1. A manufacturer has a machine that produces lollipop sticks.

The length of a lollipop stick produced by the machine is normally distributed with unknown mean \(\mu\) and standard deviation 0.2 Farhan believes that the machine is not working properly and the mean length of the lollipop sticks has decreased.
He takes a random sample of size \(n\) to test, at the 1\% level of significance, the hypotheses $$\mathrm { H } _ { 0 } : \mu = 15 \quad \mathrm { H } _ { 1 } : \mu < 15$$

1. Write down the size of this test. Given that the actual value of \(\mu\) is 14.9
2. 1. calculate the minimum value of \(n\) such that the probability of a Type II error is less than 0.05
      Show your working clearly.
   2. Farhan uses the same sample size, \(n\), but now carries out the test at a \(5 \%\) level of significance. Without doing any further calculations, state how this would affect the probability of a Type II error.