Mathematics 376 – Mathematical Statistics Midterm Exam I – March 15, 2012

Directions: Do all work in the blue exam booklet. There are 100 regular points and 10 Extra Credit points (question on back of this sheet). General Note: A “random sample” always consists of independent measurements from the indicated distribution.

I. Let X_{1}, …, X_{n} and Y_{1}, …, Y_{n} be two independent random samples of equal sizes from a population with a normal distribution having a known mean μ, but unknown variance σ^{2}. Let Σ_{X}^{2} and Σ_{Y}^{2} be the statistics:

Σ_{X}^{2} = \frac{1}{n} ∑_{i=1}^{n}(X_{i} − μ)^{2} Σ_{Y}^{2} = \frac{1}{n} ∑_{i=1}^{n}(Y_{i} − μ)^{2}.

Answer the following questions, giving complete explanations:

A) (5) What is the distribution of \frac{X_{i}−μ}{σ}?

B) (5) What is the distribution of W = \frac{nΣ_{X}^{2}}{σ^{2}} ?

C) (10) What is the distribution of V = \frac{Σ_{X}^{2}}{Σ_{Y}^{2}} ?

D) (10) What is the distribution of U = \frac{Y_{1}−μ}{Σ_{X}}?

II. Assume that the calorie content of Nature Valley granola bars is normally distributed. A random sample of 10 granola bars yields calorie content values of:

210, 220, 234, 227, 229, 240, 221, 228, 225, 219

A) (20) Find a 90% confidence interval for the population mean calorie content, μ, for

this type of granola bar. Explain how you know which method to use. B) (10) Find a 95% confidence interval for the population calorie content variance σ^{2}.

III. A researcher found that in two independent random samples, 107 out of 200 apple seeds germinated when they were planted in soil maintained at 5^{∘}C, while 123 out of 200 seeds germinated when they were planted in soil maintained at 15^{∘}C.

A) (20) Construct a 95% confidence interval for the difference in the proportions of seeds

that germinate at the two temperatures. B) (5) Do the results of this study suggest that a greater proportion of apple seeds will

germinate at the higher temperature in general? Why or why not?

IV. (15) Let Y_{1}, …, Y_{n} denote a random sample from a uniformly distributed population on an interval [θ, 2θ] where θ is unknown. Consider the estimator θ̂ = min(Y_{1}, …, Y_{n}). Is θ̂ biased or unbiased?

Extra Credit (10) In honor of π-day, here is the fact behind one of the demos that we were running in the McBrien Student Common Room yesterday. Suppose we drop a rod 1 unit long so that its midpoint lands at a point with y-coordinate uniformly distributed on the interval [0, 1/2], and the angle between the horizontal and the rod is also uniformly distributed on the interval [0, π]. Show that the probability that the rod intersects the x-axis is \frac{2}{π}.