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.

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?

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.

How come for last question answers I 95 %. Pls explain in detail

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.

Main content

Course: AP®︎/College Statistics > Unit 9

Lesson 6: Sampling distributions for sample means

Sampling distribution of a sample mean example

Google Classroom

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 mm

. The distribution of thicknesses on this part is skewed to the right with a mean of

2 mm

and a standard deviation of

0.5 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 $0.1 mm$ ‍ 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

\bar{x}

has:

\begin{aligned} μ_{\bar{x}} & = μ \\ σ_{\bar{x}} & = \frac{σ}{\sqrt{n}} \end{aligned}

Note: For this standard deviation formula to be accurate, our sample size needs to be

10 %

or less of the population so we can assume independence.

Question A (Part 2)

What is the mean of the sampling distribution of $\bar{x}$ ‍?

μ_{\bar{x}} =

mm

Question B (Part 2)

What is the standard deviation of the sampling distribution of $\bar{x}$ ‍?

σ_{\bar{x}} =

mm

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 $0.1 mm$ ‍ of the target value?

Want to join the conversation?

Sort by:

Sergey Li
Posted 6 years ago. Direct link to Sergey Li's post “I saw the Explain for que...”
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?
Button navigates to signup pageComment on Sergey Li's post “I saw the Explain for que...”
(14 votes)
Answer
- David Bryant
  Posted 6 years ago. Direct link to David Bryant's post “It's another one of those...”
  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.
  Comment on David Bryant's post “It's another one of those...”
  (40 votes)
Vikrant Jain
Posted 6 years ago. Direct link to Vikrant Jain's post “How come for last questi...”
How come for last question answers I 95 %. Pls explain in detail
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Josi S
  Posted 6 years ago. Direct link to Josi S's post “Since it is normally dist...”
  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.
  Comment on Josi S's post “Since it is normally dist...”
  (18 votes)
sanaksoomro
Posted 4 years ago. Direct link to sanaksoomro's post “I don't get one thing, in...”
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?
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- Saivishnu Tulugu
  Posted 4 years ago. Direct link to Saivishnu Tulugu's post “The thing you are confusi...”
  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.
  Comment on Saivishnu Tulugu's post “The thing you are confusi...”
  (8 votes)
Mrs. Bieler
Posted 2 months ago. Direct link to Mrs. Bieler 's post “Why does using the normal...”
Why does using the normalcdf not work for this example?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
ava wood
Posted 3 months ago. Direct link to ava wood's post “They call me Mr. Math”
They call me Mr. Math
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- Samuel Diaz Rios
  Posted 24 days ago. Direct link to Samuel Diaz Rios's post “A little bird told me tha...”
  A little bird told me that your real name is Ava Wood... but idk birds do not talk :/
  Button navigates to signup page
  (1 vote)
Horn.Caleb
Posted 3 months ago. Direct link to Horn.Caleb's post “Dude this is so frickin h...”
Dude this is so frickin hard
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
Jakeila Walton
Posted a year ago. Direct link to Jakeila Walton's post “I don't understand how th...”
I don't understand how they got 95% as the probability.
Button navigates to signup pageComment on Jakeila Walton's post “I don't understand how th...”
(0 votes)
Answer
Naresh K
Posted 4 years ago. Direct link to Naresh K's post “What is the difference be...”
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?
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer