Step by step
The question
Question: samples of 49 are taken from numbers with mean 80 and SD 14: what is the probability that the sample mean is between 78 and 83?
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, 80.
Its SD is smaller: σ divided by the square root of n.
The square root of 49 is 7.
14 divided by 7 is 2.
Careful!
The sample means have SD 2, not 14.
So we divide by 2.
2 Find z
z counts how many SDs the sample mean is from the mean: x̄ minus the mean, over the SD.
78 minus 80 is −2.
Divided by 2, that is −1.
And for 83: 83 minus 80 is 3.
Divided by 2, that is 1.5.
3 Draw and shade
The bell: the mean, 80, in the middle.
78 is left of it.
83 is right of it.
We want the sample mean between 78 and 83.
We shade that part.
4 Read the table
The z-table gives the area to the left of z.
For z = −1: 0.1587.
For z = 1.5: 0.9332.
5 The area we want
Between them: the bigger area minus the smaller one.
0.9332 minus 0.1587 is 0.7745.
6 Check
Let's check: the three parts of the bell.
Left of z = −1: 0.1587.
Right of z = 1.5: 1 minus 0.9332, 0.0668.
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.