Test the null hypothesis of independence of the two classifications, $\mathrm{A}$ and $\mathrm{B}$, in the $3 \times 3$ contingency table displayed below. Test using $\alpha=.05$.

| | | | $\mathrm{B}$ | | |--------------|-------------------|------------------| | | | $\mathrm{B}_{1}$ | $\mathrm{~B}_{2}$ | $\mathrm{~B}_{3}$ | | | $\mathrm{A}_{1}$ | 40 | 72 | 42 | | $\mathrm{A}$ | $\mathrm{~A}_{2}$ | 63 | 53 | 70 | | | $\mathrm{~A}_{3}$ | 31 | 38 | 30 |

b. Estimate the percentage of the total number of responses that constitute each of the A classification totals.

The earlier exercise shows how to calculate the power of a one-sided z-test. Power calculations for two-sided tests follow the same outline. We will find the power of a test based on a random sample of size 5, discussed in the earlier exercise. The hypotheses are

$$\begin{aligned} & H_0: \mu=8 \\ & H_u: \mu \neq 8 \end{aligned}$$

The population of all measurements is Normal with standard deviation $\sigma$ = 2, and the alternative we hope to be able to detect is $\mu$ = 10 ( the actual population mean). (If you used the Statistical Power applet for the earlier exercise, the two Normal curves for n = 5 illustrate parts (a) and (b) below.)

(a) Write the z-test statistic in terms of the sample mean $\bar{x}$. For what values of z does this two-sided test reject H$_0$ at the 5% significance level?

(b) Restate your result from part (a): what values of $\bar{x}$ lead to rejection of H$_0$?

(c) Now suppose that $\mu$ = 10. What is the probability of observing an $\bar{x}$ that leads to the rejection of H$_0$? This is the power of the test.

The following data are provided for two samples drawn from independent populations: $W=$ $545, n_1=25$, and $n_2=25$. Suppose the distribution of $W$ is approximately normal.

a. Calculate the mean and the standard deviation of the distribution of $W$.

b. Specify the competing hypotheses to determine whether the median of Population 1 differs from the median of Population 2.

c. Calculate the value of the test statistic $Z$.

d. At the $10 \%$ significance level, what is the conclusion to the hypothesis test?

Find the $p$-value using Excel (not Appendix D):

$t=1.457$, d.f. $=14$, right-tailed test

Consider the Minitab output shown below.

$$\begin{matrix} \text{Two-Sample T-Test and CI: X1, X2}\\ \text{Two-sample T for X1 vs X2}\\ \quad & \text{N} & \text{Mean} & \text{StDev} & \text{SE Mean}\\ \text{X1} & \text{20} & \text{50.19} & \text{1.71} & \text{0.38}\\ \text{X2} & \text{20} & \text{52.52} & \text{2.48} & \text{0.55}\\ \quad & \quad \\ \text{Difference = mu (X1) - mu (X2)}\\ \text{Estimate for difference: -2.33341}\\ 95\% \text{CI for difference: (-3.69547, -0.97135)}\\ \text{T-Test of difference = 0 (vs not = ):}\\ \quad T-Value = -3.47, P-Value = 0.001, DF = 38\\ \text{Both use Pooled StDev = 2.1277}\\ \end{matrix}$$

(a) Can the null hypothesis be rejected at the 0.05 level? Why? (b) Is this a one-sided or a two-sided test? (c) If the hypotheses had been $H_{0} : \mu_{1}-\mu_{2}=2$ versus $H_{1} : \mu_{1}-\mu_{2} \neq 2,$ would you reject the null hypothesis at the 0.05 level? (d) If the hypotheses had been $H_{0} : \mu_{1}-\mu_{2}=2$ versus $H_{1} : \mu_{1}-\mu_{2}<2$ would you reject the null hypothesis at the 0.05 level? Can you answer this question without doing any additional calculations? Why? (e) Use the output and the t table to find a 95% upper confidence bound on the difference in means. (f) What is the P-value if the alternative hypothesis is $H_{0} : \mu_{1}-\mu_{2}=2 \text { versus } H_{1} : \mu_{1}-\mu_{2} \neq 2 ?$

Suppose we have two groups with sample sizes of 8 and 15.

Suppose that the mean in group 1=15.4 and the mean in group 2=12.6. Based on your answer to the previous problem test whether the means in the two groups are significantly different, and report a two-sided $p$-value.

Suppose that $n=25$ and $s^2=49$. Assume normality. Test the null hypothesis $H_0: \sigma^2=25$ versus $H_a: \sigma^2>25$ at the $.05$ level of significance.

Is this sample of 25 exam scores inconsistent with the hypothesis that the true variance is 64 (i.e., $\sigma$ = 8)? Use the 5 percent level of significance in a two-tailed test. Show all steps, including the hypotheses and critical values from Appendix E.

$$\begin{array}{lllll}80 & 79 & 69 & 71 & 74 \\ 73 & 77 & 75 & 65 & 52 \\ 81 & 84 & 84 & 79 & 70 \\ 78 & 62 & 77 & 68 & 77 \\ 88 & 70 & 75 & 85 & 84\end{array}$$

To test the hypothesis that students who finish an exam first get better grades, Professor Hardtack kept track of the order in which papers were handed in. The first 25 papers showed a mean score of 77.1 with a standard deviation of 19.6, while the last 24 papers handed in showed a mean score of 69.3 with a standard deviation of 24.9. Is this a significant difference at $\alpha$ = .05? (a) State the hypotheses for a right-tailed test. (b) Obtain a test statistic and p-value assuming equal variances. Interpret these results. (c) Is the difference in mean scores large enough to be important? (d) Is it reasonable to assume equal variances? (e) Carry out a formal test for equal variances at $\alpha$ = .05, showing all steps clearly.

The mean serum-creatinine level measured in 12 patients 24 hours after they received a newly proposed antibiotic was 1.2 mg/dL.

How does your answer to

"Compute a two-sided $95 \% \mathrm{Cl}$ for the true mean serum-creatinine level in the previous problem."

relate to your answer to

"Suppose the sample standard deviation of serum creatinine in the previous problem is 0.6 mg/dL. Assume that the standard deviation of serum creatinine is not known, and perform the hypothesis test in the previous problem. Report a p-value."

Test the hypothesis that no difference exists in the distributions of response times for the two stimuli, using the Wilcoxon signed-rank test. Use a rejection region for which $\alpha$ is as near as possible to the $\alpha$ achieved using the sign-test.

Subject: 1, 2, 3, 4, 5, 6, 7, 8, 9 Stimulus 1: 9.4, 7.8, 5.6, 12.1, 6.9, 4.2, 8.8, 7.7, 6.4 Stimulus 2: 10.3, 8.9, 4.1, 14.7, 8.7, 7.1, 11.3, 5.2, 7.8

Find the $p$-value using Excel (not Appendix D):

$t=-1.847$, d.f. $=22$, left-tailed test

In previous examples we calculated a 95% confidence interval for the mean IQ u, of seventh-grade girls in a Midwestern district. In other exercise you used the same data to test the hypothesis that this population has a mean IQ significantly different from 100. Compare your two results. What did you learn from the confidence interval? What did you learn from the test P-value? Describe the relative merits of both methods.

Do a two-sample test for equality of means assuming equal variances. Calculate the $p$-value.

Comparison of GPA for randomly chosen college juniors and seniors: $\tilde{x_1}$ = 3.05, $s_1$ = .20, $n_1$ = 15,$\tilde{x_2}$ = 3.25, $s_2$ = .30, $n_2$ = 15, $\alpha$ = .025, left-tailed test.