9.7: Chapter 9 Homework
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)9.3 The F-Distribution and the F-Ratio
1. There are five basic assumptions that must be fulfilled in order to perform a one-way ANOVA test. What are they?
Use the following information to answer the next eleven exercises. Groups of men from three different areas of the country are to be tested for mean weight. The entries in Table \(\PageIndex{1}\) are the weights for the different groups.
| Group 1 | Group 2 | Group 3 |
|---|---|---|
| 216 | 202 | 170 |
| 198 | 213 | 165 |
| 240 | 284 | 182 |
| 187 | 228 | 197 |
| 176 | 210 | 201 |
Table \(\PageIndex{1}\)
2. State the null and alternative hypotheses.
3. What is the Sum of Squares Factor?
4. What is the Sum of Squares Error?
5. What is the \(df\) for the numerator?
6. What is the \(df\) for the denominator?
7. What is the Mean Square Factor?
8. What is the Mean Square Error?
9. What is the \(F\)-observed value?
10. Create an ANOVA summary table.
11. What is the \(F\)-critical value for the 95% confidence level? Make a decision about the hypothesis.
12. What is the approximate \(p\)-value for \(F\)-observed here? Make a decision about the hypothesis using \(\alpha\) = .05.
Use the following information to answer the next eleven exercises. Girls from four different soccer teams are to be tested for mean goals scored per game. The entries in Table \(\PageIndex{2}\) are the goals per game for the different teams.
| 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 |
Table \(\PageIndex{2}\)
13. State the null and alternative hypotheses.
14. What is \(SS_{Between}\)?
15. What is the \(df\) for the numerator?
16. What is \(MS_{Between}\)?
17. What is \(SS_{Within}\)?
18. What is the \(df\) for the denominator?
19. What is \(MS_{Within}\)?
20. What is the \(F\)-observed value?
21. Create an ANOVA summary table.
22. What is the \(F\)-critical value for the 95% confidence level? Make a decision about the hypothesis.
23. What is the approximate \(p\)-value for \(F\)-observed here? Make a decision about the hypothesis using \(\alpha\) = .05.
24. An \(F\)-statistic can have what values?
25. What happens to the curves as the degrees of freedom for the numerator and the denominator get larger?
Use the following information to answer the next ten exercises. Four basketball teams took a random sample of players regarding how high each player can jump (in inches). The results are shown in Table \(\PageIndex{3}\).
| 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 |
Table \(\PageIndex{3}\)
26. State the null and alternative hypotheses.
27. What is the \(df_{num}\)?
28. What is the \(df_{denom}\)?
29. What are the Sum of Squares and Mean Squares Factors?
30. What are the Sum of Squares and Mean Squares Errors?
31. What is the \(F\)-observed statistic?
32. Create an ANOVA summary table.
33. What is the \(F\)-critical value for the 95% confidence level?
34. What is the approximate \(p\)-value for \(F\)-observed here?
35. 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 ten 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 \(\PageIndex{4}\).
| Group A | Group B | Group C |
|---|---|---|
| 101 | 151 | 101 |
| 108 | 149 | 109 |
| 98 | 160 | 198 |
| 107 | 112 | 186 |
| 111 | 126 | 160 |
Table \(\PageIndex{4}\)
36. State the null and alternative hypotheses.
37. What is the \(df_{num}\)?
38. What is the \(df_{denom}\)?
39. What are the \(SS_{Between}\) and \(MS_{Between}\)?
40. What are the \(SS_{Within}\) and \(MS_{Within}\)?
41. What is the \(F\)-observed statistic?
42. Create an ANOVA summary table.
43. What is the \(F\)-critical value for the 99% confidence level? Make a decision about the hypothesis.
44. What is the approximate \(p\)-value for \(F\)-observed here? Make a decision about the hypothesis using \(\alpha\) = .01.
45. At the 1% significance level, are the scores among the different groups different? Why or why not?
Use the following information to answer the next nine 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 four 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 \(\PageIndex{5}\)
46. State the hypotheses.
\(H_0\): ____________
\(H_a\): ____________
47. sum of squares – between groups: \(SS_{Between}\) = _________
48. sum of squares – within groups: \(SS_{Within}\) = _________
49. degrees of freedom – numerator: \(df_{num}\) = _________
50. degrees of freedom – denominator: \(df_{denom}\) = ________
51. \(F\)-observed = ________
52. Approximate \(p\)-value = ______
State the decisions and conclusions (in complete sentences) for the following preconceived levels of \(\alpha\).
