1. (10 pts) For each of the following questions, determine the appropriate confidence interval or intervals.

1. For a standard normal random variable, the \(99 \%\) two-sided interval.
2. For a \(t\) -distribution with 40 degrees of freedom, the \(95 \%\) two-sided interval.
3. For an \(F\) -distribution with \((10,20)\) degrees of freedom, the \(99 \%\) one-sided interval (with the "rejection" region in the right tail).
4. For a \(\chi^{2}\) -distribution with 25 degrees of freedom, the \(95 \%\) two-sided interval.

2. (10 points) The following table summarizes statistics for the salaries of home values (in thousands) from two geographically different regions:

Let \(\mu_{1}\) denote the mean of group \(G_{1}\), and \(\mu_{2}\) denote the mean of group \(G_{2}\). Use this data to test at the \(99 \%\) interval that \(\mu_{1}-\mu_{2}=0\) with the alternative that \(\mu_{1}-\mu_{2}<0\)

3. (10 points) In a recent survey, individuals from two groups were asked a particular question from which the following table of responses were tabulated (Y=Yes, N = No)

1. Compute the table of expected values.
2. Compute the associated \(\chi^{2}\) -statistic for the hypothesis of independence of rows and columns.
3. Test the hypothesis of row and column independence at the \(95 \%\) level.

4. (10 points) Two populations are being compared to determine if the variances are equal. For the first population, a sample of size 35 has a standard deviation of 18.5, and for the second population, a sample of size 85 has a standard deviation of 15.3. Test, at the \(95 \%\) level, that the populations have equal variances.

5. (10 points) The following table summarizes statistics for the salaries of individuals from two different groups.

Let \(\mu_{1}\) denote the mean of group \(G_{1}\), and \(\mu_{2}\) denote the mean of group \(G_{2}\). Use this data to find a \(95 \%\) two-sided confidence interval for \(\mu_{1}-\mu_{2}\).

7. (10 points) Two groups of students enrolled introductory statistics classes. The first group, with 180 individuals, did not use on-line calculators and 120 students passed. The second group, with 210 individuals, did use on-line calculators and 180 students passed. Test the hypothesis that the pass rate in the second group is higher than in the first group. Carefully state the null hypothesis, the alternative hypothesis, and your computations. Test the hypothesis at the \(95 \%\) level.

8. (10 points) Individuals were asked if they preferred a blue, red or green car, as well as their residential status. The results of the survey are summarized in the following chart.

Test, at the \(90 \%\) level, that the two random variables being measures are independent.

9. (10 pts) For the following problems, provide the appropriate discussion and decision for the given hypothesis test.

1. A sample of size 10 from a normal distribution with known standard deviation \(\sigma=5\) is such that \(\bar{X}=65.745\). Test the hypothesis at the \(99 \%\) level that the true population mean is 61 . What is the \(p\) -value of the observed value of \(\bar{X}\) ?
2. Using the same data as above, suppose that the standard deviation of the population is unknown. If the sample standard deviation is 5 , then again test the hypothesis that the population mean is 61 , again at the \(99 \%\) confidence level.

10. (20 points) The Wilcoxon signed rank test is a non-parametric alternative to a paired \(t\) -test. If \(W\) denotes the Wilcoxon statistic, then for a sample size \(n \geq 15\), the statistic

\(\frac{W-n(n+1) / 4}{\sqrt{n(n+1)(2 n+1) / 24}}\)

is closely approximately by a standard normal random variable. Suppose from a sample of 15 individuals, the difference of blood pressures from before and after treatment is such that \(W=105\). Test the null hypothesis, at the \(99 \%\) level with a two-sided alternative, that the medication had no influence on blood pressure.

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