Standard +0.8 This question requires understanding of Type II errors (failing to reject H₀ when H₁ is true), working with sampling distributions under both null and alternative hypotheses, and calculating probabilities using the normal distribution with given parameters. While the calculation itself is straightforward once set up, correctly identifying what probability to find and which distribution to use requires solid conceptual understanding of hypothesis testing theory, making it moderately challenging for A-level Further Maths.
2 Amy takes a sample of size 50 from a normal distribution with mean \(\mu\) and variance 16
She conducts a hypothesis test with hypotheses:
$$\begin{aligned}
& \mathrm { H } _ { 0 } : \mu = 52 \\
& \mathrm { H } _ { 1 } : \mu > 52
\end{aligned}$$
She rejects the null hypothesis if her sample has a mean greater than 53
The actual population mean is 53.5
Find the probability that Amy makes a Type II error.
Circle your answer.
\(0.4 \% 3.9 \% 18.9 \% 15.0 \%\)
2 Amy takes a sample of size 50 from a normal distribution with mean $\mu$ and variance 16
She conducts a hypothesis test with hypotheses:
$$\begin{aligned}
& \mathrm { H } _ { 0 } : \mu = 52 \\
& \mathrm { H } _ { 1 } : \mu > 52
\end{aligned}$$
She rejects the null hypothesis if her sample has a mean greater than 53\\
The actual population mean is 53.5\\
Find the probability that Amy makes a Type II error.\\
Circle your answer.
$0.4 \% 3.9 \% 18.9 \% 15.0 \%$
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2019 Q2 [1]}}