Taxes The Internal Revenue Service reports that the mean federal

Solution

To solve for the 70th percentile of the sample mean, we need to use the concept of the Central Limit Theorem (CLT) and the properties of the 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.

Given:

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

The mean of the sample means (μx̄) is equal to the population mean (μ), and the standard deviation of the sample means (σx̄), also known as the standard error, is equal to the population standard deviation (σ) divided by the square root of the sample size (n).

So, we have:

• μx̄ = μ = $8040
• σx̄ = σ/√n = $5000/√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. For the 70th percentile, the z-score is approximately 0.52.

Finally, we use the formula for the z-score to solve for x (the 70th percentile of the sample mean):

x = μx̄ + z * σx̄

Let's calculate:

import math

mu = 8040
sigma = 5000
n = 1000
z = 0.52

sigma_x_bar = sigma / math.sqrt(n)
x = mu + z * sigma_x_bar
print(x)

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