Mathematics 376 – Mathematical Statistics Makeup Midterm Exam 2 April 27, 2012

Directions: 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{3θ^{3}}{y^{4}} & if 0 < θ ≤ y < ∞ \\ 0 & otherwise. \end{matrix}

A) (10) Determine the method of moments estimator for θ. B) (10) Determine the maximum likelihood estimator for θ. C) (5) Is U = ∑_{i} ln(Y_{i}) a sufficient statistic for θ? Why or why not?

II. In this problem, let S represent the proportion of women who agree that men are basically selfish and egotistical.

A) (15) Suppose that surveys done 20 years ago showed that S was 0.40 at that time, but a new survey finds that Y = 1260 out of a random sample of n = 3000 women said that men are basically selfish and egotistical. Is there sufficient evidence to say the proportion S has increased? Give the null and alternative hypotheses for a suitable test, carry out the analysis, estimate the attained significance level (p-value) and state your conclusion succinctly and clearly.

B) (15) Suppose that the current value of S is actually S = .43 and the test in part A was carried out to yield α = 0.05. How large would n have to be in a sample like the one from part A to ensure that β (the probability of a Type II error) is .1 or less?

III. Assume that the hardness of iron bars produced by one plant is normally distributed with mean μ and variance σ^{2}. A random sample of n = 7 bars from the output of the plant is measured to determine whether the average hardness meets the desired figure μ_{0} = 172. A) (20) The following measurements:

167, 174, 179, 164, 163, 160, 168

were obtained. Test H_{0} : μ = 172 versus the alternative H_{a} : μ < 172 using an appropriate test with α = .01.

B) (10) How is your answer from A related to the 99% lower confidence bound for μ that we learned to compute in the first section of the course?

IV. (15) Let (x_{i}, y_{i}, z_{i}), i = 1, …, n be a collection of data points. Using the matrix formulation, derive the normal equations for the least squares estimators for the coefficients β_{0}, β_{1}, β_{2} in the model Z = β_{0} + β_{1}x + β_{2}x^{2}β_{3}y + ε fitting the data. Express in the form Aβ = b where A is a matrix and b is a vector (not matrix products).

Extra Credit “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 exactly? Explain using the case of a test on a single mean.

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