Write down the number of ways of choosing 5 objects from 12 distinct objects.
Each possible set of 5 different integers selected from the integers \(1,2 , \ldots , 12\) is obtained, and for each set, the sum of the 5 integers is found. The sum \(S\) can take values between 15 and 50 inclusive. Part of the frequency distribution of \(S\) is shown in the following table, together with the cumulative frequencies.
S
15
16
17
18
19
20
21
22
23
Frequency
1
1
2
3
5
7
10
13
17
Cumulative Frequency
1
2
4
7
12
19
29
42
59
Use these numbers to determine the critical region for a 1-tail Wilcoxon rank-sum test at the \(2 \%\) significance level when \(m = 5\) and \(n = 7\).
A student says that, for a Wilcoxon rank-sum test on samples of size \(m\) and \(n\), where \(m\) and \(n\) are large, the mean and variance of the test statistic \(R _ { m }\) are 200 and \(616 \frac { 2 } { 3 }\) respectively.
Show that at least one of these values must be incorrect.