Three students, Linda, Tuan, and Javier, are given five laboratory rats each for a nutritional experiment. Each rat’s weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again and the net gain in grams is recorded.
Linda's rats
Tuan's rats
Javier's rats
43.5
47.0
51.2
39.4
40.5
40.9
41.3
38.9
37.9
46.0
46.3
45.0
38.2
44.2
48.6
Table \(\PageIndex{18}\)
Determine whether or not the variance in weight gain is statistically the same among Javier’s and Linda’s rats. Test at a significance level of 10%.
56.
A grassroots group opposed to a proposed increase in the gas tax claimed that the increase would hurt working-class people the most, since they commute the farthest to work. Suppose that the group randomly surveyed 24 individuals and asked them their daily one-way commuting mileage. The results are as follows.
Working-class
Professional (middle incomes)
Professional (wealthy)
17.8
16.5
8.5
26.7
17.4
6.3
49.4
22.0
4.6
9.4
7.4
12.6
65.4
9.4
11.0
47.1
2.1
28.6
19.5
6.4
15.4
51.2
13.9
9.3
Table \(\PageIndex{19}\)
Determine whether or not the variance in mileage driven is statistically the same among the working class and professional (middle income) groups. Use a 5% significance level.
Use the following information to answer the next two exercises. The following table lists the number of pages in four different types of magazines.
Home decorating
News
Health
Computer
172
87
82
104
286
94
153
136
163
123
87
98
205
106
103
207
197
101
96
146
Table \(\PageIndex{20}\)
57.
Which two magazine types do you think have the same variance in length?
Which two magazine types do you think have different variances in length?
59.
Is the variance for the amount of money, in dollars, that shoppers spend on Saturdays at the mall the same as the variance for the amount of money that shoppers spend on Sundays at the mall? Suppose that the Table \(\PageIndex{21}\) shows the results of a study.
Are the variances for incomes on the East Coast and the West Coast the same? Suppose that Table \(\PageIndex{22}\) shows the results of a study. Income is shown in thousands of dollars. Assume that both distributions are normal. Use a level of significance of 0.05.
East
West
38
71
47
126
30
42
82
51
75
44
52
90
115
88
67
Table \(\PageIndex{22}\)
61.
Thirty men in college were taught a method of finger tapping. They were randomly assigned to three groups of ten, with each receiving one of three doses of caffeine: 0 mg, 100 mg, 200 mg. This is approximately the amount in no, one, or two cups of coffee. Two hours after ingesting the caffeine, the men had the rate of finger tapping per minute recorded. The experiment was double blind, so neither the recorders nor the students knew which group they were in. Does caffeine affect the rate of tapping, and if so how?
King Manuel I, Komnenus ruled the Byzantine Empire from Constantinople (Istanbul) during the years 1145 to 1180 A.D. The empire was very powerful during his reign, but declined significantly afterwards. Coins minted during his era were found in Cyprus, an island in the eastern Mediterranean Sea. Nine coins were from his first coinage, seven from the second, four from the third, and seven from a fourth. These spanned most of his reign. We have data on the silver content of the coins:
First coinage
Second coinage
Third coinage
Fourth coinage
5.9
6.9
4.9
5.3
6.8
9.0
5.5
5.6
6.4
6.6
4.6
5.5
7.0
8.1
4.5
5.1
6.6
9.3
6.2
7.7
9.2
5.8
7.2
8.6
5.8
6.9
6.2
Table \(\PageIndex{24}\)
Did the silver content of the coins change over the course of Manuel’s reign?
Here are the means and variances of each coinage. The data are unbalanced.
The American League and the National League of Major League Baseball are each divided into three divisions: East, Central, and West. Many years, fans talk about some divisions being stronger (having better teams) than other divisions. This may have consequences for the postseason. For instance, in 2012 Tampa Bay won 90 games and did not play in the postseason, while Detroit won only 88 and did play in the postseason. This may have been an oddity, but is there good evidence that in the 2012 season, the American League divisions were significantly different in overall records? Use the following data to test whether the mean number of wins per team in the three American League divisions were the same or not. Note that the data are not balanced, as two divisions had five teams, while one had only four.
Three different traffic routes are tested for mean driving time. The entries in the Table \(\PageIndex{29}\) are the driving times in minutes on the three different routes.
Route 1
Route 2
Route 3
30
27
16
32
29
41
27
28
22
35
36
31
Table \(\PageIndex{29}\)
State \(SS_{between}\), \(SS_{within}\), and the \(F\) statistic.
65.
Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses.
Northeast
South
West
Central
East
16.3
16.9
16.4
16.2
17.1
16.1
16.5
16.5
16.6
17.2
16.4
16.4
16.6
16.5
16.6
16.5
16.2
16.1
16.4
16.8
\(\overline x\)=
________
________
________
________
________
\(s^2=\)
________
________
________
________
________
Table \(\PageIndex{30}\)
State the hypotheses.
\(H_0\): ____________
\(H_a\): ____________
12.3 The F Distribution and the F-Ratio
Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses.
