9231 P21 - Nov 2019 - Q11O - 14 marks
Question 11 OR alternative.
The number of puncture repairs carried out each week by a small repair shop is recorded over a period of 40 weeks. The results are shown in the following table.
| Number of repairs in a week | 0 | 1 | 2 | 3 | 4 | 5 | \(\geqslant 6\) |
|---|---|---|---|---|---|---|---|
| Number of weeks | 6 | 15 | 9 | 6 | 3 | 1 | 0 |
(i) Calculate the mean and variance for the number of repairs in a week and comment on the possible suitability of a Poisson distribution to model the data.
Records over a longer period of time indicate that the mean number of repairs in a week is 1.6. The following table shows some of the expected frequencies, correct to 3 decimal places, for a period of 40 weeks using a Poisson distribution with mean 1.6.
| Number of repairs in a week | 0 | 1 | 2 | 3 | 4 | 5 | \(\geqslant 6\) |
|---|---|---|---|---|---|---|---|
| Expected frequency | 8.076 | 12.921 | 10.337 | 5.513 | 2.205 | \(a\) | \(b\) |
(ii) Show that \(a=0.706\) and find the value of the constant \(b\).
(iii) Carry out a goodness of fit test of a Poisson distribution with mean 1.6, using a 10% significance level.