Question

According to the Internal Revenue Service, income tax returns one year averaged $1,332 in refunds for taxpayers. One explanation of this figure is that taxpayers would rather have the government keep back too much money during the year than to owe it money at the end of the year. Suppose the average amount of tax at the end of a year is a refund of $1,332, with a standard deviation of $725. Assume that amounts owed or due on tax returns are normally distributed.

(a) What proportion of tax returns show a refund greater than $1,800?
(b) What proportion of the tax returns show that the taxpayer owes money to the government?
(c) What proportion of the tax returns show a refund between $100 and $720?

Answer 1

According to the given information, the average amount of tax at the end of a year is a refund of $1,332, with a standard deviation of $725. Therefore, and and the variable of that amounts owed or due on tax returns are normally distributed , and therefore,

(a) The proportion of tax returns show a refund greater than $1,800 is determined as:

Convert it into a standard normal variate

  ( from the standard normal table)

Therefore the proportion of 25.79%  tax returns show a refund greater than $1,800 .

(b) The proportion of the tax returns show that the taxpayer owes money to the government is calculated as:

  Convert it into a standard normal variate

  ( from the standard normal table)

Therefore the proportion of 3.28% the tax returns show that the taxpayer owes money to the government.
(c) The proportion of the tax returns show a refund between $100 and $720 is calculated as:

  

  Convert it into a standard normal variate

  ( from the standard normal table)

Therefore the proportion of 15.47% of the tax returns show a refund between $100 and $720 .

Standard normal table is given as: