5 The number of letters per week received at home by Rosa may be modelled by a Poisson distribution with parameter 12.25.\\
(a) 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.\\
(b) Rosa also receives letters at work. During a 16-week period, she receives at work a total of 248 letters.\\
(i) 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.\\
(ii) Hence comment on Rosa's belief that she receives, on average, fewer letters at home than at work.

\hfill \mbox{\textit{AQA S3  Q5}}