13.6: F Distribution and One-Way ANOVA (Exercises)
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These are homework exercises to accompany the Textmap created for
"Introductory Statistics" by OpenStax.
13.1: Introduction
13.2: One-Way ANOVA
Q 13.2.1
Three different traffic routes are tested for mean driving time. The entries in the table are the driving times in minutes on the three different routes. The one-way \(ANOVA\) results are shown in Table.
| Route 1 | Route 2 | Route 3 |
|---|---|---|
| 30 | 27 | 16 |
| 32 | 29 | 41 |
| 27 | 28 | 22 |
| 35 | 36 | 31 |
State \(SS_{\text{between}}\), \(SS_{\text{within}}\), and the \(F\) statistic.
S 13.2.1
\(SS_{\text{between}} = 26\)
\(SS_{\text{within}} = 441\)
\(F = 0.2653\)
Q 13.2.2
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 | |
| \(\bar{x} =\) | ________ | ________ | ________ | ________ | ________ |
| \(s^{2} =\) | ________ | ________ | ________ | ________ | ________ |
State the hypotheses.
\(H_{0}\): ____________
\(H_{a}\): ____________
13.3: The F-Distribution and the F-Ratio
Use the following information to answer the next five exercises. There are five basic assumptions that must be fulfilled in order to perform a one-way \(ANOVA\) test. What are they?
Exercise 13.2.1
Write one assumption.
Answer
Each population from which a sample is taken is assumed to be normal.
Exercise 13.2.2
Write another assumption.
Exercise 13.2.3
Write a third assumption.
Answer
The populations are assumed to have equal standard deviations (or variances).
Exercise 13.2.4
Write a fourth assumption.
Exercise 13.2.5
Write the final assumption.
Answer
The response is a numerical value.
Exercise 13.2.6
State the null hypothesis for a one-way \(ANOVA\) test if there are four groups.
Exercise 13.2.7
State the alternative hypothesis for a one-way \(ANOVA\) test if there are three groups.
Answer
\(H_{a}: \text{At least two of the group means } \mu_{1}, \mu_{2}, \mu_{3} \text{ are not equal.}\)
Exercise 13.2.8
When do you use an \(ANOVA\) test?
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 | |
| \(\bar{x} =\) | ________ | ________ | ________ | ________ | ________ |
| \(s^{2}\) | ________ | ________ | ________ | ________ | ________ |
\(H_{0}: \mu_{1} = \mu_{2} = \mu_{3} = \mu_{4} = \mu_{5}\)
\(H_{a}\): At least any two of the group means \(\mu_{1} , \mu_{2}, \dotso, \mu_{5}\) are not equal.
Q 13.3.1
degrees of freedom – numerator: \(df(\text{num}) =\) _________
Q 13.3.2
degrees of freedom – denominator: \(df(\text{denom}) =\) ________
S 13.3.2
\(df(\text{denom}) = 15\)
Q 13.3.3
\(F\) statistic = ________
13.4: Facts About the F Distribution
Exercise 13.4.4
An \(F\) statistic can have what values?
Exercise 13.4.5
What happens to the curves as the degrees of freedom for the numerator and the denominator get larger?
Answer
The curves approximate the normal distribution.
Use the following information to answer the next seven exercise. Four basketball teams took a random sample of players regarding how high each player can jump (in inches). The results are shown in Table.
| Team 1 | Team 2 | Team 3 | Team 4 | Team 5 |
|---|---|---|---|---|
| 36 | 32 | 48 | 38 | 41 |
| 42 | 35 | 50 | 44 | 39 |
| 51 | 38 | 39 | 46 | 40 |
Exercise 13.4.6
What is the \(df(\text{num})\)?
Exercise 13.4.7
What is the \(df(\text{denom})\)?
Answer
ten
Exercise 13.4.8
What are the Sum of Squares and Mean Squares Factors?
Exercise 13.4.9
What are the Sum of Squares and Mean Squares Errors?
Answer
\(SS = 237.33; MS = 23.73\)
Exercise 13.4.10
What is the \(F\) statistic?
Exercise 13.4.11
What is the \(p\text{-value}\)?
Answer
0.1614
Exercise 13.4.12
At the 5% significance level, is there a difference in the mean jump heights among the teams?
Use the following information to answer the next seven exercises. A video game developer is testing a new game on three different groups. Each group represents a different target market for the game. The developer collects scores from a random sample from each group. The results are shown in Table
| Group A | Group B | Group C |
|---|---|---|
| 101 | 151 | 101 |
| 108 | 149 | 109 |
| 98 | 160 | 198 |
| 107 | 112 | 186 |
| 111 | 126 | 160 |
Exercise 13.4.13
What is the \(df(\text{num})\)?
Answer
two
Exercise 13.4.14
What is the \(df(\text{denom})\)?
Exercise 13.4.15
What are the \(SS_{\text{between}}\) and \(MS_{\text{between}}\)?
Answer
\(SS_{\text{between}} = 5,700.4\);
\(MS_{\text{between}} = 2,850.2\)
Exercise 13.4.16
What are the \(SS_{\text{within}}\) and \(MS_{\text{within}}\)?
Exercise 13.4.17
What is the \(F\) Statistic?
Answer
3.6101
Exercise 13.4.18
What is the \(p\text{-value}\)?
Exercise 13.4.19
At the 10% significance level, are the scores among the different groups different?
Answer
Yes, there is enough evidence to show that the scores among the groups are statistically significant at the 10% level.
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 | |
| \(\bar{x} =\) | ________ | ________ | ________ | ________ | ________ |
| \(s^{2} =\) | ________ | ________ | ________ | ________ | ________ |
Enter the data into your calculator or computer.
Exercise 13.4.20
\(p\text{-value} =\) ______
State the decisions and conclusions (in complete sentences) for the following preconceived levels of \(\alpha\).
Exercise 13.4.21
\(\alpha = 0.05\)
- Decision: ____________________________
- Conclusion: ____________________________
Exercise 13.4.22
\(\alpha = 0.01\)
- Decision: ____________________________
- Conclusion: ____________________________
Use the following information to answer the next eight exercises. Groups of men from three different areas of the country are to be tested for mean weight. The entries in the table are the weights for the different groups. The one-way \(ANOVA\) results are shown in Table .
| Group 1 | Group 2 | Group 3 |
|---|---|---|
| 216 | 202 | 170 |
| 198 | 213 | 165 |
| 240 | 284 | 182 |
| 187 | 228 | 197 |
| 176 | 210 | 201 |
Exercise 13.3.2
What is the Sum of Squares Factor?
Answer
4,939.2
Exercise 13.3.3
What is the Sum of Squares Error?
Exercise 13.3.4
What is the \(df\) for the numerator?
Answer
2
Exercise 13.3.5
What is the \(df\) for the denominator?
Exercise 13.3.6
What is the Mean Square Factor?
Answer
2,469.6
Exercise 13.3.7
What is the Mean Square Error?
Exercise 13.3.8
What is the \(F\) statistic?
Answer
3.7416
Use the following information to answer the next eight exercises. Girls from four different soccer teams are to be tested for mean goals scored per game. The entries in the table are the goals per game for the different teams. The one-way \(ANOVA\) results are shown in Table .
| Team 1 | Team 2 | Team 3 | Team 4 |
|---|---|---|---|
| 1 | 2 | 0 | 3 |
| 2 | 3 | 1 | 4 |
| 0 | 2 | 1 | 4 |
| 3 | 4 | 0 | 3 |
| 2 | 4 | 0 | 2 |
Exercise 13.3.9
What is \(SS_{\text{between}}\)?
Exercise 13.3.10
What is the \(df\) for the numerator?
Answer
3
Exercise 13.3.11
What is \(MS_{\text{between}}\)?
Exercise 13.3.12
What is \(SS_{\text{within}}\)?
Answer
13.2
Exercise 13.3.13
What is the \(df\) for the denominator?
Exercise 13.3.14
What is \(MS_{\text{within}}\)?
Answer
0.825
Exercise 13.3.15
What is the \(F\) statistic?
Exercise 13.3.16
Judging by the \(F\) statistic, do you think it is likely or unlikely that you will reject the null hypothesis?
Answer
Because a one-way \(ANOVA\) test is always right-tailed, a high \(F\) statistic corresponds to a low \(p\text{-value}\), so it is likely that we will reject the null hypothesis.
DIRECTIONS
Use a solution sheet to conduct the following hypothesis tests. The solution sheet can be found in [link] .
