Mathematics 9709 · AS & A Level · Hypothesis testing
Hypothesis testing — practice question
The faults in cloth produced on a particular machine follow a Poisson distribution with mean $2.4$ per $10\text{ m}^2$. The machine is then adjusted. At the $5\%$ significance level, the aim is to test whether the mean number of faults has gone down. A random $30\text{ m}^2$ piece of cloth is inspected and the number of faults is recorded.
(a)[1]
State appropriate null and alternative hypotheses for the test.
(b)[3]
Find the probability that a Type I error occurs.
(c)[2]
Exactly $3$ faults are observed in the randomly selected $30\text{ m}^2$ of cloth. Carry out the test at the $5\%$ significance level.
(d)[2]
At a later stage, a similar test was performed at the $5\%$ significance level using another randomly chosen $30\text{ m}^2$ of cloth. If the number of faults in fact follows a Poisson distribution with mean $0.5$ per $10\text{ m}^2$, find the probability of a Type II error.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State the hypotheses as $H_0: \mu = 7.2$ (or $2.4$), $H_1: \mu < 7.2$ (or $2.4$)” …