Step by step
The question
Question: X is normal, with mean 60 and SD 2.5: what is the probability that X is less than 63.5?
μ is the mean.
σ is the standard deviation, the SD.
1 Find z
z counts how many SDs x is from the mean: x minus the mean, over the SD.
63.5 minus 60 is 3.5.
Divided by 2.5, that is 1.4.
2 Draw and shade
The bell: the mean, 60, in the middle.
63.5 is right of it.
We want X less than 63.5.
We shade that part.
3 Read the table
The z-table gives the area to the left of z.
Row 1.4, column .00.
For z = 1.4: the row 1.4 and the column .00 meet at 0.9192.
4 The area we want
Careful!
The z-table always gives the area to the left of z.
Here that is what we want.
Less than: the area to the left.
That is just the table number, 0.9192.
5 Check
Let's check another way.
The bell is the same on both sides.
So the area left of 1.4 is the area right of −1.4.
1 minus the table at −1.4: 1 minus 0.0808 is 0.9192.
The same!
So it is right!
Let's look back.
First: find z.
Then: draw and shade.
Then: read the table.
Then: the area we want.
Last: check.