Q 13.4.1
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 |
- \(H_{0}: \mu_{L} = \mu_{T} = \mu_{J}\)
- at least any two of the means are different
- \(df(\text{num}) = 2; df(\text{denom}) = 12\)
- \(F\) distribution
- 0.67
- 0.5305
- Check student’s solution.
- Decision: Do not reject null hypothesis; Conclusion: There is insufficient evidence to conclude that the means are different.
Q 13.4.2
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 . 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 |
Q 13.4.3
Examine the seven practice laps from [link] . Determine whether the mean lap time is statistically the same for the seven practice laps, or if there is at least one lap that has a different mean time from the others.
S 13.4.3
- \(H_{0}: \mu_{1} = \mu_{2} = \mu_{3} = \mu_{4} = \mu_{5} = \mu_{6} = \mu_{T}\)
- At least two mean lap times are different.
- \(df(\text{num}) = 6; df(\text{denom}) = 98\)
- \(F\) distribution
- 1.69
- 0.1319
- Check student’s solution.
- Decision: Do not reject null hypothesis; Conclusion: There is insufficient evidence to conclude that the mean lap times are different.
Use the following information to answer the next two exercises. 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 |
Q 13.4.4
Using a significance level of 5%, test the hypothesis that the four magazine types have the same mean length.
Q 13.4.5
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?
S 13.4.6
- \(H_{a}: \mu_{d} = \mu_{n} = \mu_{h}\)
- At least any two of the magazines have different mean lengths.
- \(df(\text{num}) = 2, df(\text{denom}) = 12\)
- \(F\) distribtuion
- \(F = 15.28\)
- \(p\text{-value} = 0.001\)
- Check student’s solution.
-
- \(\alpha: 0.05\)
- Decision: Reject the Null Hypothesis.
- Reason for decision: \(p\text{-value} < \alpha\)
- Conclusion: There is sufficient evidence to conclude that the mean lengths of the magazines are different.
Q 13.4.7
A researcher wants to know if the mean times (in minutes) that people watch their favorite news station are the same. Suppose that Table 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.
Q 13.4.8
Are the means for the final exams the same for all statistics class delivery types? Table 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.
S 13.4.8
- \(H_{0}: \mu_{o} = \mu_{h} = \mu_{f}\)
- At least two of the means are different.
- \(df(\text{n}) = 2, df(\text{d}) = 13\)
- \(F_{2,13}\)
- 0.64
- 0.5437
- Check student’s solution.
-
- \(\alpha: 0.05\)
- Decision: Do not reject the null hypothesis.
- Reason for decision: \(p\text{-value} < \alpha\)
- Conclusion: The mean scores of different class delivery are not different.
Q 13.4.9
Are the mean number of times a month a person eats out the same for whites, blacks, Hispanics and Asians? Suppose that Table 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.
Q 13.4.10
Are the mean numbers of daily visitors to a ski resort the same for the three types of snow conditions? Suppose that Table 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 |
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.
S 13.4.11
- \(H_{0}: \mu_{p} = \mu_{m} = \mu_{h}\)
- At least any two of the means are different.
- \(df(\text{n}) = 2, df(\text{d}) = 12\)
- \(F_{2,12}\)
- 3.13
- 0.0807
- Check student’s solution.
-
- \(\alpha: 0.05\)
- Decision: Do not reject the null hypothesis.
- Reason for decision: \(p\text{-value} < \alpha\)
- Conclusion: There is not sufficient evidence to conclude that the mean numbers of daily visitors are different.
Q 13.4.12
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 |
Figure 13.4.1.
- 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
Example
.
- variance of the group means __________
- \(MS_{\text{between}} =\) ___________
- mean of the three sample variances ___________
- \(MS_{\text{within}} =\) _____________
- \(F\) statistic = ____________
- \(df(\text{num}) =\) __________, \(df(\text{denom}) =\) ___________
- number of groups _______
- number of observations _______
- \(p\text{-value} =\) __________ (\(P(F >\) _______\() =\) __________)
- Graph the \(p\text{-value}\).
- decision: _______________________
- conclusion: _______________________________________________________________
Q 13.4.13
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:
Figure 13.4.2.S 13.4.13
The data appear normally distributed from the chart and of similar spread. There do not appear to be any serious outliers, so we may proceed with our ANOVA calculations, to see if we have good evidence of a difference between the three groups.
\(H_{0}: \mu_{1} = \mu_{2} = \mu_{3}\);
\(H_{a}: \mu_{i} \neq \mu_{j}\) some \(i \neq j\).
