Mathematics 376 – Probability and Statistics II Midterm Exam I March 1, 2006

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

I. Assume that the calorie content of chocolate Power Bars is normally distributed. A random sample of 10 chocolate Power Bars yields calorie content values of:

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

A) (10) Estimate the population mean calorie content of the chocolate Power Bars. B) (10) The sample variance of the calorie content values above is S^{2} = 70.68 cal. Find

a 99% confidence interval for the population mean calorie content, μ. C) (10) Find a 95% confidence interval for the calorie content variance σ^{2}.

II. A researcher found that 107 out of 200 rutabaga 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) (10) 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. (15) Let Y_{1}, …, Y_{n} denote a random sample from a population whose density is given by

f(y) = \{ \begin{matrix}3β^{3}y^{−4} & if y ≥ β \\ 0 & otherwise \end{matrix}

where β is unknown. Consider the estimator β̂ = min(Y_{1}, …, Y_{n}). Is β̂ biased or unbiased?

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) Use your answer from part A to show that Σ_{X}^{2} is an unbiased estimator for σ^{2}. C) (10) 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.)