Question

(Tax fraud) Each year, the Internal Revenue Service attempts to estimate total tax compliance: whether Federal tax revenues are above or below what they should be, and by how much. For many years, this was performed by the IRS Tax Compliance Measurement Program (TCMP). To conduct the TCMP, the IRS drew a simple random sample of U.S. citizens and audited each one. In a TCMP audit, the true tax liability of the individual (or household, if the individual files Federal income taxes jointly with a spouse) is assessed by the IRS after detailed examination of the individuals (or households) total income by IRS agents. The tax liability determined in the audit is then compared to the actual tax return filed by the individual or household. The IRS extrapolates from the average tax underpayment in the TCMP sample to estimate the total personal income tax underpayment level in the U.S. Until recently, TCMP audits were performed on approximately 150 thousand randomly selected taxpayers annually.

A Congressional representative asserts that tax fraud is on the rise in the U.S. To examine the issue, the IRS provides the following (hypothetical) sample data on tax underpayment amounts from the TCMP survey for two consecutive years.

	Year 1	Year 2
Sample Avg	$423.27	$426.84
Sample SD	$359.21	$361.14
Sample Size	151,627	153,249

Some taxpayers underpay their taxes (a positive number), while others overpay their taxes (a negative number); the average of these numbers for all taxpayers in the survey is listed in the first row of the table and the SD of these numbers is listed in the second row. Assume the numbers have been adjusted for inflation and that the population variance is the same in both years.

(a) Do the data indicate that tax fraud is on the rise?

(b) Why is such a large sample used in the TCMP?

Answer 1

a) The null and alternative hypotheses are:

Since sample size is very large, we can use standard normal distribution.

The test statistic is:

For left tailed test, the p-value is:

Since p-value is less than 0.05, reject null hypothesis. We can conclude that the data indicate that tax fraud is on the rise.

------------------------------------------------------------------------------------------------------

b) Such a large sample is used in the TCMP because according to the law of averages, the TCMP can get a closer estimate to the true population mean difference.