Step by step
The question
Question: samples of 40 are taken from numbers with mean 50 and SD 12: what is the probability that the sample mean is more than 53?
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, 50.
Its SD is smaller: σ divided by the square root of n.
12 divided by the square root of 40 is about 1.8974.
Careful!
The sample means have SD 1.8974, not 12.
So we divide by 1.8974.
2 Find z
z counts how many SDs the sample mean is from the mean: x̄ minus the mean, over the SD.
53 minus 50 is 3.
Divided by 1.8974, that is about 1.58.
3 Draw and shade
The bell: the mean, 50, in the middle.
53 is right of it.
We want the sample mean more than 53.
We shade that part.
4 Read the table
The z-table gives the area to the left of z.
Row 1.5, column .08.
For z = 1.58: the row 1.5 and the column .08 meet at 0.9429.
5 The area we want
The whole bell is 1.
The right side is 1 minus the left: 1 minus 0.9429 is 0.0571.
6 Check
Let's check another way.
The bell is the same on both sides.
So the area right of 1.58 is the area left of −1.58.
The table at −1.58 gives 0.0571.
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.