Stat 11

February 27, 2006

Homework #5 (complete version) (due Friday, March 3)

This homework is due at the start of class Friday, March 3. You may work in groups (across sections if you like), consult with others, or use any references or tools that seem useful, but you must write up your solutions yourself.

Problems from Chapter 4:

4.22, 4.28, 4.46, 4.58, 4.60, 4.64a. (Do 4.64b, if you like, for extra credit.)

For 4.60 and 4.64, a sensible approach is to determine m, s², and s separately

for each of the two variables and then combine them.

Problems related to estimating a proportion (section 5.1):

Some of these problems use the formula from class:

1. A simple random sample of 100 people is selected from the 14,000 adult residents of Fort Smith, Arkansas, and those in the sample are asked whether they favor a proposed highway project. It turns out that 55 of those in the sample say yes, they favor the project. (The other 45 said no. Assume that the sampling was done perfectly and that everyone selected gave an honest answer.)

Let p represent the (unknown) fraction of adult residents that favor the project.

a. In this problem, what are and n ?

b. From the information given, what is the best estimate of p ?

c. What would you use for the standard error of that estimate?

2. Continuing the situation from problem 1…

a. Suppose the actual value of p is 0.50. (That is, exactly 7000 of the 14000 residents would say yes if asked. The difference in this problem is that you know the actual value of p.) If thousands of samples of size 100 were done, and a value of were computed for each sample, what would be the mean of all the values ? What would be the standard deviation ?

b. Same questions, but now suppose that the actual value of p is 0.80.

c. The standard deviations you gave in parts a and b are different from the standard error you gave in problem 1c. Are you still comfortable with the answer you gave to problem 1c ? That is, do you think it’s a good way to estimate the standard error in the situation given?

3. A certain university has declared that historically, 70% of its football players get degrees. You’re skeptical, so in August, 2006, you undertake a brief investigation. You obtain from the university an official list of the players that entered the program during calendar years 1996-2002, carefully select a random sample of 40 of these players, and determine , the fraction of the sample that got degrees.

a. Suppose that the university’s claim is true. If you took lots of samples of size 40 and computed for each sample, what would be the mean of these values?

b. What would be the standard deviation?

c. What would be the shape of the distribution of values?

d. What fraction of the values would be 55% or below?

e. In fact, in your sample, 55% (that is, 22 out of 40) obtained degrees. Is the university’s claim plausible?

f. Can you think of a source of bias in your survey?

(end)