S2 Hypothesis Tests AssignmentDUE: Monday 28th February

1. During busy periods at a call centre, callers either get through to an operator immediately or are put on hold. A large survey revealed that 80% of callers were put on hold.

(i) Write down the probability that a caller gets through immediately. [1]

(ii) For a random sample of 25 callers, find the probability that

(A) exactly 6 callers get through immediately,

(B) at least 2 callers get through immediately. [6]

The call centre increases the number of operators with the intention of reducing the proportion of callers who are put on hold. After the change, exactly half of a random sample of 20 callers get through immediately.

(iii) Carry out a suitable hypothesis test to examine whether the centre has been successful. Use a 5% significance level and state your hypotheses and conclusions carefully. Determine the critical region for the test. [8]

2. A roller coaster ride has a safety system to detect faults.

(i) State conditions for a Poisson distribution to be a suitable model for the number of faults occurring on a randomly selected day. [2]

Faults are detected at an average rate of 0.15 per day. You may assume that a Poisson distribution is a suitable model.

(ii) Find the probability that on a randomly chosen day there are

(A) no faults

(B) at least 2 faults. [4]

(iii) Find the probability that, in a randomly chosen period of 30 days, there are at most 3 faults.[3]

The ride is given a complete overhaul, which the operator hopes will result in a reduction in the number of faults detected.

(iv) Over a period of 30 days following the overhaul, just one fault is detected. Test, at the 5% level, whether there has been a reduction in the mean number of faults in a 30 day period.

[6]

3. Joggers produce packets of crisps. On average, 1 in every 5 packets, chosen randomly, contains a prize voucher.

A box contains 30 packets of Joggers crisps.

(i) State the expected number of packets containing a prize voucher and find the probability of exactly this number occurring. [4]

(ii) Show that it is almost certain that at least one packet will contain a voucher.[2]

Sprinters also produce packets of crisps, some of which contain a prize voucher. Jean wishes to test

whether the proportion of packets of Sprinters crisps with prize vouchers is also .

(iii) State suitable null and alternative hypotheses for the test. [2]

Jean buys 12 packets of Sprinters crisps and finds no vouchers at all.

(iv) Carry out the hypothesis test at the 5% significance level, giving the critical region for the test and stating your conclusions carefully. [5]

(v) How many packets of crisps would Jean have to buy for the critical region to have a non-empty lower tail? [2]

Total 45 marks