5. Paul takes the company bus to work. According to the bus timetable he should arrive at work at 0831. Paul believes the bus is not reliable and often arrives late. Paul decides to test the arrival time of the bus and carries out a survey. He records the values of the random variable $$X = \text { number of minutes after } 0831 \text { when the bus arrives. }$$ His results are summarised below. $$n = 15 \quad \sum x = 60 \quad \sum x ^ { 2 } = 1946$$

1. Calculate unbiased estimates of the mean, \(\mu\), and the variance of \(X\). Using the mean of Paul's sample and given \(X \sim \mathrm {~N} \left( \mu , 10 ^ { 2 } \right)\)
2. 1. calculate a 95\% confidence interval for the mean arrival time at work for this company bus.
   2. State an assumption you made about the values in the sample obtained by Paul.
3. Comment on Paul's belief. Justify your answer.