(1) (10 points) Show that the mean and variance of a binomial distribution with the number of trials n, the probability of success p, and the probability of failure q are µ = np and σ² = npq.

(a) Graph Jamal's utility function. Ts he risk averse? Explain.________

(b) Does A or B offer Jamal a higher expected prize? Explain your raasanmg with appropriate caleulations. (Hint: The expected value of a random variable is the weighted average of the possible outcomes, where the probabilities are the weights.)________

(c﹚ Does A or B offer Jamal n higher expected utility? As expected utility________ B's expected utility ________? Again, show your calculations.

(d﹚ Should Jamal pick A or B2 Why?________

(2) (10 points) To find out whether a new serum will arrest leukemia, 9 mice, all with an advanced stage of the disease, are selected. 5 mice receive the treatment and 4 do not. Survival times, in years, from the time the experiment commenced are as follows:  Treatment: 2.1 5.3 1.4 4.6 0.9 No Treatment: 1.9 0.5 2.8 3.1At the 0.05 level of significance, can the serum be said to be effective? Assume the two populations to be normally distributed with equal variances.

(3) (20 points) Consider the crushing strength of concrete cubes shown in the following table. Conduct a goodness-of-fit test at significant level 0.05 to determine whether the observed data follow a normal distribution with a mean of 7.50 and standard deviation of 0.53.

(a) Find the 96% confidence interval for the fraction of the voting population favoring the suit.

(b) How large a sample is needed if we wish to be 96% confident that our sample proportion will be within 0.02 of the true fraction of the voting population?

(a) Cumulative distribution of X.

(b) P(x＞0.25)