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 less than the permitted level of $0.5$. She records the concentration, $x$, of the pollutant at $50$ randomly selected sites in the river. The summary statistics are given below.
$n = 50 \qquad \Sigma x = 23.0 \qquad \Sigma x^2 = 13.02$
(a)[7]
At the $5\%$ significance level, carry out a test of the null hypothesis $\mu = 0.5$ against the alternative hypothesis $\mu < 0.5$.
(b)[5]
A later test is carried out at the $5\%$ significance level using another sample of size $50$ with the same hypotheses as before. Assume that the standard deviation remains unchanged. If, in fact, $\mu=0.4$, 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: “Calculate the sample mean $\hat{\mu}=\frac{23}{50}=0.46$.” …