Question 1: A lottery offers one $500 prize and five $100 prizes. One thousand tickets are sold at $3 each. Let X represent the gain of purchasing one ticket.

a) Construct the probability distribution for the random variable X.

b) What is the probability of winning?

c) Calculate the expected value of the random variable X and provide an interpretation of its value in the context of the question.

d) Find the standard deviation of the random variable X. (3 marks)

Question 2: A recent study of the lifetimes of cell phones shows that the average is 24.3 months and the standard deviation is 4.6 months. A simple random sample of 35 cell phones is selected.

a) What is the probability that the mean lifetime of these cell phones will be more than 22 months but less than 26 months?

b) Is it necessary to assume that the lifetimes of cell phones follow a normal distribution? Why or why not?

Question 3: The average weight of 40 randomly selected minivans is 4150 pounds. The standard deviation of all minivans is 480 pounds.

a) Can we apply the central limit theorem here? Briefly justify your answer.

b) Assume that the central limit theorem can be applied here. Find the 90% confidence interval for the true mean.

c) Interpret your answer in part (b) in the context of the question.

Question 4: A survey showed that among 785 randomly selected college graduates, 18% smoke.

a) Obtain a 95% confidence interval for the true proportion and interpret this interval in the context of the question.

b) Check the conditions required to validate your calculation in (a).