Refer to given Exercise, in which 12 drivers were tested after getting too little sleep and after drinking alcohol. In that exercise, we assumed that the mean difference in erroneous lane drifts during a 2-hour driving period would be 0 for the population, with a standard deviation of 5. Assume that the population of possible differences is bell-shaped.

a. Describe the sampling distribution of in this situation, including its mean and standard deviation.

b. Suppose that the sample mean difference in number of lane drifts was 8, with more lane drifts after drinking alcohol than after getting too little sleep. Would you still believe that the mean difference in the population is 0? Explain, using the standardized statistic as part of your explanation.

Exercise
Horne, Reyner, and Barrett (2003) describe a study in which 12 men took a simulated driving test after drinking alcohol at lunch, getting too little sleep the night before, neither (control condition), or both. Each of the men took the test under all four conditions. One of the measurements taken under each condition was the number of times the driver drifted out of his lane during the 2-hour test. For this problem, consider the mean difference in number of lane drifts between the condition with alcohol but enough sleep and the condition with no alcohol but too little sleep.

a. Define the parameter of interest in the context of this situation. Use appropriate notation.

b. Describe the sample statistic of interest in the context of this situation. Use appropriate notation.

c. Given that the sample size is only 12, describe what is required so that the sampling distribution for the statistic described in part (b) will be approximately normal.

d. Suppose that getting too little sleep and drinking alcohol at lunch have the same effect on the mean number of lane drifts in the 2-hour period and that the standard deviation for the population of differences is 5. Describe the sampling distribution of the statistic you gave in part (b), assuming that the situation you described in part (c) holds.