Eric runs a YouTube channel, DwarfPunk, where he gives tutorials on painting wargaming miniatures. Recently he has created a Patreon page where people have the option to make a monthly donation to support him. The Patreon page has 3 tiers: patrons in the "Hobbit" tier pay $1/month, patrons in the "Dwarf" tier pay $5/month and patrons in the "Elf" tier pay $10/month. Based on other data for similar content providers it has been found that: 80% of teenage patrons will opt for the Hobbit tier, 15% will opt for the Dwarf tier and 5% will o for the Elf tier. 50% of working age patrons will opt for the Hobbit tier, 30% will opt for the Dwarf tier and 20% will opt for the EIf tier. 30% of retired patrons will opt for the Hobbit tier, 60% will opt for the Dwarf tier, and 10% will opt for the Elf tier. 1) Of Eric's subscribers 50% are teenagers, 20% are working age and 30% are retirees. a) What proportion of Eric's patrons are expected to fall into each tier? [I mark) b) What is the expected value of each patron? [I mark] c) Hannah is in the Elf tier, what is the probability that she is a retiree? [3 marks] 2) Eric's YouTube channel has 40,000 subscribers, each subscriber has a probability of 0.02 of becoming a patron. a) What distribution does the number of patrons follow? Give all relevant parameters. [2 marks] b) Explain why it would be appropriate to approximate the distribution above by a Normal distribution? [I mark] c) Assuming it is okay, what parameters would you use for the mean and standard deviation? [1 mark] 3) To cover the costs of his time and materials Eric calculates that he needs to bring in $2400 / month. [Hint: first use the expected value from part Ib to work out how many patrons he needs] a) What is the probability that Eric can cover his costs? [1 mark) b) Draw a picture of a standard normal distribution that indicates the probability you calculated in part (a) and the corresponding z-value, 12 marks]