## 12.1 Test of Two Variances

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 |

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%.

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 |

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 |

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

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 12.21 shows the results of a study.

Saturday | Sunday | Saturday | Sunday |
---|---|---|---|

75 | 44 | 62 | 137 |

18 | 58 | 0 | 82 |

150 | 61 | 124 | 39 |

94 | 19 | 50 | 127 |

62 | 99 | 31 | 141 |

73 | 60 | 118 | 73 |

89 |

Are the variances for incomes on the East Coast and the West Coast the same? Suppose that Table 12.22 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 |

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 mg | 100 mg | 200 mg | 0 mg | 100 mg | 200 mg |
---|---|---|---|---|---|

242 | 248 | 246 | 245 | 246 | 248 |

244 | 245 | 250 | 248 | 247 | 252 |

247 | 248 | 248 | 248 | 250 | 250 |

242 | 247 | 246 | 244 | 246 | 248 |

246 | 243 | 245 | 242 | 244 | 250 |

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 |

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.

First | Second | Third | Fourth | |
---|---|---|---|---|

Mean | 6.7444 | 8.2429 | 4.875 | 5.6143 |

Variance | 0.2953 | 1.2095 | 0.2025 | 0.1314 |

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.

Division | Team | Wins |
---|---|---|

East | NY Yankees | 95 |

East | Baltimore | 93 |

East | Tampa Bay | 90 |

East | Toronto | 73 |

East | Boston | 69 |

Division | Team | Wins |
---|---|---|

Central | Detroit | 88 |

Central | Chicago Sox | 85 |

Central | Kansas City | 72 |

Central | Cleveland | 68 |

Central | Minnesota | 66 |

Division | Team | Wins |
---|---|---|

West | Oakland | 94 |

West | Texas | 93 |

West | LA Angels | 89 |

West | Seattle | 75 |

## 12.2 One-Way ANOVA

Three different traffic routes are tested for mean driving time. The entries in the Table 12.29 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 |

State *SS*_{between}, *SS*_{within}, and the *F* statistic.

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 | |

$\stackrel{\u2013}{x}=$ | ________ | ________ | ________ | ________ | ________ |

${s}^{2}=$ | ________ | ________ | ________ | ________ | ________ |

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.

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 | |

$\stackrel{\u2013}{x}=$ | ________ | ________ | ________ | ________ | ________ |

${s}^{2}=$ | ________ | ________ | ________ | ________ | ________ |

*H _{0}*:

*µ*

_{1}=

*µ*

_{2}=

*µ*

_{3}=

*µ*

_{4}=

*µ*

_{5}

*Hα*: At least any two of the group means *µ*_{1}, *µ*_{2}, …, *µ*_{5} are not equal.

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

*F* statistic = ________

## 12.4 Facts About the F Distribution

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 |

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 12.33. 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 |

*Use the following information to answer the next two exercises.* Table 12.34 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 |

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?

A researcher wants to know if the mean times (in minutes) that people watch their favorite news station are the same. Suppose that Table 12.35 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 |

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 12.36 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 |

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 mean number of times a month a person eats out the same for White, Black, Hispanic and Asian people? Suppose that Table 12.37 shows the results of a study.

White | Black | Hispanic | Asian |
---|---|---|---|

6 | 4 | 7 | 8 |

8 | 1 | 3 | 3 |

2 | 5 | 5 | 5 |

4 | 2 | 4 | 1 |

6 | 6 | 7 |

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 mean numbers of daily visitors to a ski resort the same for the three types of snow conditions? Suppose that Table 12.38 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 |

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 |

- Take a look at the data in the graph. Look at the spread of data for each group (light, medium, heavy). Does it seem reasonable to assume a normal distribution with the same variance for each group? Yes or No.
- Why is this a balanced design?
- Calculate the sample mean and sample standard deviation for each group.
- Does the weight of the paper have an effect on how far the plane will travel? Use a 1% level of significance. Complete the test using the method shown in the bean plant example in Figure 12.8.
- variance of the group means __________
*MS*= ____________{between}- mean of the three sample variances ___________
*MS*= ______________{within}*F*statistic = ____________*df(num)*= __________,*df(denom)*= ___________- number of groups _______
- number of observations _______
*p*-value = __________ (*P*(*F*> _______) = __________)- Graph the
*p*-value. - decision: _______________________
- conclusion: _______________________________________________________________

DDT is a pesticide that has been banned from use in the United States and most other areas of the world. It is quite effective, but persisted in the environment and over time became seen as harmful to higher-level organisms. Famously, egg shells of eagles and other raptors were believed to be thinner and prone to breakage in the nest because of ingestion of DDT in the food chain of the birds.

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 |

The values are the average number of eggs laid daily for each of 75 flies (25 in each group) over the first 14 days of their lives. Using a 1% level of significance, are the mean rates of egg selection for the three strains of fruitfly different? If so, in what way? Specifically, the researchers were interested in whether or not the selectively bred strains were different from the nonselected line, and whether the two selected lines were different from each other.

Here is a chart of the three groups:

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.

FL | FH | ML | MH | FL | FH | ML | MH |
---|---|---|---|---|---|---|---|

96.4 | 96.8 | 96.3 | 96.9 | 98.4 | 98.6 | 98.1 | 98.6 |

96.7 | 97.7 | 96.7 | 97 | 98.7 | 98.6 | 98.1 | 98.6 |

97.2 | 97.8 | 97.1 | 97.1 | 98.7 | 98.6 | 98.2 | 98.7 |

97.2 | 97.9 | 97.2 | 97.1 | 98.7 | 98.7 | 98.2 | 98.8 |

97.4 | 98 | 97.3 | 97.4 | 98.7 | 98.7 | 98.2 | 98.8 |

97.6 | 98 | 97.4 | 97.5 | 98.8 | 98.8 | 98.2 | 98.8 |

97.7 | 98 | 97.4 | 97.6 | 98.8 | 98.8 | 98.3 | 98.9 |

97.8 | 98 | 97.4 | 97.7 | 98.8 | 98.8 | 98.4 | 99 |

97.8 | 98.1 | 97.5 | 97.8 | 98.8 | 98.9 | 98.4 | 99 |

97.9 | 98.3 | 97.6 | 97.9 | 99.2 | 99 | 98.5 | 99 |

97.9 | 98.3 | 97.6 | 98 | 99.3 | 99 | 98.5 | 99.2 |

98 | 98.3 | 97.8 | 98 | 99.1 | 98.6 | 99.5 | |

98.2 | 98.4 | 97.8 | 98 | 99.1 | 98.6 | ||

98.2 | 98.4 | 97.8 | 98.3 | 99.2 | 98.7 | ||

98.2 | 98.4 | 97.9 | 98.4 | 99.4 | 99.1 | ||

98.2 | 98.4 | 98 | 98.4 | 99.9 | 99.3 | ||

98.2 | 98.5 | 98 | 98.6 | 100 | 99.4 | ||

98.2 | 98.6 | 98 | 98.6 | 100.8 |