Mathematics 9709 · AS & A Level · Hypothesis testing
Hypothesis testing — practice question
A biologist wants to investigate whether the mean concentration $\mu$, in suitable units, of a particular pollutant in a river is below the allowed level of $0.5$. She records the concentration, $x$, of the pollutant at $50$ randomly selected places in the river. The data are summarised below.
$n = 50 \quad \sum x = 23.0 \quad \sum x^2 = 13.02$
(a)[7]
At the $5\%$ significance level, perform a test of the null hypothesis $\mu = 0.5$ against the alternative hypothesis $\mu < 0.5$.
(b)[5]
At a later stage, a comparable test is conducted at the $5\%$ significance level using another sample of size $50$ and the same hypotheses as before. You should take the standard deviation to be unchanged. If, in reality, the value of $\mu$ is $0.4$, determine the probability of a Type II error.
Worked solution & mark scheme
This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: “The unbiased estimate of the mean is $\hat\mu=\frac{23}{50}=0.46$.” …