The coach of the Red Shirts, a Multi-dimensional League Baseball team, is recruiting new players. To do so, she is conducting a home run competition. Each batter gets 100 tries to hit an identically pitched ball over a fence that is 310 yards away; a ball hit over the fence is considered sufficient for making a home run. The coach would like to recruit the top 1% of all batters.

a) A typical batter will hit 60% of the balls; of the balls they hit, suppose that the distance

traveled is normally distributed with mean of 300 yards and standard deviation 50 yards. How many home runs should the coach set as a minimum limit for being recruited to the Red Shirts?

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b) J.D. Martian-ez hits the ball 80% of the time. When he hits the ball, the distance it travels

is normally distributed with mean 295 yards and standard deviation of 60 yards. Based on

the minimum limit set in part i., what is the probability that J.D. will make the team?

c) The coach plans to host tryouts every week, allowing players to participate at most 10 times.

J.D. plans on trying out until he makes the team. What is the expected number of times J.D.

will try out for the team?

d) If the coach allows players to try out more than once, will she actually be identifying the top

1% of all batters? Explain your answer.

Hint: For this problem, the function qbinom() may be useful. This function is used to identify the observation that corresponds to a particular probability p and has the generic structure qbinom(p, size, prob, lower.tail = TRUE)

where p is p, size is the number of trials n, and prob is the probability of success p. By default, R identifies the observation that corresponds to an area of at least p in the lower tail (i.e., to the left). To identify the observation with area p in the upper tail, specify lower.tail = FALSE.