Statistics Guide: Internal Revenue Service is Studying Charitable Contributions

This tutorial requires Microsoft EXCEL for the computation of various statistical value. In order to follow along, please make sure you have a spreadsheet ready to go.

Student Question: The Internal Revenue Service is studying the category of charitable contributions. A sample of 34 returns is selected from young couples between the ages of 20 and 35 who had an adjusted gross income of more than $100,000. Of these 34 returns, 8 had charitable contributions of more than $1,000. Suppose 7 of these returns are selected for a comprehensive audit.

You should use the hypergeometric distribution because?

You are sampling a small population without replacement

A sample size of 34 is large enough to follow the central limit theorem, but it is still considered a small sample for the purposes of a hypergeometric distribution.

Now that we know that. Let’s get to work entering the values into excel. There are three values to deal with in this problem as you can see in the image below.

excel values for irs problem

What is the probability exactly one of the seven audited had a charitable deduction of more than $1,000?

In order to solve this. We have fairly advanced statistical concept that deals with the total possible combinations in a data set. The statistical symbol nCr represents number of combinations of r objects that can come from n different objects. In excel, we use the COMBIN function to work with statistical combinations. The following formula is used to find the probability of exactly one in seven.

=COMBIN(C5,1)*COMBIN((C5*2),(C6-1))/COMBIN((C5*3),C6)

You should get a resulting probability of 0.1851 or 18.51%.

What is the probability at least one of the audited returns had a charitable contribution of more than $1,000?

In this case, we subtract the combination formula from 1 to obtain the probability

=1-COMBIN(C5,0)*COMBIN((8*2),C6)/COMBIN((C5*3),C6)

You should get a resulting probability of 0.9669 or 96.69%. 

 

  • Student: Brandy Golden
  • Textbook: Business Statistics
  • Course: QNT 351
 
 

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