2.Construct the confidence interval for μ 1 -μ 2 for the level of confidence and the data from independent samples given.
a.95% confidence,
n 1 =110 , x - 1 =77 , s 1 =15
n 2 =85 , x - 2 =79 , s 2 =21
b.90% confidence,
n 1 =65 , x - 1 =-83 , s 1 =12
n 2 =65 , x - 2 =-74 , s 2 =8
4. Construct the confidence interval for μ 1 -μ 2 for the level of confidence and the data from independent samples given.
a.99.9% confidence,
n 1 =275 , x - 1 =70.2 , s 1 =1.5
n 2 =325 , x - 2 =63.4 , s 2 =1.1
b.90% confidence,
n 1 =120 , x - 1 =35.5 , s 1 =0.75
n 2 =146 , x - 2 =29.6 , s 2 =0.80
14.Records of 40 used passenger cars and 40 used pickup trucks (none used commercially) were randomly selected to investigate whether there was any difference in the mean time in years that they were kept by the original owner before being sold. For cars the mean was 5.3 years with standard deviation 2.2 years. For pickup trucks the mean was 7.1 years with standard deviation 3.0 years.
a.Construct the 95% confidence interval for the difference in the means based on these data.
b.Test the hypothesis that there is a difference in the means against the null hypothesis that there is no difference. Use the 1% level of significance.
c.Compute the observed significance of the test in part (b).
18. An kinesiologist claims that the resting heart rate of men aged 18 to 25 who exercise regularly is more than five beats per minute less than that of men who do not exercise regularly. Men in each category were selected at random and their resting heart rates were measured, with the results shown.
n x - s
Regular exercise 40 63 1.0
No regular exercise 30 71 1.2
1. Perform the relevant test of hypotheses at the 1% level of significance.
2. Compute the observed significance of the test.
20. The Municipal Transit Authority wants to know if, on weekdays, more passengers ride the northbound blue line train towards the city center that departs at 8:15 a.m. or the one that departs at 8:30 a.m. The following sample statistics are assembled by the Transit Authority.
n x - s
8:15 a.m. train 30 323 41
8:30 a.m. train 45 356 45
a. Construct the 90% confidence interval for the difference in the mean number of daily travellers on the 8:15 train and the mean number of daily travellers on the 8:30 train.
b. Test at the 5% level of significance whether the data provide sufficient evidence to conclude that more passengers ride the 8:30 train.
c. Compute the observed significance of the test.
26. Large Data Sets 1A and 1B list the GPAs for 1,000 randomly selected students. Denote the population of all male students as Population 1 and the population of all female students as Population 2.
https://www.flatworldknowledge.com/sites/all/files/data1A.xls
https://www.flatworldknowledge.com/sites/all/files/data1B.xls
1. Restricting attention to just the males, find n1, x - 1 , and s1. Restricting attention to just the females, find n2, x - 2 , and s2.
2. Let μ 1 denote the mean GPA for all males and μ 2 the mean GPA for all females. Use the results of part (a) to construct a 95% confidence interval for the difference μ 1 -μ 2 .
3. Test, at the 10% level of significance, the hypothesis that the mean GPAs among males and females differ.
Section 2
2. Construct the confidence interval for μ 1 -μ 2 for the level of confidence and the data from independent samples given.
1. 90% confidence,
n 1 =28 , x - 1 =212 , s 1 =6
n 2 =23 , x - 2 =198 , s 2 =5
2. 99% confidence,
n 1 =14 , x - 1 =68 , s 1 =8
n 2 =20 , x - 2 =43 , s 2 =3
4.Construct the confidence interval for μ 1 -μ 2 for the level of confidence and the data from independent samples given.
1. 99.5% confidence,
n 1 =40 , x - 1 =85.6 , s 1 =2.8
n 2 =20 , x - 2 =73.1 , s 2 =2.1
2. 99.9% confidence,
n 1 =25 , x - 1 =215 , s 1 =7
n 2 =35 , x - 2 =185 , s 2 =12
14.A genetic engineering company claims that it has developed a genetically modified tomato plant that yields on average more tomatoes than other varieties. A farmer wants to test the claim on a small scale before committing to a full-scale planting. Ten genetically modified tomato plants are grown from seeds along with ten other tomato plants. At the season's end, the resulting yields in pound are recorded as below.
Sample 1(genetically modified)20232725252527232422 Sample 2(regular)21212218202018252320
a.Construct the 99% confidence interval for the difference in the population means based on these data.
b.Test, at the 1% level of significance, whether the data provide sufficient evidence to conclude that the mean yield of the genetically modified variety is greater than that for the standard variety.
