(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.

(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)

(c) The median of X.

(a) What is the probability that at least one accident will occur on a given blizzard day?

(b) Suppose there are five blizzard days this winter. What is the probability that two out of these five blizzard days are accident free? Assume that accident occurrence between blizzard days are statistically independent.

1. (10 points)A random sample of 20 students yielded a mean of = 72 and a variance of s² = 16 for scores on a college placement test in mathematics. Assuming the scores to be normally distributed, construct a 98% confidence interval for σ².

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