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), the population standard deviation (σ), and the sample size (n).

• Population mean (μ) = $8040
• Population standard deviation (σ) = $5000
• Sample size (n) = 1000

We can calculate the mean and standard deviation of the sample distribution of the mean as follows:

• Mean of the sample distribution (μx̄) = μ = $8040
• Standard deviation of the sample distribution (σx̄) = σ/√n = $5000/√1000

Now, 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 this up in a standard z-table or use a calculator that provides this functionality.

The z-score that corresponds to the 70th percentile is approximately 0.52.

Finally, we use the z-score formula to find the sample mean that corresponds to the 70th percentile:

x̄ = μx̄ + z * σx̄

Let's calculate it:

μx̄ = $8040
σx̄ = $5000/√1000
z = 0.52

Substitute these values into the formula:

x̄ = $8040 + 0.52 * ($5000/√1000)

This will give you the 70th percentile of the sample mean.