Step by step
The question
Question: samples of 4 are taken from numbers with mean 20 and SD 6: what is the probability that the sample mean is more than 23?
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, 20.
Its SD is smaller: σ divided by the square root of n.
The square root of 4 is 2.
6 divided by 2 is 3.
Careful!
The sample means have SD 3, not 6.
So we divide by 3.
2 Find z
z counts how many SDs the sample mean is from the mean: x̄ minus the mean, over the SD.
23 minus 20 is 3.
Divided by 3, that is 1.
3 Draw and shade
The bell: the mean, 20, in the middle.
23 is right of it.
We want the sample mean more than 23.
We shade that part.
4 Read the table
The z-table gives the area to the left of z.
Row 1.0, column .00.
For z = 1: the row 1.0 and the column .00 meet at 0.8413.
5 The area we want
The whole bell is 1.
The right side is 1 minus the left: 1 minus 0.8413 is 0.1587.
6 Check
Let's check another way.
The bell is the same on both sides.
So the area right of 1 is the area left of −1.
The table at −1 gives 0.1587.
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.