An engineer is monitoring the rapid charging times for a fleet of electric delivery vans. She proposes that the time, TTT minutes, taken to reach 80% charge can be modelled by a normal distribution with mean 40 minutes and standard deviation 6 minutes.
A random sample of 150 charging sessions was monitored, and the recorded times are summarised in the table below.
Time, t (min)t<3131≤t<3737≤t<4343≤t<49t≥49Frequency1233603411 \begin{array}{|l|c|c|c|c|c|} \hline \text{Time, } t \text{ (min)} & t < 31 & 31 \le t < 37 & 37 \le t < 43 & 43 \le t < 49 & t \ge 49 \\ \hline \text{Frequency} & 12 & 33 & 60 & 34 & 11 \\ \hline \end{array} Time, t (min)Frequencyt<311231≤t<373337≤t<436043≤t<4934t≥4911Stating your hypotheses clearly and using a 5% level of significance, test the engineer's proposed model. Show your working clearly and state the expected frequencies, the test statistic, and the critical value used.
The engineer's assistant calculated summary statistics for the 150 sessions:
∑t=6030and∑t2=247850 \sum t = 6030 \quad \text{and} \quad \sum t^2 = 247850 ∑t=6030and∑t2=247850Calculate unbiased estimates of the mean and variance of the charging times.
The assistant used the calculations from part (b) to conduct a χ2\chi^2χ2 goodness of fit test to see if a normal distribution is a suitable model for the times. His calculated test statistic was 3.82 (to 3 significant figures) and he did not pool any classes.
Using a 5% level of significance, complete the assistant's test, stating the critical value and the degrees of freedom used.
Estimate, to 2 significant figures, the proportion of charging sessions that take longer than 52 minutes.
