Imagine you are looking to test whether students in the morning class of a particular course get higher grades than students in the afternoon class. You take a sample of 35 students in the morning class and 12 students in the afternoon class. You test this using a t-test of students’ final grades. What are the degrees of freedom of this test?

Question

BlackTom AI · Accepted Answer

We start by restating what the question is asking: we have two independent samples, morning class with n1 = 35 students and afternoon class with n2 = 12 students, and we are conducting a t-test on their final grades to compare the two means. The key issue is which formula for degrees of freedom to use in a two-sample t-test.

Option 1: 45. If we assume equal variances and use the pooled-variance two-sample t-test, the degrees of freedom are df = n1 + n2 − 2 = 35 + 12 − 2 = 45. This is the standard df used in the classic independent two-sample t-test under the equal-variances assumption.

Option 2: 46. This would imply df = (n1 + n2) − 1, which is not the correct formula for a two-sample t-test (that would be relevant for a one-sample t-test with combined data, but not here). It does not align with the standard two-sample approach.

Option 3: 34. This would correspond to df = n1 − 1 if we were testing only the morning group, or df = n1 + n2 − something that does not match the two-sample pooled approach. It incorrectly uses n1 instead of the combined sample size minus 2.

Option 4: 11. This would be df = n2 − 1 for the afternoon group alone, or some other misapplied calculation. It misrepresents the degrees of freedom for a comparison between two independent groups.

Overall, when comparing two independent samples using a two-sample t-test with equal variances assumed, the conventional degrees of freedom is df = n1 + n2 − 2. Given n1 = 35 and n2 = 12, that yields df = 45, which matches Option 1. The other options reflect common misapplications of the df formula or misinterpretations of the sample sizes and test structure.

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