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## Statistics and probability

### Course: Statistics and probability > Unit 10

Lesson 3: Sampling distribution of a sample mean- Inferring population mean from sample mean
- Central limit theorem
- Sampling distribution of the sample mean
- Sampling distribution of the sample mean (part 2)
- Standard error of the mean
- Example: Probability of sample mean exceeding a value
- Mean and standard deviation of sample means
- Sample means and the central limit theorem
- Finding probabilities with sample means
- Sampling distribution of a sample mean example

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# Sampling distribution of a sample mean example

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 $2\text{mm}$ . The distribution of thicknesses on this part is skewed to the right with a mean of $2\text{mm}$ and a standard deviation of $0.5\text{mm}$ .

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 $0.1\text{mm}$ of the target value?

*Let's solve this problem by breaking it down into smaller parts.*

### Part 1: Establish normality

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

The sampling distribution of a sample mean $\overline{x}$ has:

Note: For this standard deviation formula to be accurate, our sample size needs to be $10\mathrm{\%}$ or less of the population so we can assume independence.

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

## 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?(13 votes)
- 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.(36 votes)

- How come for last question answers I 95 %. Pls explain in detail(2 votes)
- Since it is normally distributed you can use the empirical rule (68-95-99.7 Rule) . If you are two standard deviations away on both sides, approximately 95% will fall within 2 standard deviations.(18 votes)

- I don't get one thing, in all the video examples Sal has used sample sizes of less that 30 and proved its normal, then why are we suddenly supposed to only use CLT is the sample size exceeds 30? And what are we supposed to do for those sampling distributions which have sample size less than 30?? Are we supposed to assume that they would have the same distribution as the parent population?(7 votes)
- The thing you are confusing is that even if sample sizes are less than 30 they still CAN be normal. However, if the sample size is greater than 30 we automatically assume it will approach a normal distrn by the CLT. Therefore, if the sample size is less than 30 you have to prove whether it is normal or not but if it is over 30 no proof is neccesary.(8 votes)

- I don't understand how they got 95% as the probability.(1 vote)
- What is the difference between standard deviation of sampling distribution, and unbiased standard deviation of a sample? The formula for both are different. And which one is a better/closer estimate of the true standard deviation of the entire population?(1 vote)