9-3. Exercises

1. BASIC

Ex 1. In all exercises for this section assume that the population of differences is normal.

Ex 2. Use the following paired sample data for this exercise Population 1 35 32 35 35 36 35 36 Population 2 28 26 27 26 29 27 29

1. Compute $\bar{d}$ and $s_d$ .

2. Give a point estimate for $μ_1−μ_2=μ_d.$

3. Construct the 95% confidence interval for $μ_1−μ_2=μ_d$ from these data.

4. Test, at the 10% level of significance, the hypothesis that $μ_1−μ_2>7$ as an alternative to the null hypothesis that $μ_1−μ_2=7.$

Ex 3. Use the following paired sample data for this exercise. Population 1 : 103 127 96 110 90 118 130 106 Population 2 : 81 106 73 88 70 95 109 83

1. Compute $\bar{d}$ and $s_d$ .

2. Give a point estimate for $μ_1−μ_2=μ_d.$

3. Construct the 90% confidence interval for $μ_1−μ_2=μ_d$ from these data.

4. Test, at the 1% level of significance, the hypothesis that $μ_1−μ_2<24$ as an alternative to the null hypothesis that $μ_1−μ_2=24.$

Ex 4. Use the following paired sample data for this exercise. Population 1 : 40 27 55 34 Population 2 : 53 42 68 50

1. Compute $\bar{d}$ and $s_d$ .

2. Give a point estimate for $μ_1−μ_2=μ_d.$

3. Construct the 99% confidence interval for $μ_1−μ_2=μ_d$ from these data.

4. Test, at the 10% level of significance, the hypothesis that $μ_1−μ_2≠−12$ as an alternative to the null hypothesis that $μ_1−μ_2=−12.$

Ex 5. Use the following paired sample data for this exercise. Population 1 : 196 165 181 201 190 Population 2 : 212 182 199 210 205

1. Compute $\bar{d}$ and $s_d$ .

2. Give a point estimate for $μ_1−μ_2=μ_d.$

3. Construct the 98% confidence interval for $μ_1−μ_2=μ_d$ from these data.

4. Test, at the 2% level of significance, the hypothesis that $μ_1−μ_2≠−20$ as an alternative to the null hypothesis that $μ_1−μ_2=−20.$

2. APPLICATIONS

Ex 6. Each of five laboratory mice was released into a maze twice. The five pairs of times to escape were:

Mouse	1	2	3	4	5
First release	129	89	136	163	118
Second release	113	97	139	85	75

1. Compute $\bar{d}$ and $s_d$ .

2. Give a point estimate for $c$

3. Construct the 90% confidence interval for $μ_1−μ_2=μ_d$ from these data.

4. Test, at the 10% level of significance, the hypothesis that it takes mice less time to run the maze on the second trial, on average.

Ex 7. Eight golfers were asked to submit their latest scores on their favorite golf courses. These golfers were each given a set of newly designed clubs. After playing with the new clubs for a few months, the golfers were again asked to submit their latest scores on the same golf courses. The results are summarized below.

Golfer	1	2	3	4	5	6	7	8
Own clubs	77	80	69	73	73	72	75	77
New clubs	72	81	68	73	75	70	73	75

1. Compute $\bar{d}$ and $s_d$ .

2. Give a point estimate for $μ_1−μ_2=μ_d.$

3. Construct the 99% confidence interval for $μ_1−μ_2=μ_d$ from these data.

4. Test, at the 1% level of significance, the hypothesis that on average golf scores are lower with the new clubs.

Ex 8. A neighborhood home owners association suspects that the recent appraisal values of the houses in the neighborhood conducted by the county government for taxation purposes is too high. It hired a private company to appraise the values of ten houses in the neighborhood. The results, in thousands of dollars, are

House	County Government	Private Company
1	217	219
2	350	338
3	296	291
4	237	237
5	237	235
6	272	269
7	257	239
8	277	275
9	312	320
10	335	335

1. Give a point estimate for the difference between the mean private appraisal of all such homes and the government appraisal of all such homes.

2. Construct the 99% confidence interval based on these data for the difference.

3. Test, at the 1% level of significance, the hypothesis that appraised values by the county government of all such houses is greater than the appraised values by the private appraisal company.

Ex 9. In order to cut costs a wine producer is considering using duo or 1 + 1 corks in place of full natural wood corks, but is concerned that it could affect buyers’s perception of the quality of the wine. The wine producer shipped eight pairs of bottles of its best young wines to eight wine experts. Each pair includes one bottle with a natural wood cork and one with a duo cork. The experts are asked to rate the wines on a one to ten scale, higher numbers corresponding to higher quality. The results are:

Wine Expert	Duo Cork	Wood Cork
1	8.5	8.5
2	8.0	8.5
3	6.5	8.0
4	7.5	8.5
5	8.0	7.5
6	8.0	8.0
7	9.0	9.0
8	7.0	7.5

1. Give a point estimate for the difference between the mean ratings of the wine when bottled are sealed with different kinds of corks.

2. Construct the 90% confidence interval based on these data for the difference.

3. Test, at the 10% level of significance, the hypothesis that on the average duo corks decrease the rating of the wine.

Ex 10. Engineers at a tire manufacturing corporation wish to test a new tire material for increased durability. To test the tires under realistic road conditions, new front tires are mounted on each of 11 company cars, one tire made with a production material and the other with the experimental material. After a fixed period the 11 pairs were measured for wear. The amount of wear for each tire (in mm) is shown in the table:

Car	Production	Experimental
1	5.1	5.0
2	6.5	6.5
3	3.6	3.1
4	3.5	3.7
5	5.7	4.5
6	5.0	4.1
7	6.4	5.3
8	4.7	2.6
9	3.2	3.0
10	3.5	3.5
11	6.4	5.1

1. Give a point estimate for the difference in mean wear.

2. Construct the 99% confidence interval for the difference based on these data.

3. Test, at the 1% level of significance, the hypothesis that the mean wear with the experimental material is less than that for the production material.

Ex 11. A marriage counselor administered a test designed to measure overall contentment to 30 randomly selected married couples. The scores for each couple are given below. A higher number corresponds to greater contentment or happiness.

Couple	Husband	Wife
1	47	44
2	44	46
3	49	44
4	53	44
5	42	43
6	45	45
7	48	47
8	45	44
9	52	44
10	47	42
11	40	34
12	45	42
13	40	43
14	46	41
15	47	45
16	46	45
17	46	41
18	46	41
19	44	45
20	45	43
21	48	38
22	42	46
23	50	44
24	46	51
25	43	45
26	50	40
27	46	46
28	42	41
29	51	41
30	46	47

1. Test, at the 1% level of significance, the hypothesis that on average men and women are not equally happy in marriage.

2. Test, at the 1% level of significance, the hypothesis that on average men are happier than women in marriage.

3. LARGE DATA SET EXERCISES

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data5.xls

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data12.xls

Ex 12. Large Data Set 5 lists the scores for 25 randomly selected students on practice SAT reading tests before and after taking a two-week SAT preparation course. Denote the population of all students who have taken the course as Population 1 and the population of all students who have not taken the course as Population 2.

1. Compute the 25 differences in the order  after− before after− before, their mean $\bar{d}$ , and their sample standard deviation $s_d$ .

2. Give a point estimate for $μ_d = μ_1−μ_2$ , the difference in the mean score of all students who have taken the course and the mean score of all who have not.

3. Construct a 98% confidence interval for $μ_d.$

4. Test, at the 1% level of significance, the hypothesis that the mean SAT score increases by at least ten points by taking the two-week preparation course.

Ex 13. Large Data Set 12 lists the scores on one round for 75 randomly selected members at a golf course, first using their own original clubs, then two months later after using new clubs with an experimental design. Denote the population of all golfers using their own original clubs as Population 1 and the population of all golfers using the new style clubs as Population 2.

1. Compute the 75 differences in the order  original clubs− new clubs original clubs− new clubs, their mean $\bar{d}$ , and their sample standard deviation $s_d$ .

2. Give a point estimate for $μ_d = μ_1−μ_2$, the difference in the mean score of all golfers using their original clubs and the mean score of all golfers using the new kind of clubs.

3. Construct a 90% confidence interval for $μ_d.$

4. Test, at the 1% level of significance, the hypothesis that the mean golf score decreases by at least one stroke by using the new kind of clubs.

Ex 14. Consider the previous problem again. Since the data set is so large, it is reasonable to use the standard normal distribution instead of Student’s t-distribution with 74 degrees of freedom.

1. Construct a 90% confidence interval for μdμd using the standard normal distribution, meaning that the formula is $\bar{d} ±z_{α∕2} \frac{s_d}{\sqrt{n}}$ (The computations done in part (a) of the previous problem still apply and need not be redone.) How does the result obtained here compare to the result obtained in part (c) of the previous problem?

2. Test, at the 1% level of significance, the hypothesis that the mean golf score decreases by at least one stroke by using the new kind of clubs, using the standard normal distribution. (All the work done in part (d) of the previous problem applies, except the critical value is now zαzα instead of tαtα (or the p-value can be computed exactly instead of only approximated, if you used the p-value approach).) How does the result obtained here compare to the result obtained in part (c) of the previous problem?

3. Construct the 99% confidence intervals for $μ_d$ using both the $t-$ and $z-$ distributions. How much difference is there in the results now?