State meaning of Type II error

2 In the past, the time, in hours, for a particular train journey has had mean 1.40 and standard deviation 0.12 . Following the introduction of some new signals, it is required to test whether the mean journey time has decreased.

1. State what is meant by a Type II error in this context.
2. The mean time for a random sample of 50 journeys is found to be 1.36 hours. Assuming that the standard deviation of journey times is still 0.12 hours, test at the \(2.5 \%\) significance level whether the population mean journey time has decreased.
3. State, with a reason, which of the errors, Type I or Type II, might have been made in the test in part (b).

6 A survey taken last year showed that the mean number of computers per household in Branley was 1.66 . This year a random sample of 50 households in Branley answered a questionnaire with the following results.

1. Calculate unbiased estimates for the population mean and variance of the number of computers per household in Branley this year.
2. Test at the \(5 \%\) significance level whether the mean number of computers per household has changed since last year.
3. Explain whether it is possible that a Type I error may have been made in the test in part (ii).
4. State what is meant by a Type II error in the context of the test in part (ii), and give the set of values of the test statistic that could lead to a Type II error being made.

4 People who diet can expect to lose an average of 3 kg in a month. In a book, the authors claim that people who follow a new diet will lose an average of more than 3 kg in a month. The weight losses of the 180 people in a random sample who had followed the new diet for a month were noted. The mean was 3.3 kg and the standard deviation was 2.8 kg .

1. Test the authors' claim at the \(5 \%\) significance level, stating your null and alternative hypotheses.
2. State what is meant by a Type II error in words relating to the context of the test in part (i).

7 The weights, \(X\) kilograms, of bags of carrots are normally distributed. The mean of \(X\) is \(\mu\). An inspector wishes to test whether \(\mu = 2.0\). He weighs a random sample of 200 bags and his results are summarised as follows. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 1290$$

1. Carry out the test, at the \(10 \%\) significance level.
2. You may now assume that the population variance of \(X\) is 1.85 . The inspector weighs another random sample of 200 bags and carries out the same test at the \(10 \%\) significance level.
   1. State the meaning of a Type II error in this context.
   2. Given that \(\mu = 2.12\), show that the probability of a Type II error is 0.652 , correct to 3 significant figures.

7 In the past the time, in minutes, taken for a particular rail journey has been found to have mean 20.5 and standard deviation 1.2. Some new railway signals are installed. In order to test whether the mean time has decreased, a random sample of 100 times for this journey are noted. The sample mean is found to be 20.3 minutes. You should assume that the standard deviation is unchanged.

1. Carry out a significance test, at the \(4 \%\) level, of whether the population mean time has decreased. Later another significance test of the same hypotheses, using another random sample of size 100 , is carried out at the \(4 \%\) level.
2. Given that the population mean is now 20.1, find the probability of a Type II error.
3. State what is meant by a Type II error in this context.