Define \(\mu_{1}, \mu_{2}, \mu_{3}\), as the population mean number of eggs laid by the three groups of fruit flies.
\(F\) statistic \(= 8.6657\);
\(p\text{-value} = 0.0004\)
Figure 13.4.3.Decision: Since the \(p\text{-value}\) is less than the level of significance of 0.01, we reject the null hypothesis.
Conclusion: We have good evidence that the average number of eggs laid during the first 14 days of life for these three strains of fruitflies are different.
Interestingly, if you perform a two sample \(t\)-test to compare the RS and NS groups they are significantly different (\(p = 0.0013\)). Similarly, SS and NS are significantly different (\(p = 0.0006\)). However, the two selected groups, RS and SS are not significantly different (\(p = 0.5176\)). Thus we appear to have good evidence that selection either for resistance or for susceptibility involves a reduced rate of egg production (for these specific strains) as compared to flies that were not selected for resistance or susceptibility to DDT. Here, genetic selection has apparently involved a loss of fecundity.
Q 13.4.14
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 |
13.5: Test of Two Variances
Use the following information to answer the next two exercises. There are two assumptions that must be true in order to perform an \(F\) test of two variances.
Exercise 13.5.2
Name one assumption that must be true.
Answer
The populations from which the two samples are drawn are normally distributed.
Exercise 13.5.3
What is the other assumption that must be true?
Use the following information to answer the next five exercises. Two coworkers commute from the same building. They are interested in whether or not there is any variation in the time it takes them to drive to work. They each record their times for 20 commutes. The first worker’s times have a variance of 12.1. The second worker’s times have a variance of 16.9. The first worker thinks that he is more consistent with his commute times and that his commute time is shorter. Test the claim at the 10% level.
Exercise 13.5.4
State the null and alternative hypotheses.
Answer
\(H_{0}: \sigma_{1} = \sigma_{2}\)
\(H_{a}: \sigma_{1} < \sigma_{2}\)
or
\(H_{0}: \sigma^{2}_{1} = \sigma^{2}_{2}\)
\(H_{a}: \sigma^{2}_{1} < \sigma^{2}_{2}\)
Exercise 13.5.5
What is \(s_{1}\) in this problem?
Exercise 13.5.6
What is \(s_{2}\) in this problem?
Answer
4.11
Exercise 13.5.7
What is \(n\)?
Exercise 13.5.8
What is the \(F\) statistic?
Answer
0.7159
Exercise 13.5.9
What is the \(p\text{-value}\)?
Exercise 13.5.10
Is the claim accurate?
Answer
No, at the 10% level of significance, we do not reject the null hypothesis and state that the data do not show that the variation in drive times for the first worker is less than the variation in drive times for the second worker.
Use the following information to answer the next four exercises. Two students are interested in whether or not there is variation in their test scores for math class. There are 15 total math tests they have taken so far. The first student’s grades have a standard deviation of 38.1. The second student’s grades have a standard deviation of 22.5. The second student thinks his scores are lower.
Exercise 13.5.11
State the null and alternative hypotheses.
Exercise 13.5.12
What is the \(F\) Statistic?
Answer
2.8674
Exercise 13.5.13
What is the \(p\text{-value}\)?
Exercise 13.5.14
At the 5% significance level, do we reject the null hypothesis?
Answer
Reject the null hypothesis. There is enough evidence to say that the variance of the grades for the first student is higher than the variance in the grades for the second student.
Use the following information to answer the next three exercises. Two cyclists are comparing the variances of their overall paces going uphill. Each cyclist records his or her speeds going up 35 hills. The first cyclist has a variance of 23.8 and the second cyclist has a variance of 32.1. The cyclists want to see if their variances are the same or different.
Exercise 13.5.15
State the null and alternative hypotheses.
Exercise 13.5.16
What is the \(F\) Statistic?
Answer
0.7414
Exercise 13.5.17
At the 5% significance level, what can we say about the cyclists’ variances?
Q 13.5.1
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%.
S 13.5.1
- \(H_{0}: \sigma^{2}_{1} = \sigma^{2}_{2}\)
- \(H_{a}: \sigma^{2}_{1} \neq \sigma^{2}_{1}\)
- \(df(\text{num}) = 4; df(\text{denom}) = 4\)
- \(F_{4, 4}\)
- 3.00
- \(2(0.1563) = 0.3126\). Using the TI-83+/84+ function 2-SampFtest, you get the test statistic as 2.9986 and p -value directly as 0.3127. If you input the lists in a different order, you get a test statistic of 0.3335 but the \(p\text{-value}\) is the same because this is a two-tailed test.
