4 2. On a weekday, a garage receives telephone calls randomly, at a mean rate of 1.25 per 10 minutes. (a) 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. (2) (b) 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. (2) 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. (c) 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. (4) (d) Explain why the Poisson approximation may be reasonable in this case. (1) 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. (e) Write down the hypotheses for this test. (1) The manager found that there had been 40 telephone calls received by the garage in the first 4 hours. (f) Carry out the test using a 5% level of significance. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________