12   Question 4 continued In Shai’s game, a bundle of gems costs £5. The game also offers unlimited gems for any player spending £60 a month. (i) Discuss the appropriateness or otherwise of the normal distribution to model the spend of a whale in light of this information. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 17 marks) Turn over 13   5 Tom owns a busy cinema with two screens. Screen 1 has 120 seats, and Screen 2 has 300 seats. He shows a different film on each screen. Children and pensioners are eligible for a concession ticket that is cheaper than a standard ticket. From past experience Tom estimates that the proportion of concession tickets sold for the most popular 7 pm showing for the film on Screen 1 is 35% and the proportion for the film on Screen 2 is 15%. Both screens show films at 7 pm one night. The random variable X denotes the number of concession tickets purchased for the 7 pm showing of the film on Screen 1. The random variable Y denotes the number of concession tickets purchased for the 7 pm showing of the film on Screen 2. The random variable T denotes the total number of concessions tickets purchased for both showings of films at 7 pm that night. Tom assumes that all the tickets will be purchased, and that the probability of a concession ticket being purchased is independent from one customer to the next. (a) By modelling each of X and Y as an independent binomial distribution, (i) show that the distribution of T can be approximated as N(87, 65.55) (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (ii) estimate the probability that at least 100 concessions tickets are purchased in total. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________