53. \(\alpha = 0.05\)
a. Decision: ____________________________
b. Conclusion: ____________________________
54. \(\alpha = 0.01\)
a. Decision: ____________________________
b. Conclusion: ____________________________
55. 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 complete the test statistic approach 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. Use \(\alpha\) = .05.
| Division | Team | Wins |
|---|---|---|
| East | NY Yankees | 95 |
| East | Baltimore | 93 |
| East | Tampa Bay | 90 |
| East | Toronto | 73 |
| East | Boston | 69 |
Table \(\PageIndex{6}\)
| Division | Team | Wins |
|---|---|---|
| Central | Detroit | 88 |
| Central | Chicago Sox | 85 |
| Central | Kansas City | 72 |
| Central | Cleveland | 68 |
| Central | Minnesota | 66 |
Table \(\PageIndex{7}\)
| Division | Team | Wins |
|---|---|---|
| West | Oakland | 94 |
| West | Texas | 93 |
| West | LA Angels | 89 |
| West | Seattle | 75 |
Table \(\PageIndex{8}\)
9.2 One-Way ANOVA
56. Three different traffic routes are tested for mean driving time. The entries in Table \(\PageIndex{9}\) are the driving times in minutes on the three different routes. Conduct a hypothesis test using the test statistic approach and 5% significance, and create an ANOVA summary table of the results.
| Route 1 | Route 2 | Route 3 |
|---|---|---|
| 30 | 27 | 16 |
| 32 | 29 | 41 |
| 27 | 28 | 22 |
| 35 | 36 | 31 |
Table \(\PageIndex{9}\)
9.3 The F-Distribution and the F-Ratio
57. 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 1% and the test statistic approach, 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 \(\PageIndex{10}\) Weights of Student Lab Rats
58. 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 \(\PageIndex{11}\). Using a 5% significance level and the test statistic approach, 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 \(\PageIndex{11}\)
Use the following information to answer the next two exercises. Table \(\PageIndex{12}\) 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 \(\PageIndex{12}\)
59. Using a significance level of 5%, use test statistics test whether the four magazine types have the same mean length. Create an ANOVA summary table of the results.
60. 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. Create a new ANOVA summary table. Based on this test, are the mean lengths for the remaining three magazines statistically the same?
61. A researcher wants to know if the mean times (in minutes) that people watch their favorite news station are the same. Suppose that Table \(\PageIndex{13}\) shows the results of a study. Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use \(\alpha\) = 0.05 and the test statistic approach.
| CNN | FOX | Local |
|---|---|---|
| 45 | 15 | 72 |
| 12 | 43 | 37 |
| 18 | 68 | 56 |
| 38 | 50 | 60 |
| 23 | 31 | 51 |
| 35 | 22 |
Table \(\PageIndex{13}\)
62. Are the means for the final exams the same for all statistics class delivery types? Table \(\PageIndex{14}\) shows the scores on final exams from several randomly selected classes that used the different delivery types. 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 to test this with test statistics.
| Online | Hybrid | Face-to-Face |
|---|---|---|
| 72 | 83 | 80 |
| 84 | 73 | 78 |
| 77 | 84 | 84 |
| 80 | 81 | 81 |
| 81 | 86 | |
| 79 | ||
| 82 |
Table \(\PageIndex{14}\)
63. Are the mean number of times a month a person eats out the same for Whites, Blacks, Hispanics and Asians? Suppose that Table \(\PageIndex{15}\) shows the results of a study. 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 0.01 and the test statistic approach. Create an ANOVA summary table.
| White | Black | Hispanic | Asian |
|---|---|---|---|
| 6 | 4 | 7 | 8 |
| 8 | 1 | 3 | 3 |
| 2 | 5 | 5 | 5 |
| 4 | 2 | 4 | 1 |
| 6 | 6 | 7 |
Table \(\PageIndex{15}\)
64. Are the mean numbers of daily visitors to a ski resort the same for the three types of snow conditions? Suppose that Table \(\PageIndex{16}\) shows the results of a study. 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 and the test statistic approach.
| 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 \(\PageIndex{16}\)
65. 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 \(\PageIndex{17}\)
Figure \(\PageIndex{1}\)
- 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.
- g = _______
- n = _______
- \(SS_{Between}\)= __________
- \(MS_{Between}\)= ___________
- \(SS_{Within}\)= __________
- \(MS_{Within}\) = _____________
- \(df_{num}\) = __________, \(df_{denom}\) = ___________
- \(F\) statistic = ____________
- \(F\)-critical = ____________
- Graph the \(F\)-observed and \(F\)-critical values.
- decision: _______________________
- conclusion: _______________________________________________________________
66. 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 |
Table \(\PageIndex{18}\)
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 and the p-value approach, 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 non-selected line, and whether the two selected lines were different from each other.
Here is a chart of the three groups:
Figure \(\PageIndex{2}\)
67. 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? Conduct both the test statistic and \(p\)-value approach using 99% confidence.
| 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 |
Table \(\PageIndex{19}\)