7 In a school opinion poll a random sample of 8 pupils were asked to rate school lunches on a scale of 0 to 20 . The results were as follows.
\(\begin{array} { l l l l l l l l } 0 & 1 & 2 & 3 & 4 & 10 & 11 & 13 \end{array}\)
After a new menu was introduced, the test was repeated with a different random sample of 8 pupils. The results were as follows.
\(\begin{array} { l l l l l l l l } 7 & 8 & 9 & 14 & 15 & 17 & 19 & 20 \end{array}\)
- Carry out an appropriate Wilcoxon test at the \(5 \%\) significance level to test whether pupils' opinions of school lunches have changed.
A statistics student tells the organisers of the opinion poll that it would have been better to have asked the same 8 pupils both times.
- Explain why the statistics student's suggestion would produce a better test.
- State which test should be used if the student's suggestion is followed.
- You are given that there are 12870 ways in which 8 different integers can be chosen from the integers 1 to 16 inclusive.
Estimate the number of ways of selecting 8 different digits between 1 and 16 inclusive that have a sum less than or equal to the critical value used in the test in part (a).