Mathematics 376 – Probability and Statistics 2 Midterm Examination 2 April 21, 2006

Directions

Do all work in the blue exam booklet, and include all work necessary to justify your answers. There are 100 regular points and 10 Extra Credit points.

I. (20) Let Y_{1}, …, Y_{n} be independent samples from the distribution with pdf containing the unknown parameter θ:

f(y|θ) = \{ \begin{matrix}\frac{1}{θ}y^{1/θ−1} & if 0 < y < 1 \\ 0 & otherwise \end{matrix}

Determine the maximum likelihood estimator for θ using the Y_{i}.

II. A machine shop manufactures toggle levers. Let d be the proportion of defectives in the shop’s output. A random sample of size n = 150 toggle levers produced 6 defectives. A) (10) Find a 95% confidence interval for d using this information. B) (10) Test the hypothesis H_{0} : d = .1 versus H_{a} : d ≠ .1 using this data. Take α = .02

(probability of Type I error). Also give the p-value of your test. C) (10) For the test in part B, what is β (probability of Type II error) if the true value

of d is .03?

III. Consider the following measurements of the heat-producing capacity of the oil produced by two fields (in billions of calories per ton):

Field 1 : 8.26 8.13 8.35 8.07 8.34
Field 2 : 7.95 7.89 7.90 8.14 7.92 7.84

A) (10) Estimate the variance σ^{2} of the heat producing capacity of the oil from each field. B) (10) Test the null hypothesis H_{0} : σ_{1}^{2} = σ_{2}^{2} against the alternative H_{a} : σ_{1}^{2} ≠ σ_{2}^{2} at the

α = .05 level. State your conclusion clearly and succinctly. C) (15) Construct a two-sided 95% confidence interval for the difference of the population

mean heat producing capacities μ_{1} − μ_{2}.

IV. (15) Let (x_{i}, y_{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} in the model Y = β_{0} + β_{1}x + ε fitting the data.

Extra Credit (10) In the situation of question III part C above, suppose you had n = 50 = m measurements from each oil field instead of n_{1} = 5 and n_{2} = 6. What, if anything, would change in the method you would use? Explain.