1. A slightly positively skewed population has mean 74 and variance 128. We take a random sample of 50 observations from this population.
a. Find the probability the mean of the sample will be below 70.
b. You assumed Normality of the distribution of the sample mean to find the probability in part (a). What allows you to make this assumption?
2. Find the following table values:
a. t(40),.90 =
b. t(12),.99 =
c. t(6),.05 =
3. A case-control study is used to look at the relationship between depression and a family history of this disease. Results are given in the table below.
Depression
Yes No
|
Family history
|
Yes
|
31
|
25
|
|
of depression
|
No
|
26
|
74
|
|
|
|
57
|
99
|
a. Enter the data into SAS, then use PROC FREQ to obtain the odds ratio and corresponding 95% confidence interval. Turn in the printout.
b. Note what value of interest is (or is not) in the CI in part (a), then elaborate on what this indicates. Do so in the context of this problem.
4. An exercise science student measures the velocity of fastballs thrown by 10 college baseball pitchers. The observations are below. In this sample, the mean is 83.6, and the standard deviation is 3.7476.
82.3 78.6 80.3 83.1 88.5 85.4 87.9 86.9 78.1 84.9
a. "By hand", calculate and interpret a 95% confidence interval for the mean velocity of fastballs thrown by all college baseball pitchers.
b. Calculate a 99% CI, and briefly state how this compares to the one in part (a).
c. Using PROC MEANS, get SAS to print out the CI from part (a). Turn in the printout.
5. In settings where we calculate an odds ratio, while we seldom perform a formal test, it is possible to do so. Using the setting in problem #3, describe what a Type I error would mean, and also describe what a Type II error would mean.
6. Again use the baseball velocity data from problem #4. Suppose we now want to test if the mean velocity is greater than 80.
a. Carry out the test. In doing so, complete these steps:
i. Write the null and alternative hypotheses (note this is a one-sided test).
ii. Write the rejection region, if we use a = .05.
iii. "By hand", calculate the test statistic, t0.
iv. State your decision - either say that you reject the null hypothesis, or that you fail to reject the null hypothesis.
v. Interpret the result in the context of the problem.
b. Use PROC TTEST in SAS to obtain the test statistic and the p-value for the test in part (a). Turn in the printout.
c. State how the p-value in part (b) can be used to reach the decision of the hypothesis test in part (a).
Note: You should use the general approach in part (a) for all hypothesis testing problems (this one, the one that follows on this assignment, and those on future assignments). The only adjustment might be using the p-value from a computer program (like SAS), rather than writing a rejection region.
7. Use the REACTIONTIME data set. Use a = .05 for all inference for this problem.
a. Use SAS to obtain a confidence interval for the mean reaction time of all people who receive a visual stimulus. Turn in the printout. You do not have to interpret this CI.
b. Suppose we want to test if the mean reaction time to a visual stimulus differs from 500 (note this is a two-sided test). Using only the CI in part (a), state the decision of the test (and also explain how you reached this decision), and interpret the results.
c. Use PROC TTEST in SAS to obtain the results of a test that the mean reaction time to a tactile stimulus differs from 500. Along with turning in the printout, go through the proper steps as in part (a) of problem #6 to complete the test.