A machine fills bags with flour. The weight of flour delivered by the machine into a bag, \(X\) grams, is normally distributed with mean \(\mu\) grams and standard deviation 30 grams. To check if there is any change to the mean weight of flour delivered by the machine into each bag, Olaf takes a random sample of 10 bags. The weight of flour, \(x\) grams, in each bag is recorded and \(\bar { x } = 1020\)
Test, at the \(5 \%\) level of significance, \(\mathrm { H } _ { 0 } : \mu = 1000\) against \(\mathrm { H } _ { 1 } : \mu \neq 1000\)
Olaf decides to alter the test so that the hypotheses are \(\mathrm { H } _ { 0 } : \mu = 1000\) and \(\mathrm { H } _ { 1 } : \mu > 1000\) but keeps the level of significance at 5\%
He takes a second sample of size \(n\) and finds the critical region, \(\bar { X } > c\)
Find an equation for \(c\) in terms of \(n\)
When the true value of \(\mu\) is 1020 grams, the probability of making a Type II error is 0.0050 , to 2 significant figures.
Calculate the value of \(n\) and the value of \(c\)