Question

*show work by hand*

5) Wilcoxon Signed-Ranks T Test A statistics tutor wants to assess whether his remedial tutoring sessions have been useful for his 8 students. He records their class grades (percentages) before and after the sessions. Conduct the Wilcoxon Signed-Ranks T Test to determine whether the sessions significantly improved their grades (8 points). Difference (After - Before) Difference Student After Before Tutoring 77 Tutoring 88 Rank D.O. 72 J.I. 91 81 82 D.S. 76 79 E.C. 67 80 L.L. 75 60 J.H. 71 J.M. 78 94 84 A.M. Significant? Critical T (1-tailed, a.05) =

Answer 1

Wilcoxon Signed Rank T Test

Computation by hand:

Student	After	Before	Di=After-Before	Sign	ABS Di	Rank of ABS(Di)	#	T+	T-
D.O.	88	77	11	+	11	5		5	1
J.I.	91	72	19	+	19	8		8	2
D.S.	81	82	-1	-	1	1		6	3
E.C.	76	79	-3	-	3	2		7
L.L.	80	67	13	+	13	6		3
J.H.	75	60	15	+	15	7		4
J.M.	78	71	7	+	7	3		33
A.M.	94	84	10	+	10	4

T=min(T+,T-)=min(33,3)=3

(T+)+(T-)=33+3=36 and also n(n+1)/2= 8(8+1)/2=36.

T(one-tailed, 0.05) at N=8 is 5. [Obtained using Statistical Table]

Here, T > T(one-tailed, 0.05), fail to Reject H0.

Significant? NO

If we use Normal approximation here(n=8)
Here,take n=8 (If Di=0, then discard this values, there are no such values, thus effective sample size=8-0=8)
E(T)=n(n+1)/4=(8*9)/4= 18
V(T)=n(n+1)(2n+1)/24=(8*9*17)/24=51

Note that :Two or more observations/differences may be equal. If so, average the ranks across the tied observations and reduce the variance by (t^ 3−t)/ 48 for each group of t tied ranks.

Square root of V(T)=Square root (51)= 7.141428
Z=(3-18)/7.141428= -2.10042

Here, Z (one-tailed) test, fail to reject H0: if z < tabulated z. Z~N(0,1), tab Z at 5% level of significance= 1.64.

T - E(T) Z= 3 -18 5 -15 1100 = -2.10042 =