Step by step
The question
Question: X is normal, with mean 500 and SD 25: what is the probability that X is less than 480?
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.
480 minus 500 is −20.
Divided by 25, that is −0.8.
2 Draw and shade
The bell: the mean, 500, in the middle.
480 is left of it.
We want X less than 480.
We shade that part.
3 Read the table
The z-table gives the area to the left of z.
Row −0.8, column .00.
For z = −0.8: the row −0.8 and the column .00 meet at 0.2119.
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, 0.2119.
5 Check
Let's check another way.
The bell is the same on both sides.
So the area left of −0.8 is the area right of 0.8.
1 minus the table at 0.8: 1 minus 0.7881 is 0.2119.
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.