Question

Do students perform better on the Verbal section of the SAT compared to the Math? 10 students were randomly selected and their Math & Verbal scores are listed below. Test the claim that students perform better on the Verbal section and the 5% significance level using PHANTOMS.

Math	Verbal
640	470
660	650
550	580
560	660
600	790
560	640
550	660
600	560
540	560
540	560

Answer 1

1 denote for maths and 2 denote for Verbal

This is the case of one-tailed two samples paired t test

H0: Students perform equally for both sections. i.e. mu1 = mu2

H1:  Students perform better on the Verbal section than the Math section i.e. mu1 < mu2

alpha = 0.05

Math	Verbal	d (Math - Verbal)
640	470	170
660	650	10
550	580	-30
560	660	-100
600	790	-190
560	640	-80
550	660	-110
600	560	40
540	560	-20
540	560	-20

Test statistic =

dbar = -33

S_d = 97.75821

n = 10

after substituting, we get

t0 = -1.0675

p-value = P(t9 < t_0) = P(t9 < -1.0675) = 0.1567711

As, p-value > alpha = 0.05. So, we fail to reject H0, implies that we don't have enough evidence to claim that Students perform better in Verbal section