Which of the following is the correct definition of the F-test statistic?
a. The ratio of the smaller variance to the larger variance.
b. The ratio of the larger variance to the smaller variance.
c. The square of the difference between the two means divided by the pooled variance.
d. The difference between the two means divided by the pooled standard deviation.
The correct answer and explanation is:
Correct Answer: b. The ratio of the larger variance to the smaller variance.
The F-test is a statistical test that uses the F-distribution to compare two population variances to determine if they are significantly different from each other. The main idea behind the F-test statistic is to look at how much the two sample variances differ relative to each other. To do this, the F statistic is calculated as the ratio of the two sample variances. To ensure the F-value is greater than or equal to one, the larger sample variance is placed in the numerator and the smaller in the denominator. This ratio makes interpretation easier because an F-value close to one indicates that the two variances are similar, while an F-value much larger than one suggests that the variances differ significantly.
In practice, the F-test is commonly used in analysis of variance (ANOVA) to test the hypothesis that multiple groups have the same population variance and mean. In this context, the F statistic compares the variance between group means to the variance within the groups. However, when testing only two variances, the basic F-test simply takes the ratio of the larger variance to the smaller one. By convention, this keeps the F-value positive and greater than or equal to one, which matches the properties of the F-distribution that is always non-negative and right-skewed.
It is important not to confuse this with t-tests, which involve the means and standard deviations. Options c and d relate more to t-tests where differences between means are divided by standard deviations or pooled standard errors. The F-test focuses strictly on comparing variances, which is why option b correctly defines the F-test statistic as the ratio of the larger variance to the smaller variance. This statistic then helps in determining whether the difference in variances is statistically significant by comparing the computed F-value with critical values from an F-distribution table.