Step by step
The question
Question: X is normal, with mean 500 and SD 100: which value has 75% below it?
A value with 75 percent below it is also called the 75th percentile.
1 Shade the area
The bell, with the mean 500 in the middle.
x is the cut-off we want.
75 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.
75 percent is 0.75.
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 0.7500 inside the table.
The closest number is 0.7486: row 0.6, column .07.
So z is about 0.67.
4 Back to x
Start at the mean, then go z SDs: x is the mean plus z times the SD.
0.67 times 100 is 67.
500 plus 67 is 567.
5 Check
Let's check: the z of 567 is 567 minus 500, over 100.
That is 0.67.
The table at 0.67 gives 0.7486: about 0.75 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.