Problem #6 On average, John Jay, the number one baseball player in Major League

Question

Problem #6 On average, John Jay, the number one baseball player in Major League Baseball (Toledo Mud Hens), has .71 runs batted in (rbi) per game with a standard deviation of .2 runs per game. A baseball season consists of 162 games. The number of runs batted in for John Jay in any game is independent from the number in any other game. Let X = the total number of runs batted in for John Jay in the season. X is an integer. a) What is the expected value of X. b) What is the standard deviation of X. c) What theorem (name please) allows us to treat X as a normal random variable? d) What is the approximate probability that John Jay has more than 120 rbi in the season? (Since you are using a normal distribution to approximate a discrete distribution you should use a correction factor like the text used for the normal approximation to the binomial)

Explanation / Answer

here rbi per game =p=0.71, number of games=n=162, standard deviation=sd=0.2

answer a) X=total number of runs=162*0.71=115.02

answer b) standard deviation of X=n*sd=162*0.2=32.4

answer c) n tends to infinity and p tends to 0 while np remains constant, the binomial distribution tends normal distribution. this is due to central limit theorem and it statement is

" the central limit theorem states that the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution."

d) we use standard normal variate z=(x-mean(X))/sd(X)

E(X)=mean(X)=115.02

standard deviation of X=sd(X)=n*sd=162*0.2=32.4

for x=120, z=(120-115.02)/32.4=0.1537

P(x>120)=P(z>0.1537)=1-P(z<0.1537)=1-0.5611=0.4389

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