2. If X is a normal random variable with mean µ = 3 and variance o2 = 16, find P{X<11}, P{X>-1 }

Suppose X1,..., Xn is a random sample from an exponential distribution with mean e. If X = 17.9 with n = 50, find (a) a one-sided 95% confidence interval for 0, and (b) a two-sided 95% confidence interval for 0.

Suppose that X₁,..., Xn are i.i.d. from a normal population with mean μ and variance o². Calculate the expected value and the variance of the sample mean X and find the distribution of √n(X − µ).

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

Compute the standard score (Z-score) for x, when x = 21, the mean of the distribution = 27, and the variance (S2) = 9. Z = %3D O 2 -2 .67 -.67

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…