(3-D) (6%) Let the n random variables X₁, X₂, ..., X_nbe independent and each have the  same distribution with mean µ and variance σ². Show that σ²is the mean of the  sample variance

(2-D) the population is not normal and n is large? (3%)

(3-A) (6%) The number of customers who visit a car dealer's showroom on a Saturday  morning is a random variable with µ=18 and σ=2.5. With what probability can we assert that there will be more than 8 but fewer than 28 customers?

(3-B) (6%) Prove that for the Poisson distribution (λ is the mean value of random variable)for x = 0, 1, 2, ......

(3-C) (6%) Find k so that the following can serve as the probability density of a random variable:

(3-E) (6%) A random sample of 300 shoppers at a supermarket includes 204 who regularly use cents-off coupons. Construct a 98% confidence interval for the probability that any one shopper at the supermarket, selected at random, will regular use cents-off coupons.

(4-A) Plot the estimated regression function. Would it be reasonable to use this regression function for swimmers of age 40? Comment. (4%)

(4-B) Construct a 95% confidence interval for the curvature effect coefficient. Interpret the confidence interval? (4%)

(4-C) Test whether the curvature effect can be dropped from the quadrant regression model,controlling the α risk at 0.05. State the alternatives, the decision rule, the value of the test statistic, and the conclusion. What is the P-value of the test? (10%)

5. (12%) Suppose that we have independent random samples of n₁, n₂,..., nₖ observations from K population. Denote by..., the K group sample means and by the overall sample mean. We define the following sums of squares:Within group sum of squares (SSW) =Between groups sum of squares (SSG) =Total sum of squares (SST) = where Xᵢⱼ denotes the jᵗʰ sample observation in the iᵗʰ group. Please show the followingrelationshipSST = SSW + SSG

(6-A)Test whether opinion and gender of viewer are independent, controlling the α risk at 0.10. State the alternatives, the decision rule, the value of test statistic, and the conclusion. (12%)