Question

Income Tax and the IRS. In $2010$, the Internal Revenue Service (IRS) sampled $308,946$ tax returns to obtain estimates of various parameters. Data were published in Statistics of Income, Individual Income Tax Returns. According to that document, the mean income tax per return for the returns sampled was $\backslash (11,266$.

a. Explain the meaning of sampling error in this context.

b. If. in reality, the population mean income tax per return in 2010 was $\backslash )11,354$, how much sampling error was made in estimating that parameter by the sample means of $\backslash (11,266$ ?

c. If the IRS had sampled $400,000$ returns instead of $\backslash )308.946\$$, would the sampling error necessarily have been smaller? Explain your answer.

d. In future surveys, how can the IRS increase the likelihood of small simpling errors?

Short answer

Part (a) The sample mean income $\left(\overline{x}\right)$of $308.946$tax returns to estimate the population mean income $\left(\mu \right)$of all taxes in $2010$

Part (b) The sampling error is $-\$88$

Part (c) The sampling error does not need to be less.

Part (d) To raise the probability of modest sampling error, the Internal Revenue Service (IRS) would increase the sample size in this study.

Step-by-step solution

Part (a) Step 1: Given information

The mean income tax per return for the returns sampled was $\$11,266$

Part (a) Step 2: Explanation

The sampling error is the result of estimating the population mean income $\left(\mu \right)$of all taxes in $2010$using the sample mean income$308.946$ of tax returns.

Part (b) Step 1: Explanation

It is assumed that the population's average income tax per return in 2010 was $11,354$. In addition, the sample mean's calculated parameter is $$ 11,266$. As a result, the sampling error is

$\$11,266-\$11,354=-\$88$

Thus, the sampling error is $-\$88$

Part (c) Step 1: Explanation

The possibility of a lesser sampling error increases as the sample size is raised. To put it another way, the sampling error does not have to be less.

Part (d) Step 1: Explanation

The Internal Revenue Service (IRS) would enhance the sample size in this study to increase the possibility of small sampling error.