Three students, Linda, Tuan, and Javier, are given five laboratory rats each for a nutritional experiment. Each rat's weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again, and the net gain in grams is recorded. Using a significance level of 10%, test the hypothesis that the three formulas produce the same mean weight gain.
Linda's rats
Tuan's rats
Javier's rats
43.5
47.0
51.2
39.4
40.5
40.9
41.3
38.9
37.9
46.0
46.3
45.0
38.2
44.2
48.6
Table \(\PageIndex{32}\) Weights of Student Lab Rats
70.
A grassroots group opposed to a proposed increase in the gas tax claimed that the increase would hurt working-class people the most, since they commute the farthest to work. Suppose that the group randomly surveyed 24 individuals and asked them their daily one-way commuting mileage. The results are in Table \(\PageIndex{33}\). Using a 5% significance level, test the hypothesis that the three mean commuting mileages are the same.
Working-class
Professional (middle incomes)
Professional (wealthy)
17.8
16.5
8.5
26.7
17.4
6.3
49.4
22.0
4.6
9.4
7.4
12.6
65.4
9.4
11.0
47.1
2.1
28.6
19.5
6.4
15.4
51.2
13.9
9.3
Table 12.33
Use the following information to answer the next two exercises. Table \(\PageIndex{34}\) lists the number of pages in four different types of magazines.
Home decorating
News
Health
Computer
172
87
82
104
286
94
153
136
163
123
87
98
205
106
103
207
197
101
96
146
Table \(\PageIndex{34}\)
71.
Using a significance level of 5%, test the hypothesis that the four magazine types have the same mean length.
Eliminate one magazine type that you now feel has a mean length different from the others. Redo the hypothesis test, testing that the remaining three means are statistically the same. Use a new solution sheet. Based on this test, are the mean lengths for the remaining three magazines statistically the same?
73.
A researcher wants to know if the mean times (in minutes) that people watch their favorite news station are the same. Suppose that Table \(\PageIndex{35}\) shows the results of a study.
CNN
FOX
Local
45
15
72
12
43
37
18
68
56
38
50
60
23
31
51
35
22
Table \(\PageIndex{35}\)
Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05.
Are the means for the final exams the same for all statistics class delivery types? Table \(\PageIndex{36}\) shows the scores on final exams from several randomly selected classes that used the different delivery types.
Online
Hybrid
Face-to-Face
72
83
80
84
73
78
77
84
84
80
81
81
81
86
79
82
Table \(\PageIndex{36}\)
Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05.
75.
Are the mean number of times a month a person eats out the same for whites, blacks, Hispanics and Asians? Suppose that TableTable \(\PageIndex{38}\) shows the results of a study.
Powder
Machine Made
Hard Packed
1,210
2,107
2,846
1,080
1,149
1,638
1,537
862
2,019
941
1,870
1,178
1,528
2,233
1,382
Table \(\PageIndex{38}\)
Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05.
77.
Sanjay made identical paper airplanes out of three different weights of paper, light, medium and heavy. He made four airplanes from each of the weights, and launched them himself across the room. Here are the distances (in meters) that his planes flew.
Paper type/Trial
Trial 1
Trial 2
Trial 3
Trial 4
Heavy
5.1 meters
3.1 meters
4.7 meters
5.3 meters
Medium
4 meters
3.5 meters
4.5 meters
6.1 meters
Light
3.1 meters
3.3 meters
2.1 meters
1.9 meters
Table 12.39
Figure \(\PageIndex{8}\)
An experiment was conducted on the number of eggs (fecundity) laid by female fruit flies. There are three groups of flies. One group was bred to be resistant to DDT (the RS group). Another was bred to be especially susceptible to DDT (SS). Finally there was a control line of non-selected or typical fruitflies (NS). Here are the data:
RS
SS
NS
RS
SS
NS
12.8
38.4
35.4
22.4
23.1
22.6
21.6
32.9
27.4
27.5
29.4
40.4
14.8
48.5
19.3
20.3
16
34.4
23.1
20.9
41.8
38.7
20.1
30.4
34.6
11.6
20.3
26.4
23.3
14.9
19.7
22.3
37.6
23.7
22.9
51.8
22.6
30.2
36.9
26.1
22.5
33.8
29.6
33.4
37.3
29.5
15.1
37.9
16.4
26.7
28.2
38.6
31
29.5
20.3
39
23.4
44.4
16.9
42.4
29.3
12.8
33.7
23.2
16.1
36.6
14.9
14.6
29.2
23.6
10.8
47.4
27.3
12.2
41.7
Table \(\PageIndex{40}\)
Here is a chart of the three groups:
Figure \(\PageIndex{9}\) 79.
The data shown is the recorded body temperatures of 130 subjects as estimated from available histograms.
Traditionally we are taught that the normal human body temperature is 98.6 F. This is not quite correct for everyone. Are the mean temperatures among the four groups different?
Calculate 95% confidence intervals for the mean body temperature in each group and comment about the confidence intervals.
