Taxes The Internal Revenue Service reports that the mean federal

Solution

To solve this problem, we need to use the concept of the Central Limit Theorem (CLT) and the properties of the normal distribution. The CLT 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. From standard normal distribution tables, the z-score for the 70th percentile is approximately 0.52.

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

x̄ = μx̄ + z * σx̄

Let's calculate it:

import math

mu = 8040  # population mean
sigma = 5000  # population standard deviation
n = 1000  # sample size
z = 0.52  # z-score for 70th percentile

# calculate standard error
sigma_x_bar = sigma / math.sqrt(n)

# calculate sample mean for 70th percentile
x_bar = mu + z * sigma_x_bar
x_bar

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