12.7: Chapter Homework


12.1 Test of Two Variances

55.

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 ratsTuan's ratsJavier's rats
43.547.051.2
39.440.540.9
41.338.937.9
46.046.345.0
38.244.248.6
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-classProfessional (middle incomes)Professional (wealthy)
17.816.58.5
26.717.46.3
49.422.04.6
9.47.412.6
65.49.411.0
47.12.128.6
19.56.415.4
51.213.99.3
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 decoratingNewsHealthComputer
1728782104
28694153136
1631238798
205106103207
19710196146
Table $$\PageIndex{20}$$
57.

Which two magazine types do you think have the same variance in length?

58.

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.

SaturdaySundaySaturdaySunday
754462137
1858082
1506112439
941950127
629931141
736011873
89

Table 12.21

60.

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.

EastWest
3871
47126
3042
8251
7544
5290
11588
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?

Here are the data:

0 mg100 mg200 mg0 mg100 mg200 mg
242248246245246248
244245250248247252
247248248248250250
242247246244246248
246243245242244250

Table 12.23

62.

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 coinageSecond coinageThird coinageFourth coinage
5.96.94.95.3
6.89.05.55.6
6.46.64.65.5
7.08.14.55.1
6.69.3 6.2
7.79.2 5.8
7.28.6 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.

FirstSecondThirdFourth
Mean6.74448.24294.8755.6143
Variance0.29531.20950.20250.1314

Table 12.25

63.

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.

DivisionTeamWins
EastNY Yankees95
EastBaltimore93
EastTampa Bay90
EastToronto73
EastBoston69
Table $$\PageIndex{26}$$
DivisionTeamWins
CentralDetroit88
CentralChicago Sox85
CentralKansas City72
CentralCleveland68
CentralMinnesota66
Table $$\PageIndex{27}$$
DivisionTeamWins
WestOakland94
WestTexas93
WestLA Angels89
WestSeattle75
Table $$\PageIndex{28}$$

12.2 One-Way ANOVA

64.

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 1Route 2Route 3
302716
322941
272822
353631
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.

NortheastSouthWestCentralEast
16.316.916.416.217.1
16.116.516.516.617.2
16.416.416.616.516.6
16.516.216.116.416.8
$$\overline x$$=________________________________________
$$s^2=$$________________________________________
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.

NortheastSouthWestCentralEast
16.316.916.416.217.1
16.116.516.516.617.2
16.416.416.616.516.6
16.516.216.116.416.8
$$\overline x$$=________________________________________
$$s^2=$$________________________________________
Table $$\PageIndex{31}$$

$$H_{0} : \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}=\mu_{5}$$

$$H_a$$: At least any two of the group means $$\mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}=\mu_{5}$$ are not equal.

66.

degrees of freedom – numerator: $$df(num)$$ = _________

67.

degrees of freedom – denominator: $$df(denom)$$ = ________

68.

$$F$$ statistic = ________

12.4 Facts About the F Distribution

69.

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 ratsTuan's ratsJavier's rats
43.547.051.2
39.440.540.9
41.338.937.9
46.046.345.0
38.244.248.6
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-classProfessional (middle incomes)Professional (wealthy)
17.816.58.5
26.717.46.3
49.422.04.6
9.47.412.6
65.49.411.0
47.12.128.6
19.56.415.4
51.213.99.3

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 decoratingNewsHealthComputer
1728782104
28694153136
1631238798
205106103207
19710196146
Table $$\PageIndex{34}$$
71.

Using a significance level of 5%, test the hypothesis that the four magazine types have the same mean length.

72.

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.

CNNFOXLocal
451572
124337
186856
385060
233151
3522
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.

74.

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.

OnlineHybridFace-to-Face
728380
847378
778484
808181
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.

1,2102,1072,846
1,0801,1491,638
1,5378622,019
9411,8701,178
1,5282,233
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/TrialTrial 1Trial 2Trial 3Trial 4
Heavy5.1 meters3.1 meters4.7 meters5.3 meters
Medium4 meters3.5 meters4.5 meters6.1 meters
Light3.1 meters3.3 meters2.1 meters1.9 meters

Table 12.39

1. 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:
12.838.435.422.423.122.6
21.632.927.427.529.440.4
14.848.519.320.31634.4
23.120.941.838.720.130.4
34.611.620.326.423.314.9
19.722.337.623.722.951.8
22.630.236.926.122.533.8
29.633.437.329.515.137.9
16.426.728.238.63129.5
20.33923.444.416.942.4
29.312.833.723.216.136.6
14.914.629.223.610.847.4
27.312.241.7
Table $$\PageIndex{40}$$

Here is a chart of the three groups:

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.

FLFHMLMHFLFHMLMH
96.496.896.396.998.498.698.198.6
96.797.796.79798.798.698.198.6
97.297.897.197.198.798.698.298.7
97.297.997.297.198.798.798.298.8
97.49897.397.498.798.798.298.8
97.69897.497.598.898.898.298.8
97.79897.497.698.898.898.398.9
97.89897.497.798.898.898.499
97.898.197.597.898.898.998.499
97.998.397.697.999.29998.599
97.998.397.69899.39998.599.2
9898.397.898 99.198.699.5
98.298.497.898 99.198.6
98.298.497.898.3 99.298.7
98.298.497.998.4 99.499.1
98.298.49898.4 99.999.3
98.298.59898.6 10099.4
98.298.69898.6 100.8
Table $$\PageIndex{41}$$

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