Mathematics 376 – Probability and Statistics II Final Examination May 12, 2004

Directions

This is an open-book, open-notes exam. Do all work in the blue exam booklet, and include all work necessary to justify your answers. Be sure to read each question carefully before starting to work. There are 200 regular points and 10 Extra Credit points.

General Note: A “random sample” always consists of independent measurements from the indicated distribution.

I. Let Y_{1}, Y_{2}, Y_{3}, Y_{4}, Y_{5} be a random sample from a normal distribution with mean μ = 4 and variance σ^{2} = 100.

A) (6) What is the distribution of \overline{Y} = \frac{1}{5}(Y_{1} + Y_{2} + Y_{3} + Y_{4} + Y_{5})? Why? B) (7) What is the distribution of U = \frac{(Y_{1}−4)^{2}+(Y_{2}−4)^{2}+(Y_{3}−4)^{2}}{100} ? Why?

C) (7) What is the distribution of V = \frac{\sqrt{3}(Y_{4}−4)}{10\sqrt{U}}, where U is as in part B? Why?

II. Let Y_{1}, …, Y_{50} be a random sample of measurements of the proportion of impurities in bauxite ore samples. Each Y_{i} is a random variable with pdf

f(y) = \{ \begin{matrix}4y^{3} & if 0 ≤ y ≤ 1 \\ 0 & elsewhere \end{matrix}

A) (15) Compute E(Y_{i}) and V (Y_{i}). B) (15) Estimate P (\overline{Y} > .85) (the probability that the sample mean is at least .85) using

the Central Limit Theorem.

III. A random variable Y is said to have a log-normal distribution with parameters μ, σ^{2} if its pdf has the form

f(y) = \{ \begin{matrix}\frac{1}{y\sqrt{2πσ^{2}}} e^{−(ln(y)−μ)^{2}/2σ^{2}} & if y > 0 \\ 0 & otherwise \end{matrix}

A) (10) Compute E(ln(Y)) for a log-normal random variable. (Hint: Set up the integral,

then change variables.) B) (10) Let Y_{1}, …, Y_{n} be a random sample from a log-normal distribution with unknown

μ and σ^{2} = 1. Find the maximum-likelihood estimator for μ. C) (10) Is your estimator from part B biased or unbiased? Why?

IV. Solid copper produced by melting powdered ore is tested for “porosity” (the volume fraction due to air bubbles). A sample of n_{1} = 4 porosity measurements made in one lab has \overline{y}_{1} = .25 and s_{1}^{2} = .001. A second set of n_{2} = 5 measurements is made using identical ore in a second lab, yielding \overline{y}_{2} = .17 and s_{2}^{2} = .002.

A) (15) Find a 95% confidence interval for μ_{1} − μ_{2}, the difference of the population mean

porosity measurements from the two labs. B) (15) Is there sufficient evidence at the α = .05 level to conclude that μ_{1} ≠ μ_{2}? Explain.

V. The Rockwell hardness index for steel is determined by pressing a diamond point into the metal with a specified force and measuring the depth of penetration. For a random sample of 50 pieces of a certain grade of steel, the Rockwell hardness index averaged 62, with sample standard deviation 8.

A) (15) The manufacturer claims that this steel has average hardness index (at least) 64. Is there sufficient evidence to refute this claim? Use a lower-tail test at the α = .01 level.

B) (15) Using the rejection region you found in part A, compute the Type II error probability β if the actual mean hardness index is μ_{a} = 60.

VI. Consider the following measurements of the heat-producing capacity of the coal produced by two mines (in millions of calories per ton):

Mine 1 : 8260 8130 8350 8070 8340
Mine 2 : 7950 7890 7900 8140 7920 7840

A) (15) Estimate the variance of the heat producing capacity of the coal from each mine. B) (15) Test the null hypothesis H_{0} : σ_{1}^{2} = σ_{2}^{2} against the alternative H_{a} : σ_{1}^{2} ≠ σ_{2}^{2}.

Estimate the p-value of your test and state your conclusion clearly and succinctly.

VII. The following table gives measurements of the amount of sodium chloride which dissolved in 100 grams of water at various temperatures.

x (degrees C) y (grams)
0  7.3
15  13.0
30  23.3
45  30.7
60  39.7
75  47.7

A) (15) Find the equation of the regression line for this data set. B) (15) Is there sufficient evidence to say that β_{1} > .45? Explain, using the p-value of an

appropriate test.

Extra Credit (Short Essay – no more than a paragraph, please!) (10) Exactly what does it mean when we reject a null hypothesis in a statistical test? Are we saying that the null hypothesis is definitely false? Explain using the quantities α and β. Also, how does the “strength” of the results of a test depend on the values of α and β?

Have a safe, enjoyable, and productive summer!