Mathematics 376 – Probability and Statistics II Midterm Exam 1 – March 11, 2010

Directions:

Do all work in the blue exam booklet. There are 100 possible points and 10 Extra Credit points (Please turn over for part C of question IV and the Extra Credit problem.)

I. In a random poll of female students at BigState University, 390 out of 500 sorority women were in favor of lengthening the spring break, while 64 out of 100 non-sorority women were in favor. A) (15) Give a 99% confidence interval for the difference of the proportions favoring

lengthening spring break. B) (5) Now suppose a new poll of equal numbers of women from both groups is being

planned. How many women should be polled so that error of estimation is no larger than .08 according to a two standard error bound? (Be “conservative.”)

II. Two methods for teaching reading were applied to two randomly selected groups of school children, and compared by means of a common test. The sample means (\overline{y}) and variances (s^{2}) of the test scores for the two groups are as follows:

Statistic  Method 1 Method 2
Number of Children  11  14
\overline{y}  64  69
s^{2}  20  15

A) (10) Find a 95% confidence interval for σ_{1}^{2} (the variance of the test scores for students

taught with Method 1. What assumptions are you making? B) (20) Find a 95% confidence interval for μ_{1} −μ_{2} (the difference in the population mean

test scores with the two methods). What assumptions are you making? C) (5) Do your results support the conclusion that Method 2 is a superior method for

teaching reading? Explain.

III. A random sample of size 100 was taken from a normal population with σ^{2} = 8. A confidence interval for the population mean was given as 6±.047. What was the confidence coefficient associated with this interval (that is, the α value in the (1 − α) × 100%)?

IV. Let Y_{1}, …, Y_{n} denote a random sample of size n from a population described by the pdf:

f(y) = \{ \begin{matrix}\frac{2y}{θ^{2}} & if 0 ≤ y ≤ θ \\ 0 & otherwise. \end{matrix}

Let Y_{(n)} be the sample maximum and let U = \frac{1}{θ}Y_{(n)}. A) (10) Find a constant c so that θ̂ = cY_{(n)} is an unbiased estimator for θ.

B) (10) Show that U has CDF given by

F_{U}(u) = \{ \begin{matrix}0 & if u < 0 \\ u^{2n} & if 0 ≤ u ≤ 1 \\ 1 & if u > 1 \end{matrix}

C) (10) Note that the distribution of U does not depend on θ. Use this information to derive a 95% upper confidence bound on θ from the samples.

Extra Credit. A) (5) (Multiple Choice, and explain) A survey was conducted to determine what features

users want included in cell phone service. The results said a 95% confidence interval showed that 73% ± 4% of users wanted email services. What is meant by the ±4%?

1) As many as 4% of those polled may change their minds after the time the survey was conducted.

2) The probability is .95 that the true population percentage favoring email service is in the interval (69, 77).

3) It would be unlikely to get the observed sample proportion if the true population percentage was outside the interval (69, 77).

4) Only 4% of cellphone users were polled.

B) (5) Let Z_{1}, …, Z_{5} be independent samples from a N(0, 1) population, and define:

W = \frac{2Z_{1}}{\sqrt{Z_{2}^{2} + ⋯ + Z_{5}^{2}}} .

What is the distribution of W^{2}? Explain.