The Central Limit Theorem

Why the normal distribution keeps appearing.

12 problems · about 22 minutes · UNC-3.M

What this lesson teaches

The student applies the Central Limit Theorem to describe the sampling distribution of a mean.

For large enough n, sample means are approximately normal regardless of the population.
A guideline is n at least 30, larger for more skewed populations.
The theorem describes sample means, never individual values.

Straightforward practice. Get the method working first.

5 problems

Sigma 24 and n = 36. What is the standard error?
Answer 4
Why 4.
What does the Central Limit Theorem describe?
Answer The distribution of sample means, whatever the population shape.
Why Sample means, not individuals.
The population is already normal, n = 5. Is the sampling distribution normal? 1 yes, 0 no.
Answer 1
Why No large sample is needed.
The population is skewed, n = 50. Approximately normal? 1 yes, 0 no.
Answer 1
Why n exceeds 30.
Does the theorem make the population normal? 1 yes, 0 no.
Answer 0
Why It describes sample means.

The same ideas with more to keep track of.

3 problems

Sigma 20, n = 100. Standard error?
Answer 2
Why 20 ÷ 10.
Population mean 40. Where is the sampling distribution centred?
Answer 40
Why On the population mean.
A more skewed population needs a larger or smaller n? 1 larger, 2 smaller.
Answer 1
Why More skew, more sample.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Which statements does the theorem support?
Answer Sample means become approximately normal for large n; Normal-based inference works on skewed populations
Why The theorem is about means, not individuals.
Sigma 9, n = 9. Standard error?
Answer 3
Why 9 ÷ 3.
The Guideline: What sample size is commonly cited as large enough for the theorem?
Answer 30
Why 30.
The Standard Error: Sigma 12 and n = 36. What is the standard error?
Answer 2
Why 2.