Question 2 (a) State the null hypothesis of the Wilcoxon signed rank test and br

Question

Question 2
(a) State the null hypothesis of the Wilcoxon signed rank test and briefly describe its
meaning. [2]
(b) Explain why the Wilcoxon signed rank test is usually preferred to the sign test. [2]
(c) Give one limitation of the Wilcoxon signed rank test when it is compared with the sign
test. [2]
(d) Assume that the underlying mean for a single sample equals zero (m0=0), the sample
size is 18 and the sample data contains two 0s. Obtain the mean and variance of the
normal distribution which may be used to approximate the null distribution of the test
statistic. [5]
(e) The value of the test statistic is w+ = 101.5. Using the approximate null distribution that
you obtained in part (d), test the null hypothesis H0: m=0 using a two-sided alternative hypothesis. You should include the details of your calculations and state your
conclusions clearly. [5]
(f) A sign test is carried out using the same data. Explain whether you would expect the p
value for the sign test to be smaller or larger than that for the Wilcoxon signed rank test.
[2]

Explanation / Answer

(a). The Wilcoxon signed rank sum test is an example of a non-parametric or distribution free test . As for the sign test, the Wilcoxon signed rank sum test is used to test the null hypothesis that the median of a distribution is equal to some value.
It can be used :
1) in place of a one-sample t-test
2) in place of a paired t-test or
3) for ordered categorial data where a numerical scale is inappropriate but where it is possible to rank the observations.
When to use it:
Wilcoxon signed-rank test is used when there are two nominal variables and one measurement variable. One of the nominal variables has only two values, such as "before" and "after," and the other nominal variable often represents individuals. This is the non-parametric analogue to the paired t–test, and you should use it if the distribution of differences between pairs is severely non-normally distributed.
(b) We can analyze paired observations of a measurement variable using a paired t–test, if the null hypothesis is that the mean difference between pairs of observations is zero and the differences are normally distributed. If we have a large number of paired observations, we can plot a histogram of the differences to see if they look normally distributed. The paired t–test isn't very sensitive to non-normal data, so the deviation from normality has to be pretty dramatic to make the paired t–test inappropriate.
Sign Test is usually preferred over Wilcoxon signed rank test when the null hypothesis is equal number of differences in each direction, and we don't care about the size of the differences.
(c) Limitation of the Wilcoxon signed rank test when it is compared with the sign
test is that it simply allocates a sign to each observation, according to whether it lies above or below some hypothesized value, and does not take the magnitude of the observation into account.

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