Question

Use a Wilcoxon Signed Rank Test of the alternate hypothesis H1 : µ does not equal 7.39 and the following data:

7.02 7.35 7.34 7.17 7.28 7.77 7.09

7.22 7.45 6.95 7.40 7.10 7.32 7.14

to report the observed level of statistical significance.

Answer 1

vs

Define

D_i =X_i -3.79

Since the addition is community WLG we assume that |D₁|< |D₂|<...<|D_n|

Rank |D_i| =i

Define Z_(i)=1 if the difference whose absolute value has rank i is positive

0 o.w

we can write

Then we arrange the value in ascending order

6.95, 7.02, 7.09, 7.10, 7.14, 7.17, 7.22, 7.28, 7.32, 7.34,7.35,7.40, 7.45, 7.77 and D_i =0.44, 0.37, 0.3,0.29, 0.25, 0.22, 0.17, 0.11, 0.07, 0.05, 0.04, 0.01, 0.06, 0.38 respectively

i	Di	sign
1	0.01	+
2	0.04	-
3	0.05	-
4	0.06	+
5	0.07	-
6	0.11	-
7	0.17	-
8	0.22	-
9	0.25	-
10	0.29	-
11	0.3	-
12	0.37	-
13	0.38	+
14	0.44	-

W⁺= (1+4+13)

=18

E(W⁺)= n(n+1)/4

=14(14+1)/4

=42

Var(W⁺) =n(n+1)(2n+1)/24

=14(14+1)(2*14+1)/24

=253.75

test statistic

=(18-42)/15.93

=-1.51

p value 0.13 > 0.05 so we accept null hypothesis