In 2012 researchers at Deakin University in Australia ran a twelve-week dietary intervention study involving twins. Within each twin pair volunteering, one sibling was placed at random on either a low or high fat diet for a period of twelve weeks. At the start of the study, each subject was weighed and the weight change (in g) for each subject was recorded at the end of the study, with a positive number indicating a gain, a negative number indicating a weight loss. Suppose the data collected were as in the table below:



The researchers decide to use a sign test to test the null hypothesis that the distribution of the differences in weight gain has median zero within twin pairs.

Part a)
Would it better to adopt a one or two-sided alternative hypothesis here?

A. It does not matter.
B. Two--sided.
C. One--sided.


Part b)
The test statistic is taken as the number of pairs in which the sibling on the high fat diet gained more weight. The null hypothesis is to be rejected if this count is greater than . To three decimal places, what is the significance level of this test?


Part c)
Suppose that in fact the sibling on the high fat diet will gain more weight than their sibling in $\%$ of cases over a twelve-week period. Taking this information to define the alternative hypothesis, compute the power of the hypothesis test to three decimal places.


Part d)
Suppose that in fact the sibling on the high fat diet will gain more weight than their sibling in $\%$ of cases over a twelve-week period. Now taking this information to define the alternative hypothesis, compute the power of the hypothesis test to three decimal places.


Part e)
Which statement best describes how the test appears to behave as the median of the distribution of interest shifts upwards?

A. As the median of the within--pair difference shifts downwards so that weight gain is more likely under the high fat diet, the test has higher significance level.
B. As the median of the within--pair difference shifts upwards so that weight gain is more likely under the high fat diet, the test has lower significance level.
C. As the median of the within--pair difference shifts upwards so that weight gain is more likely under the high fat diet, the test is more powerful.
D. As the median of the within--pair difference shifts upwards so that weight gain is more likely under the high fat diet, the test is less powerful.
E. As the median of the within--pair difference shifts downwards so that weight gain is more likely under the high fat diet, the test has lower significance level.