Question

Suppose for the two exams in this course, we would like to see if there is any significant improvement from exam 1 to exam 2, i.e., testing H0 : µx ≥ µy vs HA : µx < µy for the average exam scores. Suppose we have n = 36 students, and the sample statistics are x¯ = 21, y¯ = 22, sx = sy = 3 and sxy = 4.5. Compute the p-value using paired two-sample test

Suppose we use the unpaired two-sample test instead, i.e., sxy is not used. Compute the p-value: is the p-value larger than or smaller

Suppose we do not know y¯ yet as the second exam is not held yet. Assuming everything else still holds, i.e., x¯ = 21, sx = sy = 3 and sxy = 4.5, and we use the paired test. What shall y¯ be, such that the null can be rejected at level 0.001? It suffices to express the answer by an equation that computer can solve

Answer 1

Using paired data:

Here we have

x-y=21-22

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Using unpaired two-sample test instead: