On a weekday, a garage receives telephone calls randomly, at a mean rate of 1.25 per 10 minutes.
Show that the probability that on a weekday at least 2 calls are received by the garage in a 30 -minute period is 0.888 to 3 decimal places.
Calculate the probability that at least 2 calls are received by the garage in fewer than 4 out of 6 randomly selected, non-overlapping 30-minute periods on a weekday.
The manager of the garage randomly selects 150 non-overlapping 30-minute periods on weekdays.
She records the number of calls received in each of these 30-minute periods.
Using a Poisson approximation show that the probability of the manager finding at least 3 of these 30 -minute periods when exactly 8 calls are received by the garage is 0.664 to 3 significant figures.
Explain why the Poisson approximation may be reasonable in this case.
The manager of the garage decides to test whether the number of calls received on a Saturday is different from the number of calls received on a weekday. She selects a Saturday at random and records the number of telephone calls received by the garage in the first 4 hours.
Write down the hypotheses for this test.
The manager found that there had been 40 telephone calls received by the garage in the first 4 hours.
Carry out the test using a \(5 \%\) level of significance.