Step by step
The question
Question: X is normal, with mean 70 and SD 8: which value has 90% below it?
A value with 90 percent below it is also called the 90th percentile.
1 Shade the area
The bell, with the mean 70 in the middle.
x is the cut-off we want.
90 percent is below x: we shade it, on the left of x.
2 Area to the left
The z-table works with the area to the left of z.
90 percent is 0.9.
It is already on the left.
Careful!
More than half the bell is on the left.
So x is above the mean, and z is positive.
3 Find it inside the table
Now we look for 0.9000 inside the table.
The closest number is 0.8997: row 1.2, column .08.
So z is about 1.28.
4 Back to x
Start at the mean, then go z SDs: x is the mean plus z times the SD.
1.28 times 8 is 10.24.
70 plus 10.24 is 80.24.
5 Check
Let's check: the z of 80.24 is 80.24 minus 70, over 8.
That is 1.28.
The table at 1.28 gives 0.8997: about 0.9 to the left.
So it is right!
Let's look back.
First: shade the area.
Then: area to the left.
Then: find it inside the table.
Then: back to x.
Last: check.