Mathematics 376 – Probability and Statistics II Midterm Exam Im March 1, 2004

Directions: This is an open-book, open-notes exam. Do all work in the blue exam booklet. There are 100 regular points and 10 Extra Credit points (see back of this sheet). Be sure to read each question carefully before starting to work.

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

I. (20) The amount of time (in minutes) that a dial-up computer help desk specialist spends with one caller is a random variable with a gamma distribution for α = 4 and β = 1.8. Estimate the probability that the mean call time for a random sample of 64 customers will be between 6 and 10 minutes. Give the name of the theoretical result you are using to derive your estimate.

II. In a study conducted by the Agronomy Department at Cornell University, a researcher found that 111 out of 200 rutabaga seeds germinated when they were planted in soil maintained at 5^{∘}C, while 119 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) Can we rely on the results of this study to conclude that a greater proportion of

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

III. A random sample of 10 healthy adult males’ total serum cholesterol levels yielded values of

210, 220, 175, 227, 240, 192, 221, 201, 186, 219

A) (5) Estimate the mean cholesterol level of this population. B) (20) Assuming the cholesterol levels are normally distributed, find a 90% confidence

interval for the mean cholesterol level. Explain how you decided which method to use. C) (5) Can we assume that the true mean cholesterol level of this population is contained

in the interval you found in B? Why or why not?

IV. Let X_{1}, …, X_{n} and Y_{1}, …, Y_{n} be two independent random samples of equal sizes from a normal distribution with 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}

A) (10) What is the distribution of nΣ_{X}^{2}/σ^{2}? Explain. B) (5) What is the distribution of W = \frac{X_{1}−μ}{Σ_{Y}} = _{\frac{(X_{1}−μ)/σ}{\sqrt{Σ_{Y}^{2} /σ^{2}}}} ? Explain.

C) (10) Is Σ_{X}^{2} biased or unbiased as an estimator for σ^{2}?

Extra Credit (10) The median of a sample is the “middle” value, that is the value with the property that half the sample values are greater than or equal to the median and half are less than or equal to the median. Give formulas for computing the sample median in terms of the order statistics of the sample. (Note: the cases n even and n odd are different.)