A certain factory consumes on average 1000 m3 of water per day. A random sample of 100 days was taken to test if the mean daily water intake remains 1000 m3 against the alternative that the mean water consumption has increased. We know that the sample mean equals = 1005 m3 and the sample quasi-variance is s2 = 400 m6.
(a) Define the null and alternative hypotheses for this test.
(b) Perform the test at the significance level α = 0.05.
(c) What is the smallest value of α for which the null hypothesis can be rejected?
(d) Calculate the power of the test in two scenarios: If the true mean water consumption equals
µ1 = 1000 m3 and µ1 = 1008 m3.
(e) Indicate which of the following statements are true/false and justify your answer:
i. If we reject the hypothesis at the level α = 0.05, we can also reject H0 at the level α = 0.1.
ii. The Type 1 error is the probability to reject the null hypothesis when H1 is true.
iii. If the p-value equals 0.15, we can reject the null hypothesis at the level 10%.