Step by step
The question
Question: samples of 64 are taken from numbers with mean 12 and SD 2: what is the probability that the sample mean is between 11.6 and 12.5?
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, 12.
Its SD is smaller: σ divided by the square root of n.
The square root of 64 is 8.
2 divided by 8 is 0.25.
Careful!
The sample means have SD 0.25, not 2.
So we divide by 0.25.
2 Find z
z counts how many SDs the sample mean is from the mean: x̄ minus the mean, over the SD.
11.6 minus 12 is −0.4.
Divided by 0.25, that is −1.6.
And for 12.5: 12.5 minus 12 is 0.5.
Divided by 0.25, that is 2.
3 Draw and shade
The bell: the mean, 12, in the middle.
11.6 is left of it.
12.5 is right of it.
We want the sample mean between 11.6 and 12.5.
We shade that part.
4 Read the table
The z-table gives the area to the left of z.
For z = −1.6: 0.0548.
For z = 2: 0.9772.
5 The area we want
Between them: the bigger area minus the smaller one.
0.9772 minus 0.0548 is 0.9224.
6 Check
Let's check: the three parts of the bell.
Left of z = −1.6: 0.0548.
Right of z = 2: 1 minus 0.9772, 0.0228.
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.