- Check student't solution.
- Decision: Do not reject the null hypothesis; Conclusion: There is insufficient evidence to conclude that the variances are different.
Q 13.5.2
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.
Q 13.5.3
Refer to the data from [link] .
Examine practice laps 3 and 4. Determine whether or not the variance in lap time is statistically the same for those practice laps.
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 |
S 13.5.3
- \(H_{0}: \sigma^{2}_{1} = \sigma^{2}_{2}\)
- \(H_{a}: \sigma^{2}_{1} \neq \sigma^{2}_{1}\)
- \(df(\text{n}) = 19, df(\text{d}) = 19\)
- \(F_{19,19}\)
- 1.13
- 0.786
- Check student’s solution.
-
- \(\alpha: 0.05\)
- Decision: Do not reject the null hypothesis.
- Reason for decision: \(p\text{-value} > \alpha\)
- Conclusion: There is not sufficient evidence to conclude that the variances are different.
Q 13.5.4
Which two magazine types do you think have the same variance in length?
Q 13.5.5
Which two magazine types do you think have different variances in length?
S 13.5.5
The answers may vary. Sample answer: Home decorating magazines and news magazines have different variances.
Q 13.5.6
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 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 |
Q 13.5.7
Are the variances for incomes on the East Coast and the West Coast the same? Suppose that Table 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 |
S 13.5.7
- \(H_{0}: \sigma^{2}_{1} = \sigma^{2}_{2}\)
- \(H_{a}: \sigma^{2}_{1} \neq \sigma^{2}_{1}\)
- \(df(\text{n}) = 7, df(\text{d}) = 6\)
- \(F_{7,6}\)
- 0.8117
- 0.7825
- Check student’s solution.
-
- \(\alpha: 0.05\)
- Decision: Do not reject the null hypothesis.
- Reason for decision: \(p\text{-value} > \alpha\)
- Conclusion: There is not sufficient evidence to conclude that the variances are different.
Q 13.5.8
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 |
Q 13.5.9
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 |
S 13.5.9
Here is a strip chart of the silver content of the coins:
Figure 13.5.1.While there are differences in spread, it is not unreasonable to use \(ANOVA\) techniques. Here is the completed \(ANOVA\) table:
| Source of Variation | Sum of Squares (\(SS\)) | Degrees of Freedom (\(df\)) | Mean Square (\(MS\)) | \(F\) |
|---|---|---|---|---|
| Factor (Between) | 37.748 | \(4 - 1 = 3\) | 12.5825 | 26.272 |
| Error (Within) | 11.015 | \(27 - 4 = 23\) | 0.4789 | |
| Total | 48.763 | \(27 - 1 = 26\) |
\(P(F > 26.272) = 0\);
Reject the null hypothesis for any alpha. There is sufficient evidence to conclude that the mean silver content among the four coinages are different. From the strip chart, it appears that the first and second coinages had higher silver contents than the third and fourth.
Q 13.5.10
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 |
S 13.5.10
Here is a stripchart of the number of wins for the 14 teams in the AL for the 2012 season.
Figure 13.5.2.While the spread seems similar, there may be some question about the normality of the data, given the wide gaps in the middle near the 0.500 mark of 82 games (teams play 162 games each season in MLB). However, one-way \(ANOVA\) is robust.
Here is the \(ANOVA\) table for the data:
| Source of Variation | Sum of Squares (\(SS\)) | Degrees of Freedom (\(df\)) | Mean Square (\(MS\)) | \(F\) |
|---|---|---|---|---|
| Factor (Between) | 344.16 | 3 – 1 = 2 | 172.08 | 26.272 |
| Error (Within) | 1,219.55 | 14 – 3 = 11 | 110.87 | 1.5521 |
| Total | 1,563.71 | 14 – 1 = 13 |
\(P(F > 1.5521) = 0.2548\)
Since the \(p\text{-value}\) is so large, there is not good evidence against the null hypothesis of equal means. We decline to reject the null hypothesis. Thus, for 2012, there is not any have any good evidence of a significant difference in mean number of wins between the divisions of the American League.