Calculate probability of 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.

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. \(0.4 \% 3.9 \% 18.9 \% 15.0 \%\)

The random variable \(R\) has the distribution Po\((\lambda)\). A significance test is carried out at the 1% level of the null hypothesis H\(_0\): \(\lambda = 11\) against H\(_1\): \(\lambda > 11\), based on a single observation of \(R\). Given that in fact the value of \(\lambda\) is 14, find the probability that the result of the test is incorrect, and give the technical name for such an incorrect outcome. You should show the values of any relevant probabilities. [6]