Mathematics 376 – Probability and Statistics II Midterm Exam I February 27, 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. 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. A random sample of 10 chocolate Power Bars yielded calorie content values of

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

A) (5) Estimate the mean calorie content of this type of chocolate Power Bar. B) (15) The sample variance of the calorie content values above is S^{2} = 70.68 cal. Assum-

ing the calorie content is normally distributed, find a 99% confidence interval for the mean calorie content of this type of chocolate Power Bar. Explain how you decided which method to use.

C) (5) Can we assume that the true mean calorie content of this type of chocolate Power Bar is contained in the interval you found in B? Why or why not?

II. In a study conducted by the Horticulture Department at Virginia Tech, a researcher found that 107 out of 200 broccoli 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 90% 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

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

III. The amount of time (in minutes) that a drive-through bank teller takes to serve a single customer is a random variable with a gamma distribution for α = 1.6 and β = 2. A) (20) Estimate the probability that the mean service time for a random sample of

64 customers will be between 2 and 3 minutes. Explain how you are deriving your estimate.

B) (5) If 100 customers were included in the sample instead of 64, would your estimate from part A increase, decrease, or stay the same?

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) (10) Use your answer from part A to show that Σ_{X}^{2} is an unbiased estimator for σ^{2}. C) (5) What is the distribution of Σ_{X}^{2}/Σ_{Y}^{2} ? Explain.

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.)