Mathematics 376 – Probability and Statistics 2 Midterm Examination 2 April 22, 2010

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. Let Y_{1}, …, Y_{n} be independent samples from the distribution with pdf containing the unknown parameter θ > 0:

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

A) (10) Determine the method of moments estimator for θ. B) (15) Determine the maximum likelihood estimator for θ.

II. A shop manufactures O-rings for the Space Shuttle booster rockets for NASA. Let d be the proportion of defectives in the shop’s output. A random sample of size n = 70 O-rings produced 6 defectives. A) (15) 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. B) (15) For the test in part A, 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 natural gas produced by two fields of gas wells (in calories per cubic meter):

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

Let μ_{i} (i = 1, 2) be the population mean heat-producing capacity of the natural gas from field i. A) (10) What assumptions do you need to make in order to use the appropriate test of

H_{0} : μ_{1} = μ_{2} versus H_{a} : μ_{1} ≠ μ_{2}? B) (15) Carry out the test from part A with α = .05 and state your conclusion clearly

and succinctly. C) (5) Construct a two-sided 95% confidence interval for the difference of the population

mean heat producing capacities μ_{1} − μ_{2}. How is this related to your answer in B?

IV. (15) Let (x_{i}, y_{i}), i = 1, …, n be a collection of data points in the plane. Using the matrix formulation, derive the normal equations for the least squares estimators for the coefficients β_{0}, β_{1}, β_{2}, β_{3} in the model Y = β_{0} + β_{1}x + β_{2}x^{2} + β_{3}x^{3} + ε fitting the data. (Note: You do not need to try to solve the equations!)

Extra Credit (10) In the situation of question I, suppose you set up a test of the hypothesis H_{0} : θ = 1 versus the alternative H_{a} : θ > 1 using Y_{1}, Y_{2} (exactly two of the samples). If you reject H_{0} when Y_{1} + Y_{2} > 1.8, what is the Type I error probability, α?