Question

Two independent random samples, each of size 7, from two populations yield the following values: \[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline \text{Population 1} & 18 & 20 & 16 & 20 & 17 & 18 & 14 \\ \hline \text{Population 2} & 17 & 18 & 14 & 20 & 14 & 13 & 16 \\ \hline \end{array} \] If Mann–Whitney \( U \) test is performed at 5% level of significance to test the null hypothesis \( H_0: \) Distributions of the populations are same, against the alternative hypothesis \( H_1: \) Distributions of the populations are not same, then the value of the test statistic \( U \) (in integer) for the given data, is ________

Hint:

For the Mann–Whitney U test, remember to rank all observations together and use the formula \( U = n_1n_2 + \frac{n_1(n_1+1)}{2} - R_1 \).

Answer 1

Correct Answer: 15

The Mann–Whitney \( U \) test is based on the ranks of the combined data from both samples. First, combine all 14 observations from both populations and assign ranks in ascending order: \[ 13, 14, 14, 14, 16, 16, 17, 17, 18, 18, 18, 20, 20, 20 \] After assigning average ranks for ties: \[ 13 (1), \, 14 (2,3,4) \rightarrow 3, \, 16 (5,6) \rightarrow 5.5, \, 17 (7,8) \rightarrow 7.5, \, 18 (9,10,11) \rightarrow 10, \, 20 (12,13,14) \rightarrow 13 \] Now, assign ranks to each sample: Population 1: 18(10), 20(13), 16(5.5), 20(13), 17(7.5), 18(10), 14(3) Sum of ranks for Population 1, \( R_1 = 10 + 13 + 5.5 + 13 + 7.5 + 10 + 3 = 62 \) Population 2: 17(7.5), 18(10), 14(3), 20(13), 14(3), 13(1), 16(5.5) Sum of ranks for Population 2, \( R_2 = 43 \) The test statistic \( U \) is calculated as: \[ U_1 = n_1n_2 + \frac{n_1(n_1 + 1)}{2} - R_1 \] \[ U_1 = 7(7) + \frac{7(8)}{2} - 62 = 49 + 28 - 62 = 15 \] Thus, the test statistic \( U \) is \( \boxed{15} \).