Question 4. Paul is my hairdresser and each time I get a haircut we talk about...

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Question 4. Paul is my hairdresser and each time I get a haircut we talk about statistics. (This is absolutely true and also totally unbelievable, I know.) Last time I estimated his income from 1 tips (before taxes) and he was in awe of how close to the actual number my estimate was. The following numbers are made up and not the ones that Paul told me: On average Paul gives ten haircuts a day, six days a week, 36 weeks a year. As a simplification, suppose that the tips (in USD) are random and follow Bin(20,0.6). (a) Use the weak law of large numbers to approximate Paul's income from tips (before taxes) in one year. (b) Use the central limit theorem to approximate the probability that Paul's income from tips (before taxes) in one year is less than USD 25,000. (c) Use R to verify your calculations from parts (a) and (b) using a simulation, i.e. simulate 1000 yearly incomes from tips (before taxes): plot the simulated incomes in a histogram; compute the mean of the simulated incomes, add a vertical line with the mean to the histogram; compute the percentage of simulated incomes that are less than 25,000. Question 4. Paul is my hairdresser and each time I get a haircut we talk about statistics. (This is absolutely true and also totally unbelievable, I know.) Last time I estimated his income from 1 tips (before taxes) and he was in awe of how close to the actual number my estimate was. The following numbers are made up and not the ones that Paul told me: On average Paul gives ten haircuts a day, six days a week, 36 weeks a year. As a simplification, suppose that the tips (in USD) are random and follow Bin(20,0.6). (a) Use the weak law of large numbers to approximate Paul's income from tips (before taxes) in one year. (b) Use the central limit theorem to approximate the probability that Paul's income from tips (before taxes) in one year is less than USD 25,000. (c) Use R to verify your calculations from parts (a) and (b) using a simulation, i.e. simulate 1000 yearly incomes from tips (before taxes): plot the simulated incomes in a histogram; compute the mean of the simulated incomes, add a vertical line with the mean to the histogram; compute the percentage of simulated incomes that are less than 25,000.