Questions: Taxes: The Internal Revenue Service reports that the mean federal income tax paid in the year 2010 was 8040. Assume that the standard deviation is 4500. The IRS plans to draw a sample of 1000 tax returns to study the effect of a new tax law. Do you think it would be unusual for an individual to pay a tax of less than 7800? Explain. Assume the variable is normally distributed. Round the answer to at least four decimal places.

Solution Steps

To determine if it is unusual for an individual to pay a tax of less than \$7800, we first calculate the Z-score using the formula:

\[ z = \frac{X - \mu}{\sigma} \]

where:

• \(X = 7800\)
• \(\mu = 8040\)
• \(\sigma = 4500\)

Substituting the values, we have:

\[ z = \frac{7800 - 8040}{4500} = \frac{-240}{4500} \approx -0.0533 \]

Thus, the Z-score for an individual paying a tax of \$7800 is approximately \(-0.0533\).

Next, we calculate the probability that an individual pays less than \$7800. This is given by:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-0.0533) - \Phi(-\infty) \]

Using the cumulative distribution function (CDF) for the standard normal distribution, we find:

\[ P \approx 0.4787 \]

This means that the probability that an individual pays less than \$7800 is approximately \(0.4787\).

To determine if it is unusual for an individual to pay less than \$7800, we compare the calculated probability to a common threshold of \(0.05\):

Since \(0.4787 > 0.05\), it is not unusual for an individual to pay less than \$7800.

Final Answer