Step by step
The question
Question: X is normal, with mean 50 and SD 10: what is the probability that X is more than 40?
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.
40 minus 50 is −10.
Divided by 10, that is −1.
2 Draw and shade
The bell: the mean, 50, in the middle.
40 is left of it.
We want X more than 40.
We shade that part.
3 Read the table
The z-table gives the area to the left of z.
Row −1.0, column .00.
For z = −1: the row −1.0 and the column .00 meet at 0.1587.
4 The area we want
Careful!
0.1587 is the area to the left.
We want the right side.
The whole bell is 1.
The right side is 1 minus the left: 1 minus 0.1587 is 0.8413.
5 Check
Let's check another way.
The bell is the same on both sides.
So the area right of −1 is the area left of 1.
The table at 1 gives 0.8413.
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.