6. For healthy individuals, the level of prothrombin in the blood is approximately normally distributed with mean 20 mg/100 mL and standard deviation 4 mg/100 mL. Low levels indicate low clotting ability. In studying the effect of gallstones on prothrombin, the level of each patient in a sample is measured to see if there is a deficiency. Let u be the true average level of prothrombin for gallstone patients. (a) What are the appropriate null and alternative hypotheses? (b) Let X denote the sample average level of prothrombin in a sample of n = 20 randomly selected gallstone patients. Consider the test procedure with test statistic X and rejection region x S 17.92. What is the probability distribution of the test statistic when Ho is true (i.e. determine center, spread, and shape of X )? What is the probability of a type I error for this test procedure? (c) What is the probability distribution of the test statistic, X when u = 16.7? Using the test procedure of part (b), what is the probability that gallstone patients will be judged not deficient in prothrombin, when in fact u = 16.7 (a type II error)? (d) How would you change the test procedure of part (b) to obtain a test with significance level 0.05? What impact would this change have on the error probability of part (c)