34. The mean squared error of an estimator Ô is MSE (Ô) = E(ô – ĝ)². If Ô is unbiased, then MSE(Ô) = V(Ô), but in general MSE(Ô) = V(Ô) + (bias)². Consider the estimator ²=KS2, where S² = sample variance. What value of K minimizes the mean squared error of this estima- tor when the population distribution is normal? [Hint: It can be shown that E[(S²)²] = (n + 1)σơ³/(n − 1) In general, it is difficult to find Ô to minimize MSE(Ô), which is why we look only at unbiased estimators and minimize V().]

1. Suppose that a variance is thought to be σ² = 3. A random sample of n=25 yields a sample standard deviation of s=2.1. α=0.05. (a) Test the hypothesis that the variance is not equal to 3 by using critical regions. (b) Test the hypothesis using the P-value.

X. is found to be 19.1, and the A simple random sample of sizen is drawn from a population that is normally distributed. The sample mean, sample standard deviation, s, is found to be 4.9. (a) Construct a 96% confidence interval about u if the sample size, n, is 39. (b) Construct a 96% confidence interval about u if the sample size, n, is 68. How does increasing the sample size affect the margin of error, E? (c) Construct a 98% confidence interval about u if the sample size, n, is 39. How does increasing the level of confidence affect the size of the margin of error, E? (d) If the sample size is 14, what conditions must be satisfied to compute the confidence interval? (a) Construct a 96% confidence interval about u if the sample size, n, is 39. Lower bound: Upper bound: (Round to two decimal places as needed.) (b) Construct a 96% confidence interval about u if the sample size, n, is 68. Lower bound: ; Upper bound: (Round to two decimal places as needed.) How does increasing the sample…

If X is a negative binomial rv, then Y= r+Xis the total number of trials necessary to obtainr S’s. Obtain the mgf of Y and then its mean valueand variance. Are the mean and variance intuitively consistent with the expressions for E(X)and V(X)? Explain

Let S1^2 be the sampling variance for a random sample of twelve values (amount of mercury in the blood) and let S2^2 be the sampling variance for a random sample of ten values (amount of lead in the blood); samples from the same population were used. The population variance for mercury measurements is assumed to be twice the corresponding population variance for lead measurements. We will further assume that S1^2 is independent of S2^2. 1. Find a number b such that P [(S1^2/S2^2)<=b]=0.95 enter such a b to three decimal places b= 2. Consider the number b calculated above, find a number a such that P[a<=(S1^2/S2^2)<=b]=0.90 enter said a to three decimal places. a=

A random sample of n = 19 winter days in Denver gave a sample mean pollution index x1 = 43. Previous studies show that o1 = 10. For Englewood (a suburb of Denver), a random sample of n2 = 18 winter days gave a sample mean pollution index of x2 = 34. Previous studies show that o2 = 13. Assume the pollution index is normally distributed in both Englewood and Denver. Do these data indicate that the mean population pollution index of Englewood is different (either way) from that of Denver in the winter? Use a 1% level of significance. (a) What is the level of significance? State the null and alternate hypotheses. O Ho: H1 H2 O Ho: H1 = l2; H1: H1 < µ2 (b) What sampling distribution will you use? What assumptions are you making? O The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations. O The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations. O The Student's…

Suppose that we would like to test the following hypotheses on the population correlation coefficient ρ: H0: ρ ≤ 0 and H1: ρ > 0. The sample correlation coefficient (r) for a sample with 7 observations (n = 7) on x and y is 0.35 . Then the test statistic t has a value of