Problem
2. It is claimed that $10 \%$ of men can distinguish between butter and margarine, but some people feel that this percentage is too low. A hypothesis test is carried out on a random sample of ten men. What is the null hypothesis, $\mathrm{H}_{0}$, which is used? What is the alternative hypothesis, $\mathrm{H}_{1}$ which is used? Out of the ten men who were tested, three correctly distinguished between butter and margarine. What is the $p$-value for this test? What is the outcome of the test at the $5 \%$ significance level? In another sample, $N$ men from a sample of 12 men correctly distinguished between butter and margarine, and $\mathrm{H}_{0}$ was rejected at the $5 \%$ significance level What are the possible values of $N$ ?
Solution
Step 1 ：\(H_0: p = 0.1\)
Step 2 ：\(H_1: p > 0.1\)
Step 3 ：\(p\text{-value} = \binom{10}{3}(0.1)^3(0.9)^7 + \binom{10}{4}(0.1)^4(0.9)^6 + \cdots + \binom{10}{10}(0.1)^{10}(0.9)^0 = 0.057\)
Step 4 ：\(\text{Since } p\text{-value} > 0.05, \text{ do not reject } H_0\)
Step 5 ：\(\text{Possible values of } N: \boxed{5, 6, 7, 8, 9, 10, 11, 12}\)
