Question 3. Luxlait sells milk in cartons of 1 liter. It is supposed that the amount
of milk in one carton has a normal distribution with a mean of 1.00 liter and a
2
standard deviation of 0.06 liter. To test whether the filling machine still puts 1 liter
in a carton on average, a quality assurance officer decides to take 9 cartons of milk
from the production line at random times during a day. Of these 9 cartons, the
content is measured carefully and the sample average $X$ is determined. The quality
assurance officer wants to use a significance level of 5%.
(a) Formulate appropriate hypotheses that the quality assurance officer can test
based on his sample of 9 cartons.
(b) What is the distribution of $X$ under the null hypothesis that you formulated in
the previous question?
(c) What is the critical region for the statistical test?
(d) Suppose that the actual mean amount of milk per carton is not 1.00 liter but
1.03 liter. What is the probability that the statistical test discovers that the
mean amount of milk per carton has changed?
(e) Suppose that you would like to be able to detect a shift in the mean from 1 liter
to 1.03 liter with a probability of at least 80%. How large should the sample
size be to achieve this?
