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 $2,300?
(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 $120 and $640?

Answer 1

Answer: (a) From the table, the area to the right of $\( z = 1.335 \)$ is approximately 0.0918.

(b) From the table, the area to the left of $\( z = 0 \)$ (mean) is 0.5.

(c) From the table, we find the areas to the left of $\( z = -1.586 \) and \( z = -0.834 \)$, then subtract the smaller area from the larger area to find the proportion between them.

Step-by-step explanation:

To solve this problem, we'll use the z-score formula to standardize the values and then use the standard normal distribution table (or calculator) to find the proportions.

Given data:

- Average refund (mean), $\( \mu = \ 1,332 \)$

- Standard deviation, $\( \sigma = \ 725 \)$

We'll calculate the z-scores for each value using the formula:

$\[ z = \frac{x - \mu}{\sigma} \]$

Where:

- \( x \) is the value

- \( \mu \) is the mean

- \( \sigma \) is the standard deviation

(a) To find the proportion of tax returns with a refund greater than $2,300:

$\[ z = \frac{2300 - 1332}{725} \]$

$\[ z = \frac{968}{725} \approx 1.335 \]$

Using the standard normal distribution table, we find the area to the right of \( z = 1.335 \), which represents the proportion of returns with a refund greater than $2,300.

(b) To find the proportion of tax returns where the taxpayer owes money:

Since the average refund is $1,332, any amount below this would mean the taxpayer owes money.

To find the proportion, we can find the area to the left of $\( z = 0 \)$ (since it represents the mean).

(c) To find the proportion of tax returns with a refund between $120 and $640:

We need to find the z-scores for $120 and $640, then find the area between these two z-scores using the standard normal distribution table.

Let's calculate each:

(a) $\( z = 1.335 \)$

(b) $\( z = \frac{0 - 1332}{725} \approx -1.838 \)$

(c) $\( z_{120} = \frac{120 - 1332}{725} \approx -1.586 \) and \( z_{640} = \frac{640 - 1332}{725} \approx -0.834 \)$

Now, we'll use a standard normal distribution table to find the proportions:

(a) From the table, the area to the right of $\( z = 1.335 \)$ is approximately 0.0918.

(b) From the table, the area to the left of $\( z = 0 \)$ (mean) is 0.5.

(c) From the table, we find the areas to the left of $\( z = -1.586 \) and \( z = -0.834 \)$, then subtract the smaller area from the larger area to find the proportion between them.

These calculations will give us the answers to the respective proportions.