Question

I have an example problem: please use the answer to answer the next problem! Thanks. Use these answers as an example to solve the following problem. The Internal Revenue Service reports that the mean federal income tax paid in the year 2010 was $8040. Assume that the standard deviation is $4700.

The IRS plans to draw a sample of 1000 tax returns to study the effect of a new tax law.
(a) What is the probability that the sample mean tax is less than $8000? = 0.3939.
(b) What is the probability that the sample mean tax is between $7600 and $8100? = 0.2706
(c) Find the 30th percentile of the sample mean. = 7962.06.
(d) Would it be unusual if the sample mean were less than $7600?
It is unusual because the probability of the sample mean being less than $7600 is 0.0015.
(e) Do you think it would be unusual for an individual to pay a tax of less than $7600? Explain. Assume the variable is normally distributed.
No, because the probability that an individual to pays a tax less than $7600 is 0.4629.
THE PROBLEM: 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.
(a) What is the probability that the sample mean tax is less than $8000? Round the answer to at least four decimal places.
(b) What is the probability that the sample mean tax is between $7400 and $8000? Round the answer to at least four decimal places.
(c) Find the 60th percentile of the sample mean. Round the answer to at least two decimal places.
(d) Would it be unusual if the sample mean were less than $7600? Round the answer to at least four decimal places.
(e) Would it be unusual for an individual to pay a tax of less than $7600? Explain. Yes or No, Assume the variable is normally distributed. Round the answer to at least four decimal places.

Answer 1

Answer&explanation:

(a) To calculate the probability that the sample mean tax is less than $8000, we need to use the z-score formula. The z-score is calculated by subtracting the sample mean from the population mean and then dividing it by the standard deviation divided by the square root of the sample size.

In this case, the population mean is $8040, the standard deviation is $4500, and the sample size is 1000.

By plugging these values into the formula, we can calculate the z-score. Then, we can use a standard normal distribution table or a calculator to find the probability associated with that z-score.

(b) To calculate the probability that the sample mean tax is between $7400 and $8000, we use the same method as in part (a). However, this time we need to find the z-scores for both values and subtract the probability associated with the lower z-score from the probability associated with the higher z-score.

(c) To find the 60th percentile of the sample mean, we need to find the z-score that corresponds to the cumulative probability of 0.60. We can use a standard normal distribution table or a calculator to find the z-score and then calculate the sample mean using the z-score formula.

(d) To determine if it would be unusual for the sample mean to be less than $7600, we need to calculate the probability of the sample mean being less than $7600 using the z-score formula. If the probability is below a certain threshold (e.g., 0.05 or 5%), we can consider it to be unusual.

(e) To determine if it would be unusual for an individual to pay a tax of less than $7600, we need to calculate the probability of an individual paying a tax of less than $7600 using the z-score formula. If the probability is below a certain threshold (e.g., 0.05 or 5%), we can consider it to be unusual.

Please note that the calculations require the correct values for the population mean and standard deviation. In the given problem, the population mean is $8040, and the standard deviation is $4500. Make sure to use these values in your calculations.