Wilcoxon Rank Sum Statistic Expected Value

The question asks for the expected value of the Wilcoxon rank sum statistic, denoted as R, based on the ranks of the X observations in a combined sample of X and Y observations.

We are given two independent and identically distributed (i.i.d.) sets of observations:

• X observations: $X_1, X_2, ..., X_{15}$. The sample size for X is $n_1 = 15$.
• Y observations: $Y_1, Y_2, ..., Y_{20}$. The sample size for Y is $n_2 = 20$.

The total number of observations in the combined sample is $N = n_1 + n_2 = 15 + 20 = 35$.

The Wilcoxon rank sum statistic R (when calculated as the sum of ranks of the first sample, X in this case) is the sum of the ranks assigned to the $n_1$ observations from the first sample when all $N$ observations from both samples are combined and ranked from 1 to N.

The formula for the expected value of the Wilcoxon rank sum statistic R, based on the sum of ranks of the sample with size $n_1$, is given by:

$$E(R) = \frac{n_1(N + 1)}{2}$$

Where:

• $n_1$ is the size of the first sample (X).
• $N$ is the total sample size ($N = n_1 + n_2$).

Calculating the Expected Value

Using the given values:

• $n_1 = 15$
• $n_2 = 20$
• $N = n_1 + n_2 = 15 + 20 = 35$

Substitute these values into the formula for the expected value of R:

$$E(R) = \frac{15(35 + 1)}{2}$$

$$E(R) = \frac{15(36)}{2}$$

$$E(R) = 15 \times \frac{36}{2}$$

$$E(R) = 15 \times 18$$

Now, perform the multiplication:

$$15 \times 18 = 270$$

So, the expected value of the Wilcoxon rank sum statistic R is 270.

This value represents the average sum of ranks one would expect for the X sample if the two populations from which X and Y are drawn are identical (under the null hypothesis).

Comparing this result with the given options, we find that 270 matches Option 1.