Identify which error type was made

6 Last year Samir found that the time for his journey to work had mean 45.7 minutes and standard deviation 3.2 minutes. Samir wishes to test whether his journey times have increased this year. He notes the times, in minutes, for a random sample of 8 journeys this year with the following results. $$\begin{array} { l l l l l l l l } 46.2 & 41.7 & 49.2 & 47.1 & 47.2 & 48.4 & 53.7 & 45.5 \end{array}$$ It may be assumed that the population of this year's journey times is normally distributed with standard deviation 3.2 minutes.

1. State, with a reason, whether Samir should use a one-tail or a two-tail test.
2. Show that there is no evidence at the \(5 \%\) significance level that Samir's mean journey time has increased.
3. State, with a reason, which one of the errors, Type I or Type II, might have been made in carrying out the test in part (ii).

5 The numbers of basketball courts in a random sample of 70 schools in South Mowland are summarised in the table.

1. Calculate unbiased estimates for the population mean and variance of the number of basketball courts per school in South Mowland.
   The mean number of basketball courts per school in North Mowland is 1.9 .
2. Test at the \(5 \%\) significance level whether the mean number of basketball courts per school in South Mowland is less than the mean for North Mowland.
3. State, with a reason, which of the errors, Type I or Type II, might have been made in the test in part (ii).

8 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 8 ^ { 2 } \right)\). A test is carried out, at the \(5 \%\) significance level, of \(\mathrm { H } _ { 0 } : \mu = 30\) against \(\mathrm { H } _ { 1 } : \mu > 30\), based on a random sample of size 18 .

1. Find the critical region for the test.
2. If \(\mu = 30\) and the outcome of the test is that \(\mathrm { H } _ { 0 }\) is rejected, state the type of error that is made. On a particular day this test is carried out independently a total of 20 times, and for 4 of these tests the outcome is that \(\mathrm { H } _ { 0 }\) is rejected. It is known that the value of \(\mu\) remains the same throughout these 20 tests.
3. Find the probability that \(\mathrm { H } _ { 0 }\) is rejected at least 4 times if \(\mu = 30\). Hence state whether you think that \(\mu = 30\), giving a reason.
4. Given that the probability of making an error of the type different from that stated in part (ii) is 0.4 , calculate the actual value of \(\mu\), giving your answer correct to 4 significant figures. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

7 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 8 ^ { 2 } \right)\). The mean of a random sample of 12 observations of \(X\) is denoted by \(\bar { X }\). A test is carried out at the \(1 \%\) significance level of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 80\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu < 80\). The test is summarised as follows: 'Reject \(\mathrm { H } _ { 0 }\) if \(\bar { X } < c\); otherwise do not reject \(\mathrm { H } _ { 0 } { } ^ { \prime }\).

1. Calculate the value of \(c\).
2. Assuming that \(\mu = 80\), state whether the conclusion of the test is correct, results in a Type I error, or results in a Type II error if:
   1. \(\bar { X } = 74.0\),
   2. \(\bar { X } = 75.0\).
   3. Independent repetitions of the above test, using the value of \(c\) found in part (i), suggest that in fact the probability of rejecting the null hypothesis is 0.06 . Use this information to calculate the value of \(\mu\).