Consider the following two samples of test scores:

Which of the following options is a valid randomization of them?

The Mann-Whitney $U$ test is a nonparametric test used to compare two independent samples. It is equivalent to the Wilcoxon rank-sum test and leads to the same conclusions when applied appropriately. The test statistic is calculated as:

$z=\frac{U-\frac{n_1{n}_2}{2}}{\sqrt{\frac{n_1{n}_2(n_1+n_2+1)}{12}}}$

where:

$U=n_1{n}_2+\frac{n_1(n_1+1)}{2}-R$

$n_1=$ size of Sample $1$

$n_2=$ size of Sample $2$

$R=$sum of the ranks for Sample $1$

Use the following pulse rate data (in units of beats per minute) for two groups of athletes to calculate the $z$-test statistic for the Mann-Whitney $U$ test.

Sample $1$ (athletes) : $72,78,80,85,90$

Sample $2$ (long-distance runners): $60,65,68,70,75$

Suppose we have two treatments, X and Y, each with only two quantitative observations. Because the sample sizes are small (fewer than $10$), the Wilcoxon signed-ranks test is not applicable.

The combined data have been ranked from $1$ (lowest) to $4$ (highest). Below is one possible assignment of treatments to ranks.

Which of the following is the correct list of all possible rank sums for treatment X when there are two values from treatment X and two from treatment Y, ranked from $1$ (lowest) to $4$ (highest)?