Directions: Show all your work. Indicate clearly the methods you use, because you will be graded on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. In a report to the department of transportation of a western state, a large trucking firm stated that the distribution of weights of its fully loaded tractor trailer trucks is approximately normal with a mean of 19,016 pounds and a standard deviation of 2,324 pounds. The state police decided to check a sample of 40 of the company's trucks to test the company's claim concerning the mean weight and standard deviation of the weights of its trucks.

a) Assume that the company's claim is true. Describe the distribution of the sample mean weight for random samples, each consisting of 40 trucks.

b) At the company's large terminal, a state police crew selects a random sample of 40 fully loaded trucks and finds that the mean weight of those trucks is 19,168 pounds. What is the probability that a random sample of 40 of the company's fully loaded trucks would have a mean weight of 19,168 pounds or more if the company's claim is true?

c) A second state police crew is assigned to check trucks at the same terminal as in part (b) but on a different day. However, the second crew believes that the instructions to carry out a random sample are too complicated and too time-consuming. Instead, the crew weighs the first 40 fully loaded trucks as they leave the terminal and finds that the mean weight of the selected trucks is 18,894 pounds. Why is the lack of random selection in using the first 40 trucks a potential problem?