Step by step
The question
Question: samples of 36 are taken from numbers with mean 30 and SD 9: what is the probability that the sample mean is less than 32?
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, 30.
Its SD is smaller: σ divided by the square root of n.
The square root of 36 is 6.
9 divided by 6 is 1.5.
Careful!
The sample means have SD 1.5, not 9.
So we divide by 1.5.
2 Find z
z counts how many SDs the sample mean is from the mean: x̄ minus the mean, over the SD.
32 minus 30 is 2.
Divided by 1.5, that is about 1.33.
3 Draw and shade
The bell: the mean, 30, in the middle.
32 is right of it.
We want the sample mean less than 32.
We shade that part.
4 Read the table
The z-table gives the area to the left of z.
Row 1.3, column .03.
For z = 1.33: the row 1.3 and the column .03 meet at 0.9082.
5 The area we want
Less than: the area to the left.
That is just the table number, 0.9082.
6 Check
Let's check another way.
The bell is the same on both sides.
So the area left of 1.33 is the area right of −1.33.
1 minus the table at −1.33: 1 minus 0.0918 is 0.9082.
The same!
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.