The Internal Revenue Service reports that the mean federal income tax paid in 20

Question

The Internal Revenue Service reports that the mean federal income tax paid in 2007 was $7908 with a standard deviation of $5000.

a. Find the probability that if an individual tax payer is randomly selected, the amount they paid in taxes in 2007 is less than $8000. Round your probability to three significant figures.

b. In this situation, could the Central Limit Theorem be applied to a sample of 1000 taxpayers? Justify your answer.

c. Find the probability that 1000 randomly selected taxpayers will have a mean that is less than $8000. Round your probability to three significant figures.

d. Suppose you want the standard deviation of the sample mean to be 1 dollar or less. How large of a sample is required?

Explanation / Answer

a)P(X<8000)=P(Z<(8000-7908)/5000)=P(Z<0.02)=0.508 (please try 0.507 if this comes wrong)

b)Yes as for Central Limit Theorem to be applicable ; sample size should be greater than or equal to 30,

c)here for n=1000 ; std error of mean =std deviation/sqrt(n)=5000/sqrt(1000)=158.114

hence P(Xbar<8000)=P(Z<(8000-7908)/158.114)=P(Z<0.58)=0.719 ( please try 0.720 if this comes wrong)

d)

for sample size =(std deviation/std error)2 =(5000/1)2 =25000000

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