Final answer:
To calculate an F test variance, estimates of variance due to treatment and error variance are used. Assumptions such as normal distribution, equal variances, and independent observations must be met. The F test compares the null hypothesis of no difference with an alternative hypothesis of a significant difference.
Step-by-step explanation:
To calculate the value of a variance, such as when determining whether there is a significant difference between two variances, we use an F test.
In the scenario provided, if we are to calculate the variance between samples, also known as the variance due to treatment or explained variation, we must consider whether the data is balanced (equal sample sizes) or unbalanced (different sample sizes).
In balanced data, the calculation is straightforward and involves comparing the variance between group means. For unbalanced data, a weighted approach is needed. Once the variance between samples is determined, it can be compared against the within-sample variance (error variance) to compute the F ratio.
The F test requires certain assumptions to be met:
The populations from which the samples are drawn must be normally distributed.
Population variances must be equal (homogeneity of variance).
The observations must be independent.
When conducting an F test for two variances, the null hypothesis (H0) typically states that there is no difference in variances between the two groups.
The alternative hypothesis (H1 or Ha) states the opposite that there is a significant difference. For the example comparing math and English department grades, H0 would be that the variances are equal, and H1 would be that there is greater variance in math department grades.