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.
The central limit theorem: the means of many samples make a bell.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The center
Its middle is the same as the numbers' mean: 70.
2 The spread
Means of samples are closer together than single numbers: in a mean, high and low numbers cancel out part of the way.
Their SD is σ divided by the square root of n, where n is the sample size.
The square root of 16 is 4.
8 divided by 4 is 2.
Careful!
We divide by the square root of 16, not by 16.
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!
4 Pick the choice
Our answer is μx̄ = 70, σx̄ = 2.
That is choice C.
5 Why not the others
A is mean 70, SD 8.
Wrong: the sample mean varies less: divide the SD by √16.
B is mean 70, SD 0.5.
Wrong: divided by 16: divide by √16.
D is mean 17.5, SD 2.
Wrong: divided the mean too: it stays 70.
Let's look back.
First: the center.
Then: the spread.
Then: check.
Then: pick the choice.
Last: why not the others.