Homework Assignment#5

due Wednesday, October 29, 2008

Complete the following exercises from your text: pages 263-265, #1, 2, 5a-e, 9, and practice problems 3.1.10, (page 195), 3.1.11 (page 199).

These exercises are reproduced below.

#1) Suppose that you want to estimate the proportion of vehicles on the road in your hometown that are sport utility vehicles (SUVs). You decide to stand at the intersection closest to your home between 7 and 8 A.M. every morning for a week, keeping track of how many vehicles go by and how many of them are SUVs.

(a) Identify the population, sample, parameter, and statistic (all in words) in this study.

(b) Give some reasons why your sampling method is probably not unbiased.

#2) Reconsider the previous question. Now suppose that you change your sampling plan and go to a local car dealership. You ask the manager to let you inspect a random sample of vehicles sold in the past year.

(a) Identify the sampling frame in this study.

(b) Is this sampling method likely to be unbiased for estimating the proportion of SUVs among all vehicles on the road in your town? Explain.

#5) Suppose that you want to estimate the average length of a word (measured by the number of letters) in this book.

(a) Is this number a parameter or a statistic? Explain. Also indicate what symbol would be commonly used to denote it.

Suppose that your sampling plan is to open the book haphazardly, set your finger on that page, haphazardly, record the number of letters in the word that your finger lands on, and then repeat this process.
(b) Does this constitute a simple random sample? Explain.
(c) Is this sampling plan likely to be unbiased? If so, explain why. If not, indicate whether the sample mean is likely to over- or underestimate the population mean, and explain your answer.

Now suppose that you decide to select a page at random, examine all of the words on that page, and calculate the average length of the words on that page.

(d) Is this number a parameter or a statistic? Explain. Also indicate what symbol would be commonly used to denote it.

(e) Answer (c) for this sampling plan.

#9) On a recent trip one of the authors took 4 flights and encountered 8 pilots (2 per flight). Seven of these 8 pilots were men, and 1 was a woman. Let the random variable X represent the number of women in a random sample of 8 commercial pilots.

(a) Would it be reasonable to model the distribution of X with a binomial distribution? Explain and clearly define the parameters of the distribution in words.

(b) Suppose that ½ of all commercial pilots are women. Determine the probability of encountering 1 or fewer women in a random sample of 8 commercial pilots. (Show the details of your calculations, or Minitab, or applet output, to support your answer.)

(c) Is the probability in (b) small enough to convince you that fewer than ½ of all commercial pilots are women, using the 0.05 level of significance?

(d) According to Women in Aviation International (WAI), women comprised 5% of all commercial pilots in 2001. Use this parameter value to calculate and graph the probability distribution of X. (Hint: List all possible values of X and their probabilities. You should use Minitab.)

(e) Assuming that the WAI parameter value is correct and using the results from (d), is 1 the most common value for the number of women in a random sample of 8 commercial pilots? Would you say that it is a surprising value? Explain.

Practice Problem #3.1.10 Sampling Words (cont.)

Recall that in part (o) in Investigation 3.1.4, you found that the probability of obtaining at most 1 noun in a random sample of ten words from the Gettysburg Address, assuming 50 nouns in the population, is .414.

a) Continue to suppose that there are 50 nouns in the population (π = .187) but a student obtains a random sample of 20 words that contained 2 nouns (= .10). Use the hypergeometric distribution and Minitab to decide if obtaining at most 2 nouns would be a surprising outcome. How does this p-value compare to that of part (0) in Investigation 3.1.4? Explain why this relationship makes sense.

b) Suppose we were sampling from 4 copies of the Gettysburg Address so that the population size was 1072 words. Suppose we thought there were 200 nouns in this population (π = .187) but our random sample of 10 words contained only 1 noun (= .10). Use the hypergeometric distribution and Minitab to decide if obtaining at most 1 noun would be a surprising outcome. Comment on how this probability compares to that from part (0) of Investigation 3.1.4.

c) Now redo part (a) of this exercise for this larger population (of 1072 words). How does this p-value compare to that from part (a)?

d) Does the population size appear to affect these p-value calculations considerably? Explain.

Practice Problem 3.1.11 Freshmen Voting Patterns (cont.)

Reconsider Investigation 3.1.5. Let k equal the number of Kerry supporters in the population of all 705 first-year students at this school.

(a) Determine the largest value of k such that the probability of getting 22 or more Kerry supporters in a random sample of 30 freshmen would be less than 0.05 (You will need to use Minitab and trial-and-error.)

(b) Determine the smallest value of k such that the probability of getting 22 or fewer Kerry supporters in a random sample of 30 freshmen would be less than 0.05.

(c) Explain the sense in which the values of k between these two extremes are plausible values for the number of Kerry supporters in the population in light of the sample results.

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