Taxes The Internal Revenue Service reports that the mean federal

Solution

To find the 70th percentile of the sample mean, we need to use the concept of the Central Limit Theorem (CLT) and the properties of a normal distribution.

The Central Limit Theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually n > 30).

If the population is normal, then the theorem holds true even for samples smaller than 30.

In this case, we know the population mean (μ) is $8040, the population standard deviation (σ) is $5000, and the sample size (n) is 1000.

The mean of the sampling distribution of the sample mean (also known as the expected value of the sample mean) is equal to the population mean (μ).

The standard deviation of all possible sample means, also called the standard error (SE), is equal to the standard deviation of the population (σ) divided by the square root of the sample size (n).

So, we can calculate the standard error (SE) as follows:

SE = σ / sqrt(n)
SE = 5000 / sqrt(1000)

Next, we need to find the z-score that corresponds to the 70th percentile. The z-score is a measure of how many standard deviations an element is from the mean.

We can look up this value in a standard normal distribution table, or use a calculator or statistical software to find that the z-score that corresponds to the 70th percentile is approximately 0.52.

Finally, we can find the 70th percentile of the sample mean by using the following formula:

X = μ + Z(SE)

Where:

• X is the 70th percentile of the sample mean
• μ is the population mean
• Z is the z-score that corresponds to the 70th percentile
• SE is the standard error

Substitute the known values into the formula to find the 70th percentile of the sample mean.

Remember to round your final answer to at least 2 decimal places.