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 sample means will be the population mean, μ.

The standard deviation of the sample means, also known as the standard error (SE), is the population standard deviation divided by the square root of the sample size.

So,

SE = σ / √n

We can use the Z-score formula to find the value at the 70th percentile.

Z = (X - μ) / SE

Where:

• Z is the Z-score,
• X is the value at the 70th percentile,
• μ is the mean, and
• SE is the standard error.

We know that the Z-score for the 70th percentile is approximately 0.52. We can solve the Z-score formula for X to find the 70th percentile.

X = Z * SE + μ

Let's calculate:

import math

# given values
mu = 8040  # mean
sigma = 5000  # standard deviation
n = 1000  # sample size

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

# Z-score for 70th percentile
Z = 0.52

# calculate X
X = Z * SE + mu

Conclusion

The 70th percentile of the sample mean is the value of X calculated above.