Challenging +1.2 This question requires understanding of Type II errors, working with normal distributions under both null and alternative hypotheses, and calculating probabilities using the standard normal distribution. While it involves multiple conceptual steps (finding critical region, then probability under true distribution), it's a standard S2 textbook exercise with clear setup and straightforward calculation once the method is understood.
3 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). A hypothesis test is carried out of \(\mathrm { H } _ { 0 } : \mu = 20.0\) against \(\mathrm { H } _ { 1 } : \mu < 20.0\), at the \(1 \%\) level of significance, based on the mean of a sample of size 16. Given that in fact \(\mu = 15.0\), find the probability that the test results in a Type II error.
3 The random variable $X$ has the distribution $\mathrm { N } \left( \mu , 5 ^ { 2 } \right)$. A hypothesis test is carried out of $\mathrm { H } _ { 0 } : \mu = 20.0$ against $\mathrm { H } _ { 1 } : \mu < 20.0$, at the $1 \%$ level of significance, based on the mean of a sample of size 16. Given that in fact $\mu = 15.0$, find the probability that the test results in a Type II error.
\hfill \mbox{\textit{OCR S2 2011 Q3 [7]}}