This document provides instructions for a statistics assignment involving hypothesis testing. It includes several hypothesis tests related to call center wait times, fault detection on a roller coaster ride, prize vouchers in packets of crisps, and determining the necessary sample size for a critical region to have a non-empty lower tail.

1. S2 Hypothesis Tests Assignment<br />DUE: Monday 28th February<br />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.<br />Write down the probability that a caller gets through immediately.[1]<br />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]<br />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.<br />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]<br />A roller coaster ride has a safety system to detect faults.<br />State conditions for a Poisson distribution to be a suitable model for the number of faults occurring on a randomly selected day.[2]<br />Faults are detected at an average rate of 0.15 per day. You may assume that a Poisson distribution is a suitable model.<br />Find the probability that on a randomly chosen day there are(A)no faults(B)at least 2 faults.[4]<br />Find the probability that, in a randomly chosen period of 30 days, there are at most 3 faults.[3]<br />The ride is given a complete overhaul, which the operator hopes will result in a reduction in the number of faults detected.<br />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]<br />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.<br />State the expected number of packets containing a prize voucher and find the probability of exactly this number occurring.[4]<br />Show that it is almost certain that at least one packet will contain a voucher.[2]<br />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 .<br />State suitable null and alternative hypotheses for the test.[2]<br />Jean buys 12 packets of Sprinters crisps and finds no vouchers at all.<br />Carry out the hypothesis test at the 5% significance level, giving the critical region for the test and stating your conclusions carefully.[5]<br />How many packets of crisps would Jean have to buy for the critical region to have a non-empty lower tail?[2]<br />Total 45 marks<br />