Step by step
The question
Question: samples of 16 are taken from numbers with mean 70 and SD 8: find the mean and the SD of the sample means.
μ = 70, σ = 8, n = 16: mean and SD of x̄?
The central limit theorem: the means of many samples make a bell.
many samples of 16: their means make a bell
1 The center
Its middle is the same as the numbers' mean: 70.
μx̄ = μ = 70
2 The spread
Means of samples are closer together than single numbers.
Their SD is σ divided by the square root of n.
means are less spread out: SD = σ ÷ √n
The square root of 16 is 4.
8 divided by 4 is 2.
σx̄ = σ√n = 8√16 = 84 = 2
Careful!
We divide by the square root of 16, not by 16.
Careful: divide by √n, not by n
The bell of the sample means: middle 70, one SD is 2.
3 Check
Let's check: times the square root of 16 takes us back.
2 times 4 is 8, the SD σ.
So it is right!
2 × √16 = 2 × 4 = 8
Let's look back.
First: the center.
Then: the spread.
Last: check.
Answer
Careful: divide by √n, not by n