Problem

The manager of a supermarket records the number of complaints, $x$, the supermarket receives each day over a 50-day period. The results are summarised below.

$\sum x = 99$		and		$\sum x^2 = 975$

The manager believes that the number of complaints received by the supermarket each day could be modelled by a Poisson distribution.

(a) Show how the results above support the manager in his belief.

The manager uses a Poisson distribution with mean 2 to model the number of complaints received each day.

(b) For a randomly selected day, find using the manager’s model, the probability that there are

(i) at least 4 complaints received,

(ii) more than 2 but less than 7 complaints received.

A working week consists of 5 consecutive days.

In a randomly selected working week, the supermarket received 17 complaints.

(c) Test, at the 5% level of significance, whether there is significant evidence that the mean number of complaints received is greater than 2 per day.

State your hypotheses clearly.

(Total for Question 2 is 12 marks)