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# Sampling distribution of a sample mean example

AP.STATS:
UNC‑3 (EU)
,
UNC‑3.Q (LO)
,
UNC‑3.Q.1 (EK)
,
UNC‑3.Q.2 (EK)
,
UNC‑3.R (LO)
,
UNC‑3.R.1 (EK)
,
UNC‑3.R.2 (EK)
,
UNC‑3.S (LO)
,
UNC‑3.S.1 (EK)
Here's the type of problem you might see on the AP Statistics exam where you have to use the sampling distribution of a sample mean.

## Example: Means in quality control

An auto-maker does quality control tests on the paint thickness at different points on its car parts since there is some variability in the painting process. A certain part has a target thickness of 2, start text, space, m, m, end text. The distribution of thicknesses on this part is skewed to the right with a mean of 2, start text, space, m, m, end text and a standard deviation of 0, point, 5, start text, space, m, m, end text.
A quality control check on this part involves taking a random sample of 100 points and calculating the mean thickness of those points.
Assuming the stated mean and standard deviation of the thicknesses are correct, what is the probability that the mean thickness in the sample of 100 points is within 0, point, 1, start text, space, m, m, end text of the target value?
Let's solve this problem by breaking it down into smaller parts.

### Part 1: Establish normality

What is the shape of the sampling distribution of the sample mean thickness?

### Part 2: Find the mean and standard deviation of the sampling distribution

The sampling distribution of a sample mean x, with, \bar, on top has:
\begin{aligned} \mu_{\bar x}&=\mu \\\\ \sigma_{\bar x}&=\dfrac{\sigma}{\sqrt n} \end{aligned}
Note: For this standard deviation formula to be accurate, our sample size needs to be 10, percent or less of the population so we can assume independence.
Question A (Part 2)
What is the mean of the sampling distribution of x, with, \bar, on top?
mu, start subscript, x, with, \bar, on top, end subscript, equals
start text, m, m, end text

Question B (Part 2)
What is the standard deviation of the sampling distribution of x, with, \bar, on top?
sigma, start subscript, x, with, \bar, on top, end subscript, equals
start text, m, m, end text

### Part 3: Use normal calculations to find the probability in question

Assuming the stated mean and standard deviation of the thicknesses are correct, what is the approximate probability that the mean thickness in the sample of 100 points is within 0, point, 1, start text, space, m, m, end text of the target value?

## Want to join the conversation?

• I saw the Explain for question 1 saying n=100 ≥ 30, the central limit theorem applies. I don't understand where the 30 comes from?
• It's another one of those "rules of thumb". The experience of statisticians with many different populations and many different sample sizes over a large number of years led them to adopt this particular rule.

The CLT tells us that as the sample size n approaches infinity, the distribution of the sample means approaches a normal distribution. Experience shows us that most of the time 30 is close enough to infinity for us to employ the normal approximation and get good results.