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The New York Choral Society divides male singers up into four categories from highest voices to lowest: Tenor1, Tenor2, Bass1, Bass2. In the table are heights of the men in the Tenor1 and Bass2 groups. One suspects that taller men will have lower voices, and that the variance of height may go up with the lower voices as well. Do we have good evidence that the variance of the heights of singers in each of these two groups (Tenor1 and Bass2) are different?
| Tenor1 | Bass2 | Tenor 1 | Bass 2 | Tenor 1 | Bass 2 |
|---|---|---|---|---|---|
| 69 | 72 | 67 | 72 | 68 | 67 |
| 72 | 75 | 70 | 74 | 67 | 70 |
| 71 | 67 | 65 | 70 | 64 | 70 |
| 66 | 75 | 72 | 66 | 69 | |
| 76 | 74 | 70 | 68 | 72 | |
| 74 | 72 | 68 | 75 | 71 | |
| 71 | 72 | 64 | 68 | 74 | |
| 66 | 74 | 73 | 70 | 75 | |
| 68 | 72 | 66 | 72 |
The histograms are not as normal as one might like. Plot them to verify. However, we proceed with the test in any case.
Subscripts: T1= tenor1 and B2 = bass 2
The standard deviations of the samples are s T 1 = 3.3302 and s B 2 = 2.7208.
The hypotheses are
and (two tailed test)
The F statistic is 1.4894 with 20 and 25 degrees of freedom.
The p -value is 0.3430. If we assume alpha is 0.05, then we cannot reject the null hypothesis.
We have no good evidence from the data that the heights of Tenor1 and Bass2 singers have different variances (despite there being a significant difference in mean heights of about 2.5 inches.)
“MLB Vs. Division Standings – 2012.” Available online at http://espn.go.com/mlb/standings/_/year/2012/type/vs-division/order/true.
The F test for the equality of two variances rests heavily on the assumption of normal distributions. The test is unreliable if this assumption is not met. If both distributions are normal, then the ratio of the two sample variances is distributed as an F statistic, with numerator and denominator degrees of freedom that are one less than the samples sizes of the corresponding two groups. A test of two variances hypothesis test determines if two variances are the same. The distribution for the hypothesis test is the F distribution with two different degrees of freedom.
F has the distribution F ~ F ( n 1 – 1, n 2 – 1)
F =
If σ 1 = σ 2 , then F =
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.
Name one assumption that must be true.
The populations from which the two samples are drawn are normally distributed.
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.
State the null and alternative hypotheses.
H 0 : σ 1 = σ 2
H a : σ 1 <σ 2
or
H 0 :
H a :
What is s 1 in this problem?
What is n ?
What is the p -value?
Is the claim accurate?
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.
State the null and alternative hypotheses.
What is the p -value?
At the 5% significance level, do we reject the null hypothesis?
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.
State the null and alternative hypotheses.
At the 5% significance level, what can we say about the cyclists’ variances?
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