Step by step
The question
Question: X is normal, with mean 60 and SD 5: which value has 95% below it?
A value with 95 percent below it is also called the 95th percentile.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Shade the area
The bell, with the mean 60 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.
95 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.
95 percent is 0.95.
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 0.9500 inside the table.
0.9500 is halfway between 0.9495 and 0.9505.
So z is halfway between 1.64 and 1.65: 1.645.
4 Back to x
Start at the mean, then go z SDs: x is the mean plus z times the SD.
1.645 times 5 is 8.225.
60 plus 8.225 is 68.225.
5 Check
Let's check: the z of 68.225 is 68.225 minus 60, over 5.
That is 1.645.
The table at 1.65 gives 0.9505: about 0.95 to the left.
So it is right!
6 Pick the choice
Our answer is x = 68.225.
That is choice B.
7 Why not the others
A is 69.8.
Wrong: used 1.96: that leaves 2.5% above.
C is 51.775.
Wrong: went below the mean.
D is 65.
Wrong: used 1 standard deviation.
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.