Mathematics 376 – Probability and Statistics II Midterm Exam II April 16, 2004

Directions: This is an open-book, open-notes exam. Do all work in the blue exam booklet. There are 100 regular points and 10 Extra Credit points. Be sure to read each question carefully before starting to work.

I. Let Y_{1}, …, Y_{n} be a random sample from a population with density function

f(y|θ) = \{ \begin{matrix}\frac{3y^{2}}{θ^{3}} & if 0 ≤ y ≤ θ \\ 0 & otherwise \end{matrix}

containing the parameter θ > 0. A) (10) Determine the method of moments estimator for θ. B) (10) Determine the maximum likelihood estimator for θ.

II. In this problem, let p represent the proportion of women who agree that men are basically selfish and egotistical. Suppose that in the past p was 0.40, but we assert that p has now increased. A) (15) State the null and alternative hypotheses, test statistic and rejection region for

a test of this assertion with α = .01, assuming a large sample size. Sketch a standard normal pdf and illustrate the rejection region.

B) (10) The Detroit Free Press reported in 1990 that Y = 1260 women out of a random sample of n = 3000 agreed with the statement that men are basically selfish and egotistical. What conclusion do you draw from your test in part A with this data?

C) (10) Suppose that the current value of p is actually p = .43. Determine β (= probability of a Type II error in your test) if n = 3000 as in part B.

III. Let X be the Brinell hardness measurement of an iron bar, and assume X is normally distributed with mean μ and variance σ^{2}. A random sample of 7 bars is measured to determine whether the average strength meets the desired figure μ_{0} = 172. A) (20) The following measurements: 167, 174, 179, 164, 163, 160, 168 were obtained. Test

whether the data indicate μ < 172 using α = .1 and state your conclusion. B) (15) Test the null hypothesis H_{0} : σ^{2} = 40 against the alternative hypothesis H_{a} :

σ^{2} ≠ 40 using an appropriate test with α = .1, and state your conclusion.

IV. “Thought Questions” – Answer in 2 or 3 complete sentences: A) (5) Exactly why is it the case that a Z-test is not appropriate when the sample size

is small and population variance is not known? Explain using the case of a test on a single mean.

B) (5) Suppose you do two different tests: an upper-tail test and a two-tail test. How would the p-values of those tests compare if you use the same data for each? Explain.

Extra Credit (10) What is the standard error for the method of moments estimator from problem I part A? How would you use that information to determine which of the two estimators in problem I is superior?