5 The number of hours that Mrs Hughes spends on her business in a week is normally distributed with mean \(\mu\) and standard deviation 4.8. In the past the value of \(\mu\) has been 49.5.
- Assuming that \(\mu\) is still equal to 49.5 , find the probability that in a random sample of 40 weeks the mean time spent on her business in a week is more than 50.3 hours.
Following a change in her arrangements, Mrs Hughes wishes to test whether \(\mu\) has decreased. She chooses a random sample of 40 weeks and notes that the total number of hours she spent on her business during these weeks is 1920.
- (a) Explain why a one-tail test is appropriate.
(b) Carry out the test at the 6\% significance level.
(c) Explain whether it was necessary to use the Central Limit theorem in part (ii) (b).