5 The flight time between two airports is known to be Normally distributed with mean 3.75 hours and standard deviation 0.21 hours. A new airline starts flying the same route. The flight times for a random sample of 12 flights with the new airline are shown in the spreadsheet (Fig. 5), together with the sample mean.
\begin{table}[h]
| A | B | C | D | E | F | G | H | I | J | K | L |
| 1 | 3.595 | 3.723 | 3.584 | 3.643 | 3.669 | 3.697 | 3.550 | 3.674 | 3.924 | 3.563 | 3.330 | 3.706 |
| 2 | | | | | | | | | | | | |
| 3 | Mean | 3.638 | | | | | | | | | | |
| | | | | | | | | | | | |
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{table}
\section*{(i) In this question you must show detailed reasoning.}
You should assume that:
- the flight times for the new airline are Normally distributed,
- the standard deviation of the flight times is still 0.21 hours.
Carry out a test at the \(5 \%\) significance level to investigate whether the mean flight time for the new airline is less than 3.75 hours.
(ii) If both of the assumptions in part (i) were false, name an alternative test that you could carry out to investigate average flight times, stating any assumption necessary for this test.
(iii) If instead the flight times were still Normally distributed but the standard deviation was not known to be 0.21 hours, name another test that you could carry out.