1. *The inside diameter of a randomly selected piston ring is a random variable with mean value 12 cm and standard deviation 0.04 cm. a. If X is the sample mean diameter for a random sample of n = 16 rings, where is the sampling distribution of X centered, and what is the standard deviation of the X distribution? b. Answer the questions posted in part (a) for a sample size of n =64 rings. c. For which of the two random samples, the one of part (a) or the one of part (b), is X more likely to be within 0.01 cm of 12 cm? Explain your reasoning. 2. *Refer to the previous problem (3). Suppose the distribution of the diameter is normal. 12.01 ) when n = 16. a. Calculate P ( 11.99 X b. How likely is it that the sample mean diameter exceeds 12.01 when n = 25? 3. * The first assignment in a statistical computing class involves running a short program. If past experience indicates that 40% of all students will make no programming errors, compute the (approximate) probability that in a class of 50 students a. At least 50% will make no error. b. Between 30% and 50% will make no errors. 4. John's projected annual income after graduation is normally distributed with mean $80,000 and standard deviation $10,000. Sally's is normally distributed with mean $85,000 and standard deviation $12,000. If their incomes are independent, what is the probability that their combined income exceeds $180,000? 5. Refer to #4 above What is the standard deviation of their combined income if their incomes have a bivariate normal distribution and the correlation between them is 0.4? What is it if the correlation is 0.4? 6. *Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and the alcohol content of each bottle is determined. Let denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the resulting 95% confidence interval is (7.8, 9.4). a. Would a 90% confidence interval calculated from this same sample have been narrower or wider than the given interval? Explain your reasoning. b. Consider the following statement: There is a 95% chance that is between 7.8 and 9.4. Is this statement correct? Why or why not? c. Consider the following statement: We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 7.8 and 9.4. Is this statement correct? Why or why not? d. Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 95% interval is repeated 100 times, 95 of the resulting intervals will include . Is this statement correct? Why or why not