Question

As part of a recent? survey, self-reported? heights, x, and measured? heights, y, were obtained for males aged 12?16. Use the Wilcoxon? signed-ranks test to test the claim that the matched pairs have differences that come from a population with median equal to zero at a significance level of alpha equals 0.1. Find the null and alternate hypothesis, test statistic, critival value and state the conclusion.

x	y
67	68.6
61	63
68	61.1
67	61
64	67.7
64	68.3
63	59.9
67	58.3
69	59.7
63	60.1
69	69.9
60	62.6

Answer 1

Hypothesis:

H0 : The population of differences has a median equal to 0.

H1 : The population of differences has a median not equal to 0.

Calculations for test statistics :

Note : For large samples with n>10 paired observations the W-statistics approximates a Normal Distribution.

Step 1: Calculate the differences of the repeated measurements
         and to calculate the absolute differences.

Step 2 : Order increasing absolute differences. Then rank them .
        If the original difference < 0 then the rank is multiplied by -1;
         if the difference is positive( > 0) the rank stays positive.

Note : Note :For the Wilcoxon signed rank test we can ignore cases where the difference is zero.
    For all other cases we assign their relative rank.
        In case of tied ranks the average rank is calculated.
         That is if rank 10 and 11 have the same observed differences both are assigned rank 10.

Sum of positive rank :

Sum of negative ranks :

Here we use normal appeoximation :

Since the Wilcoxon signed rank test has the null hypothesis that there is on average no difference between the two measurements, it is assumed that

mean =

Test statistics :

Critical value :

Here test is two tailed test ,and     then

So here we get two critical values , at 0.05 area .

In excel use command , =NORMSINV(0.05) ,then hit enter you will get z-score as -1.96.

So here we get critical values,     and

Decision : Test statistics value =Z= -1.098 > -1.96 , since we fail to reject H0 .

Conclusion : Fail to reject H0.

                   There is insufficient evidence to warrent rejection of the claim of no difference