Step by step
The question
Question: samples of 100 are taken from numbers with mean 3.2 and SD 0.8: what is the probability that the sample mean is between 3.1 and 3.3?
x̄, x bar, is the mean of one sample.
1 The sample means
The central limit theorem: the means of many samples make a bell.
Its middle is the mean, 3.2.
Its SD is smaller: σ divided by the square root of n.
The square root of 100 is 10.
0.8 divided by 10 is 0.08.
Careful!
The sample means have SD 0.08, not 0.8.
So we divide by 0.08.
2 Find z
z counts how many SDs the sample mean is from the mean: x̄ minus the mean, over the SD.
3.1 minus 3.2 is −0.1.
Divided by 0.08, that is −1.25.
And for 3.3: 3.3 minus 3.2 is 0.1.
Divided by 0.08, that is 1.25.
3 Draw and shade
The bell: the mean, 3.2, in the middle.
3.1 is left of it.
3.3 is right of it.
We want the sample mean between 3.1 and 3.3.
We shade that part.
4 Read the table
The z-table gives the area to the left of z.
For z = −1.25: 0.1056.
For z = 1.25: 0.8944.
5 The area we want
Between them: the bigger area minus the smaller one.
0.8944 minus 0.1056 is 0.7888.
6 Check
Let's check: the three parts of the bell.
Left of z = −1.25: 0.1056.
Right of z = 1.25: 1 minus 0.8944, 0.1056.
They add up to 1, the whole bell.
So it is right!
Let's look back.
First: the sample means.
Then: find z.
Then: draw and shade.
Then: read the table.
Then: the area we want.
Last: check.