of the season, eight people are chosen at random from those guests who agree to take part in the experiment. For each of these guests, Brand \(\bar{X}\) is randomly applied to one arm and Brand \(\mathrm{Y}\) is applied to the other arm. These guests fish for 4 hours, then the owner counts the number of bites on each arm. The table below shows the number of bites on the arm with Brand \(X\) and those on the arm with Brand \(Y\) for each guest. \begin{tabular}{l|rrrrrrrr} \hline Guest & A & B & C & D & E & F & G & H \\ \hline Brand X & 12 & 23 & 18 & 36 & 8 & 27 & 22 & 32 \\ \hline Brand Y & 9 & 20 & 21 & 27 & 6 & 18 & 15 & 25 \\ \hline \end{tabular} a. Construct a \(95 \%\) confidence interval for the mean \(\mu_{d}\) of population paired differences, where a paired difference is defined as the number of bites on the arm with Brand \(X\) minus the number of bites on the arm with Brand \(Y\). b. Test at a \(5 \%\) significance level whether the mean number of bites on the arm with Brand \(\mathrm{X}\) and the mean number of bites on the arm with Brand \(Y\) are different for all such guests.