ES9 Additional Exercises

Chapter 9

1. There seems to be no end to how large the signing bonuses professional athletes can obtain when they start their careers. When the Indianapolis Colts gave Peyton Manning $11.6 million and the San Diego Chargers awarded Ryan Leaf $11.25 million as signing bonuses in 1998, both these amounts exceeded what the 1989 first round draft pick, Troy Aikman, earned in his first five National Football League seasons combined while playing for the Dallas Cowboys.

Source: Sports Illustrated, “Inside the NFL: Powerball Numbers”, August 10, 1998.

Suppose a sample of 18 new NFL players report their signing bonuses at the start of the 1998 season, and the results show a mean of $3.81 million and a standard deviation of $1.7 million.

a. Estimate with 95% confidence the mean signing bonus based on the report. [Specify the population parameter of interest, the criteria, the sample evidence, and the interval limits.]

b. Discuss how this situation does or does not satisfy the assumptions for the inference.

2. Ten randomly selected shut-ins were each asked to list how many hours of television they watched per week. The results are

82 66 90 84 75 88 80 94 110 91

Determine the 90% confidence interval estimate for the mean number of hours of television watched per week by shut-ins. Assume the number of hours is normally distributed.

3. The weights of the drained fruit found in 21 randomly selected cans of peaches packed by Sunny Fruit Cannery were (in ounces)

11.0 11.6 10.9 12.0 11.5 12.0 11.2 10.5 12.2 11.8 12.1

11.6 11.7 11.6 11.2 12.0 11.4 10.8 11.8 10.9 11.4

Using a computer or a calculator,

a. Calculate the sample mean and standard deviation.

b. Assume normality and construct the 98% confidence interval for the estimate of the mean weight of drained peaches per can.

4. "Obesity raises heart-attack risk" according to a study published in the March 1990 issue of the New England Journal of Medicine. "Those about 15 to 25 percent above desirable weight had twice the heart disease rate." Suppose the data listed below are the percentages above desired weight for a sample of patients involved in a similar study.

18.3 19.7 22.1 19.2 17.5 12.7 22.0 17.2 21.1 16.2 15.4

19.9 21.5 19.8 22.5 16.5 13.0 22.1 27.7 17.9 22.2 19.7

18.1 22.4 17.3 13.3 22.1 16.3 21.9 16.9 15.4 19.3

Use a computer or calculator to test the null hypothesis,  = 18%, versus the alternative hypothesis,   18%. Use = 0.05.

5. A telephone survey was conducted to estimate the proportion of households with a personal computer. Of the 350 households surveyed, 75 had a personal computer.

a. Give a point estimate for the proportion in the population who have a personal computer.

b. Give the maximum error of estimate with 95% confidence.

7. "Parents should spank children when they think it is necessary, said 51% of adult respondents to a survey-though most child-development experts say spanking is not appropriate. The survey of 7225 adults . . . was co-sponsored by Working Mother magazine and Epcot Center at Walt Disney World." This statement appeared in the Rochester Democrat & Chronicle (12-20-90). Find the 99% confidence maximum error of estimate for the parameter p, P(should spank when necessary), for the adult population.

8. In a survey of 12,000 adults aged 19 to 74, National Cancer Institute researchers found that 9% in the survey ate at least the recommended two servings of fruit or juice and three servings of vegetables per day (Ladies Home Journal, April 1991). Use this information to determine a 95% confidence interval for the true proportion in the population who follow the recommendation.

9. a. Calculate the maximum error of estimate for p for the 95% confidence interval for each of the situations listed in the table.

Approximate Value of p

Sample Size n	0.1	0.3	0.5	0.7	0.9
100
500
1000
1500

b. Explain the relationship between answers in columns 0.1 and 0.9; 0.3 and 0.7.

10. According to the June 1994 issue of Bicycling, only 16% of all bicyclists own helmets. You wish to conduct a survey in your city to determine what percent of the bicyclists own helmets. Use the national figure of 16% for your initial estimate of p.

a. Find the sample size if you want your estimate to be within 0.02 with 90% confidence.

b. Find the sample size if you want your estimate to be within 0.04 with 90% confidence.

c. Find the sample size if you want your estimate to be within 0.02 with 98% confidence.

d. What effect does changing the maximum error have on the sample size? Explain.

e. What effect does changing the level of confidence have on the sample size? Explain.

11. A bank believes that approximately 2/5 of its checking-account customers have used at least one other service provided by the bank within the last six months. How large a sample will be needed to estimate the true proportion to within 5% at the 98% level of confidence?

