Question

(In R) There were two materials (A and B )that were randomly assigned to the left and right shoes of 10boys. the data are available in R ,is a list called shoes(MASS) a)Test the Hypothesis of no difference in mean shoe wear for materials A and B. Repeat using a non parametric test. Do the two tests agree? b) Assume now that there is in fact no difference between A and B. Test the hypothesis that the mean shoe wear from right feet is greater than that from left feet. Your answer should explain how you have dealt with the L-R information for boy 4 ,which is missing.

BOY	A	B
1	13.2(L)	14.0(R)
2	8.2(L)	8.8(R)
3	10.9(R)	11.2(L)
4	14.3	14.2
5	10.7(R)	11.8(L)
6	6.6(L)	6.4(R)
7	9.5(L)	9.8(R)
8	10.8(L)	11.3(R)
9	8.8(R)	9.3(L)
10	13.3(L)	13.6(R)

Answer 1

$let~\eta~be~population~median~of~difference~between~materials~A~and~B.\\ The~null~hypothesis,~H_0:\eta=0~vs.~Alternative~hypothesis,~H_1:\eta\neq0.$

$where,~D_i=difference~between~left~and~right~feet.\\ Arrange~the~absolute~values~of~|D_i|~in~increasing~order~then\\ Wilcoxon~sign~rank~statistic=w=sum~of~the~rank~of~positive~D_i\\ =1&plus;2=3\\ p-value= 0.014<0.05\\ Hence~we~reject~H_0~at~5\%~level~of~significance~and~conclude~that~\\ there~is~significant~difference~between~materials~A~and~B.$

We randomly assign (R, L) for boy y.

Case: 1 we assign 14.3 for R and 14.2 for L and then perform Wilcoxon sign rank test.

$let~\theta~be~population~median~of~difference~between~left~and~right~feet.\\ The~null~hypothesis,~H_0:\theta=0~vs.~Alternative~hypothesis,~H_1:\theta<0.$

$where,~D_i=difference~between~left~and~right~feet.\\ Arrange~the~absolute~values~of~|D_i|~in~increasing~order~then\\ Wilcoxon~sign~rank~statistic=w=sum~of~the~rank~of~positive~D_i\\ =4&plus;1&plus;10&plus;2&plus;6.5&plus;6.5=30\\ p-value=0.620>0.05\\ Hence~we~fail~to~reject~H_0~at~5\%~level~of~significance~and~conclude~that~\\ no~difference~between~means~shoe~wear~from~right~feet~and~left~feet.$

Case: 2 we assign 14.3 for L and 14.2 for R and then perform Wilcoxon sign rank test.

$Wilcoxon~sign~rank~statistic=w=sum~of~the~rank~of~positive~D_i\\ =4&plus;10&plus;2&plus;6.5&plus;6.5= 26.5\\ p-value=0.480>0.05\\ Hence~we~fail~to~reject~H_0~at~5\%~level~of~significance~and~conclude~that~\\ no~difference~between~means~shoe~wear~from~right~feet~and~left~feet.\\ Hence~two~tests~have~same~conclusion.$