Mathematics 9709 · AS & A Level · Hypothesis testing
Hypothesis testing — practice question
A biscuit maker says that, on average, $1$ out of every $3$ packets of biscuits includes a prize offer. Gerry thinks the proportion of packets with the prize offer is below $\frac{1}{3}$. To check the maker’s statement, he buys $20$ packets chosen at random. He discovers that exactly $2$ of those packets include the prize offer.
(a)[5]
Perform the test at the $10\%$ significance level.
(b)[3]
Maria also believes that the proportion of packets containing the prize offer is less than $1$ in $3$. She carries out a significance test at the $10\%$ level using $20$ packets chosen at random. She will reject the manufacturer’s claim if she finds that there are $3$ or fewer packets containing the prize offer.
Find the probability of a Type II error in Maria’s test if the proportion of packets containing the prize offer is actually $1$ in $7$.
(c)[1]
Explain the meaning of a Type II error in this context.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “The correct hypotheses are $H_0:p=\frac{1}{3},\;H_1:p<\frac{1}{3}$” …