Step by step
The question
Question: X is normal, with mean 100 and SD 15: which value has 10% above it?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Shade the area
The bell, with the mean 100 in the middle.
x is the cut-off we want.
This time we go backwards: we know the area, and we look for the value.
10 percent is above x: we shade it, on the right of x.
2 Area to the left
The z-table works with the area to the left of z.
10 percent is 0.1.
It is on the right.
So the area to the left is 1 minus 0.1: 0.9.
Careful!
0.1 is on the right.
The table needs the area to the left, 0.9.
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.
The row gives z up to the tenths, the column adds the hundredths.
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 15 is 19.2.
100 plus 19.2 is 119.2.
5 Check
Let's check: the z of 119.2 is 119.2 minus 100, over 15.
That is 1.28.
The table at 1.28 gives 0.8997: about 0.9 to the left.
So it is right!
6 Pick the choice
Our answer is x = 119.2.
That is choice B.
7 Why not the others
A is 80.8.
Wrong: went below the mean: the top 10% is above it.
That is the Careful mistake from before.
C is 115.
Wrong: used z = 1.
D is 110.
Wrong: added 10, not 1.28 × 15.
Let's look back.
First: shade the area.
Then: area to the left.
Then: find it inside the table.
Then: back to x.
Then: check.
Then: pick the choice.
Last: why not the others.