ES9 Additional Exercises

Chapter 14

1. Find the 75% confidence interval for the median swim time for a swimmer whose recorded times are

24.7 24.7 24.6 25.5 25.7 25.8 26.5 24.5 25.3

26.2 25.5 26.3 24.2 25.3 24.3 24.2 24.2

2. Each year elementary students are given achievement tests in mathematics. The following table lists for 26 of the states the percentage of eighth grade students who in 1992 and 1996 scored at or above the proficient level in national tests:

a. Construct a table showing the sign of the difference between the 1992 and 1996 achievement tests.

b. Find a 95% confidence interval for the median difference in achievement test scores.

3. USA Today reported on May 8, 1991, “Teacher’s pay increases 5.4%.” The following sample of average teacher salary increases was taken from the annual report of the National Education Association.

1.6 11.4 4.2 8.7 4.5 6.4 5.9 3.7 4.5 4.6 4.4 4.1

7.6 5.5 10.6 2.8 6.8 3.0 3.5 7.9 8.0 4.3 2.4 5.0

Do the sample data suggest that the claim “median pay increase is 5.4%” be rejected at the 0.05 level of significance?

4. In order to test the null hypothesis that there is no difference is the ages of husbands and wives, the following data were collected.

Husbands 28 45 40 37 25 42 21 22 54 47 35 62 29 44 45 38 59

Wives 26 36 40 42 28 40 20 24 50 54 37 60 25 40 34 42 49

5. Each year elementary students are given achievement tests in mathematics. The percentage of eighth grade students who in 1992 and 1996 scored at or above the proficient level in national tests are given in Exercise 14.3. Use the sign test to determine at the 0.025 level of significance if the math achievement test scores improved between 1992 and 1996.

6. An article titled “Venocclusive Disease of the Liver: Development of a Model for Predicting Fatal Outcome After Marrow Transplantation” (Journal of Clinical Oncology, September 1993) gives the median age of 355 patients who underwent marrow transplantation at the Fred Hutchinson Cancer Research Center as 30 years. A sample of 100 marrow transplantation patients were recently selected for a study, and it was found that 40 of the patients were over 30 and 60 were under 30 years of age. Test the null hypothesis that the median age of the population from which the 100 patients were selected equals 30 years versus the alternative that the median does not equal 30 years. Use = 0.05.

7. An article titled “Naturally Occurring Anticoagulants and Bone Marrow Transplantation: Plasma Protein C Predicts the Development of Venocclusive Disease of the Liver” (Blood, June 1993) compared baseline values for antithrobin III with antithrobin II values 7 days after a bone marrow transplant for 45 patients. The differences were found to be nonsignificant. Suppose 17 of the differences were positive and 28 were negative. The null hypothesis is that the median difference is zero, and the alternative hypothesis is that the median difference is not zero. Use the 0.05 level of significance. Complete the test and carefully state your conclusion.

8. According to an article in a Newsweek special issue (Fall/Winter 1990), 51.1% of 17-year-olds answered the following question correctly:

If 7X + 4 = 5X + 8, then X = ___ a) 1 b) 2 c) 4 d) 6

Suppose we wished to test the null hypothesis that one-half of all 16-year-olds could solve the problem above against the alternative hypothesis, “the proportion who can solve differs from one-half.” Furthermore, suppose we asked 75 randomly selected 17-year-olds to solve the problem. Let + represent a correct solution and - represent an incorrect solution. Do we have sufficient evidence to show the proportion who can solve is different than one-half? Explain.

a. If we obtain 20 (+) signs and 55 (–) signs.

b. If we obtain 27 (+) signs and 48 (–) signs.

c. If we obtain 30 (+) signs and 45 (–) signs.

d. If we obtain 33 (+) signs and 42 (–) signs.

9. The following set of data represents the ages of drivers involved in automobile accidents. Do these data present sufficient evidence to conclude that there is a difference in the average age of men and women drivers involved in accidents? Use a two-tailed test at = 0.05.

Men 70 60 77 39 36 28 19 40 23 23 63 31 36 55 24 76

Women 62 46 43 28 21 22 27 42 21 46 33 29 44 29 56 70

a. State the null hypothesis that is being tested.

b. Complete the test using a computer or calculator.

10. Commercial airlines are often evaluated on the basis of two major performance categories: on-time arrivals and baggage handling. In 1997 and 1998, Delta Airlines received the following competitive ratings (the lower the better) on each of these dimensions over a 13-month period:

a. Convert the table to a table of ranks of the on-time arrivals (A) and baggage handling (B) for Delta.

b. Use the Mann-Whitney U test to test the hypothesis that baggage handling obtained higher ratings than on-time arrivals during the period. Use the 0.05 level of significance.

11. Stock quotations are influenced by numerous variables, including the earnings of the companies, economic conditions, stock market trading volume, press releases, and splits. The table below shows the August 18, 1998 closing prices of 20 stocks selected from the New York Stock Exchange. These were the first 20 stocks whose company names started with the letter B. All prices have been rounded to the nearest dollar.

a. Determine the median closing price and the number of runs above and below the median.

b. Use the runs test to test whether these stock prices are listed in a random sequence about the median.

c. State your conclusion.

12. An article titled “Clintonomics Hurt Middle Class, Poor” (USA Today, 10-17-94) states that the median income for 1993 equals $36,959. A random sample of 250 incomes has a median value different from any of the 250 incomes in the sample. The data contains 105 runs above and below the median. Use the above information to test the null hypothesis that the incomes in the sample form a random sequence with respect to the two properties above and below the median value versus the alternative that the sequence is not random at = 0.05.

13. The December 30, 1998 issue of the USA Today gives the Nielsen rankings for America’s favorite prime-time television programs, for the last week of 1998. The top ten programs are listed below along with the rankings assigned by a panel of educators.

Monday Night Football 1.0 8.0

60 Minutes 2.0 1.5

Touched by an Angel 3.0 3.0

Barbara Walters Presents 4.0 4.5

Dateline NBC-Monday 5.0 1.5

Home Improvement 6.0 4.5

CBS Sunday Movie 7.0 9.0

Walker Texas Ranger 8.0 10.0

Everybody Loves Raymond 9.5 6.0

Compute the Spearman rank coefficient. Test the null hypothesis that there is no relationship between the Nielsen rankings and the panel rankings versus the alternative that there is a relationship between them. Use = 0.05.

14. While trying to decide on the best time to harvest his crop, a commercial apple farmer recorded the day on which the first apple on the top half and the first apple on the bottom half of 20 randomly selected trees were ripe. The variable x was assigned a value of 1 on the first day that the first ripe apple appeared on 1 of the 20 trees. The days were then numbered sequentially. The observed data are shown in the following table. Do these data provide convincing evidence that the apples on the top of the trees start to ripen before the apples on the bottom half? Use = 0.05.

Tree

Position 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Top 5 6 1 4 5 3 6 7 8 5 8 6 4 7 8 10 3 2 9 7

15. A sample of 32 students received the following grades on an exam.

41 42 48 46 50 54 51 42 51 50 45 42 32 45 43 56

55 47 45 51 60 44 57 57 47 28 41 42 54 48 47 32