Mathematics 400 – Directed Readings in 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 using the Y_{i}. B) (15) Determine the maximum likelihood estimator for θ using the Y_{i}. C) (5) Suppose you have n = 5 data values y_{1}, …, y_{5} = .78, .80, .70, .84, .47. What are

the two estimates of θ given by your answers from parts A and B? D) (10) 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} > .8, what is the Type I error probability, α?

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) (15) 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}? How could you determine whether it is reasonable to assume those assumptions are satisfied? (You do not need to carry out a test here – just say how the test would be set up.)

B) (10) 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?

Extra Credit (10) In the situation of question I, is there one of the two estimators that you derived that is clearly “better” to use? Explain how you could answer a question like this.