Chai packs china mugs into cardboard boxes. Chai's manager suspects that breakages occur at random times and that the number of breakages may follow a Poisson distribution. He takes a small sample of observations and finds that the number of breakages in a one-hour period has a mean of 2.4 and a standard deviation of 1.5.
(a) Explain how this information tends to support the manager's suspicion.

The manager now takes a larger sample and claims that the numbers of breakages in a one-hour period follow a Poisson distribution. The numbers of breakages in a random sample of 180 one-hour periods are summarised in the following table.

The mean number of breakages calculated from this sample is 2.5 .
(b) Use the data from this larger sample to carry out a goodness of fit test, at the \(10 \%\) significance level, to test the claim.