Step by step
The question
Question: samples of 50 are taken from numbers with mean 200 and SD 30: what is the probability that the sample mean is less than 195?
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, 200.
Its SD is smaller: σ divided by the square root of n.
30 divided by the square root of 50 is about 4.2426.
Careful!
The sample means have SD 4.2426, not 30.
So we divide by 4.2426.
2 Find z
z counts how many SDs the sample mean is from the mean: x̄ minus the mean, over the SD.
195 minus 200 is −5.
Divided by 4.2426, that is about −1.18.
3 Draw and shade
The bell: the mean, 200, in the middle.
195 is left of it.
We want the sample mean less than 195.
We shade that part.
4 Read the table
The z-table gives the area to the left of z.
Row −1.1, column .08.
For z = −1.18: the row −1.1 and the column .08 meet at 0.1190.
5 The area we want
Less than: the area to the left.
That is just the table number, 0.1190.
6 Check
Let's check another way.
The bell is the same on both sides.
So the area left of −1.18 is the area right of 1.18.
1 minus the table at 1.18: 1 minus 0.8810 is 0.1190.
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.