Question

In a certain experiment to compare two types of sheep food A and B the following results of increase in weights were observed:

Sheep No 1 2 3 4 4 6 7 8
Food A 49 53 51 52 47 50 52 53
Food B 52 55 52 53 50 54 54 53

Assuming that the two samples of sheep are independent; can we conclude that Food B is better than Food A?

Answer 1

We can use the sign test in this situation. We suppose that random variable $X$ corresponds to the values in row "Food A" and $Y$ to the values in row "Food B". We formulate two hypotheses: $H_0: P(X>Y)=\frac{1}{2}$ and $H_1:P(X>Y)<\frac{1}{2}$ . As we can see from the values, for all samples holds $y_i<x_i$ . Thus, we accept the alternative hypothesis $H_1$ . It means that Food B is better than Food A.

We took the data from the table. As we can see, there are 8 entries. The differences between Food B and Food A are always positive. For the sign test we use the values from the table for binomial distribution (https://www.statisticshowto.com/tables/binomial-distribution-table/). We calculate the probability of 8 successes in 8 trials with the probability 0.5. It is equal to approximately 0.004. It means that the hypothesis $H_0:P(X>Y)=\frac12$ can be rejected. Thus, the Food B is better than Food A