You wish to conduct a study of extreme weather forecast accuracy by comparing local forecasts with actual weather conditions occurring in your city.

a. What is the best estimate available for the probability of accuracy in predicting extreme weather events.

b. Find the sample size if you want your estimate to be within 0.02 with 90% confidence.

c. Find the sample size if you want your estimate to be within 0.04 with 95% confidence.

d. Find the sample size if you want your estimate to be within 0.06 with 99% confidence.

e. If the level of confidence remains constant, what happens to the required sample size if you wish to double the maximum error of your estimate?

13. According to the May 1990 issue of Good Housekeeping, only about 14% of lung cancer patients survive for five years after diagnosis. Suppose you wanted to see if this survival rate were still true. How large a sample would you need to take to estimate the true proportion surviving for five years after diagnosis to within 1% with 95% confidence? (Use the 14% as the value of p.)

14. The article "Making Up for Lost Time" (U.S. News & World Report, July 30, 1990) reported that more than half of the country's workers aged 45 to 64 want to quit work before they reach age 65. Suppose you conduct a survey of 1000 randomly chosen workers in order to test H_o: p = 0.5 versus H_a: p < 0.5, where p represents the proportion who want to quit before they reach age 65. 460 of the 1000 sampled want to quit work before age 65. Use = 0.01.

a. Calculate the value of the test statistic.

b. Solve using the p-value approach. c. Solve using the classical approach.

15

Many students work full-time or part-time. Listed below is the amount earned last month by each in a sample of 35 college students.

0 362 276 158 409 0 0 534 449 281 37 338

240 0 0 0 142 0 519 356 280 161 0
Use this sample data to describe the amount earned by working college students.

a. How many of the students in the sample above are working?

b. Describe the variable, amount earned by a working college student last month, using one graph, one measure of central tendency, one measure of dispersion.

c. Find evidence to show that the assumptions for use of Student's t-distribution have been satisfied.

d. Estimate the mean amount earned by a college student per month using a point estimate and a 95% confidence interval.

e. A Statistical Snapshot® suggests the average amount earned each month by college students is approximately $350. Does the sample show sufficient reason to reject that claim?

Source: Family Circle, “The Great Grocery Challenge”, September 15, 1998.

Suppose the mean of all coupon sales at supermarkets in the U.S. is $10. A random sample of 25 shoppers with annual household incomes exceeding $75,000 is taken and reveals a mean redemption of $15 and a standard deviation of $7. Do shoppers from the higher-income group redeem coupons worth more than those redeemed by the rest of the nation? Use  = 0.01.

17. Home schooling became legal in all 50 states in 1993, thus allowing parents to take charge of their kids’ education from kindergarten to college. Researchers estimate that as many as 1.5 million children and teenagers in 1998 were being taught primarily by their mothers and fathers; about five times as many as there were ten years earlier. This number is rather remarkable considering that the number of two-income households also rose during the same period. (i.e., Who’s staying at home teaching the kids?) The average ACT score for a home-schooled in 1998 was 23, whereas the average for traditionally schooled students was 21.

Source: Newsweek, “Learning at Home: Does it Pass the Test?”, October 5, 1998.

Suppose a recent survey of 22 home schoolers in your state revealed a mean ACT score of 23.2 and a standard deviation of 4.1. Do the ACT scores of home schoolers in your state exceed the scores for the traditionally schooled? Use the 0.05 level of significance.

18. The LEXIS, a national law journal, found from a survey conducted on April 6-7, 1991, that nearly two-thirds of the 800 people surveyed said doctors should not be prosecuted for helping people with terminal illnesses commit suicide. The poll carries a margin of error of plus or minus 3.5%.

a. Describe how this survey of 800 people fits the properties of a binomial experiment. Specifically identify: n, a trial, success, p, and x.

b. Exactly what is the "two-thirds" reported? How was it obtained? Is it a parameter or a statistic?

c. Calculate the 95% confidence maximum error of estimate for the population proportion of all people who believe doctors should not be prosecuted.

d. How is the maximum error, found in (c), related to the 3.5% mentioned in the survey report?

19. Prevention magazine reported in its latest survey that 64% of adult Americans, or 98 million people, were overweight. The telephone survey of 1254 randomly selected adults was conducted November 8-29, 1990, and had a margin of error of three percentage points.

a. Calculate the maximum error of estimate for 0.95 confidence with p' = 0.64.

b. How is the margin of error of three percentage points related to answer (a)?

c. How large a sample would be needed to reduce the maximum error to 0.02 with 95% confidence?

20. "Two of five Americans believe the country should rely on nuclear power more than other energy sources for energy in the 1990s, according to a poll released yesterday  .The telephone poll, taken April 10 - 11, has a margin of error of plus or minus 3 points." This statement appeared in the Rochester Democrat & Chronicle on April 21, 1991. Forty percent plus or minus three points sounds like a confidence interval.

a. What is another name for the "margin of error of plus or minus 3 points"?

b. If we assume a 95% level of confidence, how large a sample is needed for a maximum error of 0.03?