16.The coaching staff of professional football team believes that the rushing offense has become increasingly potent in recent years. To investigate this belief, 20 randomly selected games from one year's schedule were compared to 11 randomly selected games from the schedule five years later. The sample information on passing yards per game (pypg) is summarized below.
n x - s
pypg previously 20 203 38
pypg recently 11 232 33
1. Construct the 95% confidence interval for the difference in the population means based on these data.
2. Test, at the 5% level of significance, whether the data on passing yards per game provide sufficient evidence to conclude that the passing offense has become more potent in recent years.
Section 3
2.Use the following paired sample data for this exercise.
Population 1Population 2 10381 127106 9673 11088
Population 1Population 2 9070 11895 130109 10683
3. Compute d - and sd.
4. Give a point estimate for μ 1 -μ 2 =μ d .
5. Construct the 90% confidence interval for μ 1 -μ 2 =μ d from these data.
6. 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.
6.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 d - and sd.
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.
8.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
7. Give a point estimate for the difference between the mean ratings of the wine when bottled are sealed with different kinds of corks.
8. Construct the 90% confidence interval based on these data for the difference.
9. Test, at the 10% level of significance, the hypothesis that on the average duo corks decrease the rating of the wine.
12.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.
https://www.flatworldknowledge.com/sites/all/files/data12.xls
1. Compute the 75 differences in the order original clubs- new clubs , their mean d - , and their sample standard deviation sd.
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.
Section 4
1. Construct the confidence interval for p 1 -p 2 for the level of confidence and the data given. (The samples are sufficiently large.)
1. 98% confidence,
n 1 =750 , p ˆ 1 =0.64
n 2 =800 , p ˆ 2 =0.51
2. 99.5% confidence,
n 1 =250 , p ˆ 1 =0.78
n 2 =250 , p ˆ 2 =0.51
14. To investigate a possible relation between gender and handedness, a random sample of 320 adults was taken, with the following results:
Men Women
Sample size, n 168 152
Number of left-handed, x 24 9
1. Give a point estimate for the difference in the proportion of all men who are left-handed and the proportion of all women who are left-handed.
2. Construct the 95% confidence interval for the difference, based on these data.
3. Test, at the 5% level of significance, the hypothesis that the proportion of men who are left-handed is greater than the proportion of women who are.
4. Compute the p-value of the test.
16.In professional basketball games, the fans of the home team always try to distract free throw shooters on the visiting team. To investigate whether this tactic is actually effective, the free throw statistics of a professional basketball player with a high free throw percentage were examined. During the entire last season, this player had 656 free throws, 420 in home games and 236 in away games. The results are summarized below.
Home Away
Sample size, n 420 236
Free throw percent, p ˆ 81.5% 78.8%
1. Give a point estimate for the difference in the proportion of free throws made at home and away.
2. Construct the 90% confidence interval for the difference, based on these data.
3. Test, at the 10% level of significance, the hypothesis that there exists a home advantage in free throws.
4. Compute the p-value of the test.
20.Large Data Set 11 records the results of samples of real estate sales in a certain region in the year 2008 (lines 2 through 536) and in the year 2010 (lines 537 through 1106). Foreclosure sales are identified with a 1 in the second column. Let all real estate sales in the region in 2008 be Population 1 and all real estate sales in the region in 2010 be Population 2.
https://www.flatworldknowledge.com/sites/all/files/data11.xls
1. Use the sample data to construct point estimates p ˆ 1 and p ˆ 2 of the proportions p1 and p2 of all real estate sales in this region in 2008 and 2010 that were foreclosure sales. Construct a point estimate of p 1 -p 2 .
2. Use the sample data to construct a 90% confidence for p 1 -p 2 .
3. Test, at the 10% level of significance, the hypothesis that the proportion of real estate sales in the region in 2010 that were foreclosure sales was greater than the proportion of real estate sales in the region in 2008 that were foreclosure sales. (The default is that the proportions were the same.)
Section 5
2.Estimate the common sample size n of equally sized independent samples needed to estimate μ 1 -μ 2 as specified when the population standard deviations are as shown.
1. 80% confidence, to within 2 units, σ 1 =14 and σ 2 =23
2. 90% confidence, to within 0.3 units, σ 1 =1.3 and σ 2 =0.8
3. 99% confidence, to within 11 units, σ 1 =42 and σ 2 =37
4.Estimate the number n of pairs that must be sampled in order to estimate μ d =μ 1 -μ 2 as specified when the standard deviation sd of the population of differences is as shown.
1. 90% confidence, to within 20 units, σ d =75.5
2. 95% confidence, to within 11 units, σ d =31.4
3. 99% confidence, to within 1.8 units, σ d =4
8. A university administrator wishes to estimate the difference in mean grade point averages among all men affiliated with fraternities and all unaffiliated men, with 95% confidence and to within 0.15. It is known from prior studies that the standard deviations of grade point averages in the two groups have common value 0.4. Estimate the minimum equal sample sizes necessary to meet these criteria.