Question

Please use excel

b. A rookie is brought to a baseball club on the assumption that he will have a .300 batting average. (Batting average is the ratio of the number of hits to the number of times at bat.) In the first year, he comes to bat 300 times and his batting average is .267. Could such a low average be considered just bad luck or should he be sent back to the minor leagues? Assume that his at bats can be considered a binomial distribution with probability 0.3 for success. Explain your answer

2a. If we assume that each of the digits 0-9 has an equal chance of appearing, i.e. they are uniformly distributed, what is the probability that a tax return with 30,000 total numbers (after doing our counts) that the digit 3 appears no more than 2,900 times. (Note: The IRS actually does this but with more sophisticated methods. The actual distribution of, so-called, financial numbers is not uniformly distributed but instead follows Benford’s Law if you want to read more about it)

Hint: for the first question, treat this as a Binomial Distribution with the probability of getting a 3 equal to 1/10 and then use the Central Limit Theorem.

b.If the cutoff for an audit is having a probability of less than 0.05, do we audit this particular tax return?

Answer 1

Solution:

We know the batting average=0.267 and n=300 times. So number of hits=0.267*300=80

Let X be the number of times Rookie hits.

So, X has a Binomial distribution with p=0.3 and n=300

The probability to get a batting average of 0.267 or less is then the probability to have 80 hits or less in 300 times at bats.

P(X<=80)=B300,0.3(80)

Since n=300, which is too large to find its probability in the table, we use normal approximaion.

B300,0.3(80)N300,0.3(80)N(300*0.3),(300*0.3*0.7)N90,63(80)

Therefore, P(X<=80)=N90,63(80)=N0,1(80-90)/63

=N0,1(-1.26)

=1-N0,1(1.26)

=1-0.9131(from the normal distribution table)

=0.0869=0.09

So the probability is 9% to show the batting average of 0.267.