Step by step
The question
Question: samples of 100 are taken from numbers with mean 500 and SD 50: what is the probability that the sample mean is more than 510?
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, 500.
Its SD is smaller: σ divided by the square root of n.
The square root of 100 is 10.
50 divided by 10 is 5.
Careful!
The sample means have SD 5, not 50.
So we divide by 5.
2 Find z
z counts how many SDs the sample mean is from the mean: x̄ minus the mean, over the SD.
510 minus 500 is 10.
Divided by 5, that is 2.
3 Draw and shade
The bell: the mean, 500, in the middle.
510 is right of it.
We want the sample mean more than 510.
We shade that part.
4 Read the table
The z-table gives the area to the left of z.
Row 2.0, column .00.
For z = 2: the row 2.0 and the column .00 meet at 0.9772.
5 The area we want
The whole bell is 1.
The right side is 1 minus the left: 1 minus 0.9772 is 0.0228.
6 Check
Let's check another way.
The bell is the same on both sides.
So the area right of 2 is the area left of −2.
The table at −2 gives 0.0228.
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.