Problem 4 According to the Internal Revenue Service, 75% of all tax returns lead

Question

Problem 4 According to the Internal Revenue Service, 75% of all tax returns lead to a refund. An auditor decided to examine a random sample of 100 tax returns.

• Describe the sampling distribution of the sample proportion of tax returns that lead to a refund in any sample of 100. In the specific, answer the following questions: (i) Is the distribution Normal? If so, why? (ii) What is the mean of this distribution? What is the standard deviation of this distribution?

• Suppose the auditor claims that in the sample of 100 tax returns 80 led to a refund: was he very unlucky?

• What is the probability that there are less than 70 tax returns leading to a refund in a random sample of 100?

• What is the probability that, in a sample of 100 tax returns, the refund proportion will be within 5% of the reported national proportion of 75%?

Explanation / Answer

here p=0.75

(i)( Yes this is approximately normal cause np(1-p)>10

hence mean =np=100*0.75=75

std deviation =(np(1-p))1/2 =4.3301

a)P(X>=80) = 1-P(X<=79) =1-P(Z<(79.5-75)/4.3301)= 1-P(Z<1.0392)=1-0.8507 =0.1493

as the probabilty of 80 or above was not less then 0.05 ; it is not unlucky

b) P(X<=70) =P(Z<-1.0392)=0.1493

c) P(70<=X<=80)=0.8507-0.1493 =0.7013

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