Question

The Internal Revenue Service (IRS) reports that the mean federal income tax paid in a recent year was $8,040. Assume the standard deviation is $5,000. The IRS plans to draw a sample of 1000 tax returns.

a. What is the probability that the sample mean tax return is less than $8,000?

b. What is the probability that the sample mean tax return is between $7,600 and $7,900?

c. Would it be unusual if the sample mean were less than $7,500? (Recall unusual is the top and bottom 5%)

Answer 1

Given that mean =8040 and standard deviation=5000 and a sample of 1000 is taken hence considered as normal distribution

a) Now , is calculated using Z score and by Z table as

so,

=P(Z>0.25) is calculated using Z table shown below

=0.4013

b) is calculated using Z scores at respective means as

Z at mean =7600

Z at mean= 7900

So,

=0.1867-0.0027

=0.184

c) Z at mean =7500

Since the Z score at Mean =7500 is -3.42 which does not lie in the middle 90% usual region hence it is usual that the mean is less than 7500.