Members of the Town of Colonie Golfers’ Loop frequently raise concerns to Mr. Lange, as its leader, about how the league is run. The most frequent concern raised is that the teams competing each week are not statistically even. In other words, members raise concerns that one or more teams have a statistical advantage over other teams. To evaluate the validity of these claims, Mr. Lange has decided to use a mix of statistical hypothesis tests discussed in this course to evaluate the teams from Outing #14 in calendar-year 2016 (2016, #14). One is the Equal Vari- ances F Test, one is Welch’s T Test, and the other is the 2-Sample T Test for Equal Means. In this particular outing, seven teams were competing for the win. A random sample of eight prior scores from each team member was collected, and the summary statistics are shown below (correct to three decimal places).
Members of the Town of Colonie Golfers’ Loop frequently raise concerns to Mr. Lange, as its leader, about how the league is run. The most frequent concern raised is that the teams competing each week are not statistically even. In other words, members raise concerns that one or more teams have a statistical advantage over other teams.
To evaluate the validity of these claims, Mr. Lange has decided to use a mix of statistical hypothesis tests discussed in this course to evaluate the teams from Outing #14 in calendar-year 2016 (2016, #14). One is the Equal Vari- ances F Test, one is Welch’s T Test, and the other is the 2-Sample T Test for Equal Means.
In this particular outing, seven teams were competing for the win. A random sample of eight prior scores from each team member was collected, and the summary statistics are shown below (correct to three decimal places).
| team | sample mean | sample variance | |
| 1 | 24 |
76.500 |
14.522 |
| 2 | 32 |
79.656 |
18.233 |
| 3 | 32 |
79.500 |
13.097 |
| 4 | 32 |
82.375 |
47.984 |
| 5 | 32 |
80.406 |
18.314 |
| 6 | 32 |
80.281 |
11.047 |
| 7 | 32 |
80.438 |
14.383 |
To test for statistical evenness among these teams using the mix of tests we described above, however, we will have to adjust the significance levels we are accustomed to using. Use the table on the next page to help you:
| if you are using | use |
| 5% | 0.238% |
| 1% | 0.048% |
| 0.1% | 0.005% |
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For each pair of teams, do all of the following:
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(a) Compute the f∗ statistic for the Equal Variances F Test and determine the numerator and denominator degrees of freedom on the F distribution. Make sure to have f∗ ≥ 1.
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(b) Compute α0 for the Equal Variances F Test, as discussed in class. Can we conclude that the two teams’ population variances differ?
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(c) Using your answer to (b), are we allowed to use the 2-Sample T Test for Equal Means to determine whether the two teams are statistically even? Or must we use Welch’s T Test?
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(d) Compute the t∗ test statistic for the T test chosen in (c). Also compute the degrees of freedom on the T distribution needed for the chosen test.
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(e) Compute α0 for the chosen T test. Can we conclude that the two teams are not statistically even?
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