1. Alan's journey time to work can be modelled by a normal distribution with standard deviation 6 minutes. Alan measures the journey time to work for a random sample of 5 journeys. The mean of the 5 journey times is 36 minutes.

1. Construct a \(95 \%\) confidence interval for Alan's mean journey time to work, giving your values to one decimal place.
2. Alan claims that his mean journey time to work is 30 minutes. State, with a reason, whether or not the confidence interval found in part (a) supports Alan's claim.
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   \includegraphics[max width=\textwidth, alt={}, center]{db093bfd-a08d-4554-ba5c-5204b6045d0e-2_344_1657_1025_246} \section*{2.} Indre works on reception in an office and deals with all the telephone calls that arrive. Calls arrive randomly and, in a 4-hour morning shift, there are on average 80 calls.
3. Using a suitable model, find the probability of more than 4 calls arriving in a particular 20-minute period one morning. Indre is allowed 20 minutes of break time during each 4-hour morning shift, which she can take in 5-minute periods. When she takes a break, a machine records details of any call in the office that Indre has missed. One morning Indre took her break time in 4 periods of 5 minutes each.
4. Find the probability that in exactly 3 of these periods there were no calls. On another occasion Indre took 1 break of 5 minutes and 1 break of 15 minutes.
5. Find the probability that Indre missed exactly 1 call in each of these 2 breaks.