Question

Suppose the students in a school district are assigned to different schools by lottery. You are interested in knowing if the test scores upon graduation differ significantly across schools more than random errors. You are considering an ANOVA test on the equality of test scores across the three schools.

(a) Three independent student samples have been drawn, with the mean and SD within each sample reported above. What are the assumptions for applying the ANOVA test? Do they hold in this setting?

Mean test score SD test score Number of student
school 1 82 8 60
school 2 79 5 80
school 3 78 6 50

Answer 1

To apply a one-way ANOVA test, assumptions of normality, independence, equal variances, a categorical factor, and a numerical response must be met. In the provided scenario, differing standard deviations across schools indicate a violation of the equal variance assumption.

Step-by-step explanation:

The question pertains to the application of a one-way ANOVA test to determine if there is a significant difference in test scores upon graduation across three different schools. For a one-way ANOVA to be appropriately used, certain assumptions must be met:

• Each population from which a sample is taken is assumed to be normal.

• All samples must be randomly selected and independent.

• The populations are assumed to have equal standard deviations (or variances).

• The factor, in this case the school, is a categorical variable.

• The response, namely the test score, is a numerical variable.

In the given scenario, the samples appear to be independently and randomly selected, and the outcome variable (test score) is numerical. However, the reported standard deviations are different across the three schools, suggesting that the assumption of equal variances may not hold. This violation means further statistical testing, such as Levene's test or Brown-Forsythe test, may be required to validate the use of ANOVA or to consider alternative methods that do not assume equal variances.