Confidence interval for Poisson mean

1 The number of telephone calls per hour to an out-of-hours doctors' service may be modelled by a Poisson distribution. The total number of telephone calls received during a random sample of 12 weekday night shifts, all of the same duration, was 392.

1. Calculate an approximate \(98 \%\) confidence interval for the mean number of calls received per weekday night shift.
2. The mean number of calls received during weekend shifts of 48 hours' total duration is 136.8 . Comment on a claim that the mean number of calls per hour during weekend shifts is greater than that during weekday night shifts, which are each of \(\mathbf { 1 4 }\) hours' duration.
   (3 marks)

5 The number of letters per week received at home by Rosa may be modelled by a Poisson distribution with parameter 12.25.

1. Using a normal approximation, estimate the probability that, during a 4 -week period, Rosa receives at home at least 42 letters but at most 54 letters.
2. Rosa also receives letters at work. During a 16-week period, she receives at work a total of 248 letters.
   1. Assuming that the number of letters received at work by Rosa may also be modelled by a Poisson distribution, calculate a \(98 \%\) confidence interval for the average number of letters per week received at work by Rosa.
   2. Hence comment on Rosa's belief that she receives, on average, fewer letters at home than at work.