Section Exercises
One-Way ANOVA
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?
1. Write one assumption.
3. Write a third assumption.
5. Write the final assumption.
7. State the alternative hypothesis for a one-way ANOVA test if there are three groups.
9. 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 |
10. State SSbetween, SSwithin, and the F statistic.
| 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 | |
| [latex]\overline{x}[/latex]= | ________ | ________ | ________ | ________ | ________ |
| [latex]{s}_{2}[/latex]= | ________ | ________ | ________ | ________ | ________ |
12. State the hypotheses.
H0: ____________
Ha: ____________
The F Distribution and the F-Ratio
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 |
13. What is the Sum of Squares Factor?
15. What is the df for the numerator?
17. What is the Mean Square Factor?
19. What is the F statistic?
| 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 |
20. What is SSbetween?
21. What is the df for the numerator?
23. What is SSwithin?
25. What is MSwithin?
27. Judging by the F statistic, do you think it is likely or unlikely that you will reject the null hypothesis?
| 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 | |
| [latex]\overline{x}[/latex]= | ________ | ________ | ________ | ________ | ________ |
| [latex]{s}_{2}[/latex] | ________ | ________ | ________ | ________ | ________ |
28. H0: µ1 = µ2 = µ3 = µ4 = µ5
29. Hα: At least any two of the group means µ1, µ2, …, µ5 are not equal.
30. degrees of freedom – numerator: df(num) = _________
31. degrees of freedom – denominator: df(denom) = ________
Facts About the F Distribution
33. An F statistic can have what values?
34. What happens to the curves as the degrees of freedom for the numerator and the denominator get larger?
| 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 |
35. What is the df(num)?
36. What is the df(denom)?
38. What are the Sum of Squares and Mean Squares Errors?
40. What is the p-value?
| Group A | Group B | Group C |
|---|---|---|
| 101 | 151 | 101 |
| 108 | 149 | 109 |
| 98 | 160 | 198 |
| 107 | 112 | 186 |
| 111 | 126 | 160 |
42. What is the df(num)?
44. What are the SSbetween and MSbetween?
46. What is the F Statistic?
48. At the 10% significance level, are the scores among the different groups different?
| 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 | |
| [latex]\overline{x}[/latex]= | ________ | ________ | ________ | ________ | ________ |
| [latex]{s}_{2}[/latex]= | ________ | ________ | ________ | ________ | ________ |
49. Enter the data into your calculator or computer.
50. p-value = ______
State the decisions and conclusions (in complete sentences) for the following preconceived levels of α.
51. α = 0.05
a. Decision: ____________________________
b. Conclusion: ____________________________
52. α = 0.01
a. Decision: ____________________________
b. Conclusion: ____________________________
Use a solution sheet to conduct the following hypothesis tests. The solution sheet can be found in Appendix E.
53. 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 |
54. 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 |
55. Examine the seven practice laps from Appendix C. 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.
| 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 |
56. Using a significance level of 5%, test the hypothesis that the four magazine types have the same mean length.
57. 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?
| CNN | FOX | Local |
|---|---|---|
| 45 | 15 | 72 |
| 12 | 43 | 37 |
| 18 | 68 | 56 |
| 38 | 50 | 60 |
| 23 | 31 | 51 |
| 35 | 22 |
59. 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.
60. 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 |
61. 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.
| White | Black | Hispanic | Asian |
|---|---|---|---|
| 6 | 4 | 7 | 8 |
| 8 | 1 | 3 | 3 |
| 2 | 5 | 5 | 5 |
| 4 | 2 | 4 | 1 |
| 6 | 6 | 7 |
63. 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.
64. 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 |
65. 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.
| 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 Example.
- variance of the group means __________
- MSbetween= ___________
- mean of the three sample variances ___________
- MSwithin = _____________
- F statistic = ____________
- df(num) = __________, df(denom) = ___________
- number of groups _______
- number of observations _______
- p-value = __________ (P(F > _______) = __________)
- Graph the p-value.
- decision: _______________________
- conclusion: _______________________________________________________________
67. 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.
68. 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 |
69. 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:
70. The data shown is the recorded body temperatures of 130 subjects as estimated from available histograms.
71. 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?
72. Calculate 95% confidence intervals for the mean body temperature in each group and comment about the confidence intervals.
99.198.699.5 99.198.6 99.298.7 99.499.1 99.999.3 10099.4 100.8
| 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 | ||||
| 98.2 | 98.4 | 97.8 | 98 | ||||
| 98.2 | 98.4 | 97.8 | 98.3 | ||||
| 98.2 | 98.4 | 97.9 | 98.4 | ||||
| 98.2 | 98.4 | 98 | 98.4 | ||||
| 98.2 | 98.5 | 98 | 98.6 | ||||
| 98.2 | 98.6 | 98 | 98.6 |
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.
73. Name one 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.
75. State the null and alternative hypotheses.
77. What is s2 in this problem?
79. What is the F statistic?
81. Is the claim accurate?
82. State the null and alternative hypotheses.
83. What is the F Statistic?
85. At the 5% significance level, do we reject the null hypothesis?
86. State the null and alternative hypotheses.
87. What is the F Statistic?
89. 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 |
90. 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%.
| 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 |
92. 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.
Refer to the data from Appendix C.
93. 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 |
95. Which two magazine types do you think have different variances in length?
| 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 |
97. 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 |
98. 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 |
99. 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:
6.2 5.8 5.8
| 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 | ||
| 7.7 | 9.2 | ||
| 7.2 | 8.6 | ||
| 6.9 | |||
| 6.2 |
100. Did the silver content of the coins change over the course of Manuel’s reign?
101. 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 |
102. 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 |