Worked examples
The question
Question: X is normal, with mean 50 and SD 10: which value has 90% below it?
μ = 50, σ = 10: x with 90% below it?
A value with 90 percent below it is also called the 90th percentile.
90% below it = the 90th percentile
1 Shade the area
The bell, with the mean 50 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.
z-table: area to the left of z
90 percent is 0.9.
It is already on the left.
area to the left = 0.9
Careful!
More than half the bell is on the left.
So x is above the mean, and z is positive.
Careful: more than half on the left: 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.
x = μ + z × σ
1.28 times 10 is 12.8.
50 plus 12.8 is 62.8.
x = 50 + 1.28 × 10 = 50 + 12.8 = 62.8
5 Check
Let's check: the z of 62.8 is 62.8 minus 50, over 10.
That is 1.28.
z = 62.8 − 5010 = 1.28
The table at 1.28 gives 0.8997: about 0.9 to the left.
So it is right!
table at z = 1.28: 0.8997 ≈ 0.9
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.
Answer x = 50 + 1.28 × 10 = 50 + 12.8 = 62.8
▶ Watch it solved
The question
Question: X is normal, with mean 60 and SD 8: which value has 25% below it?
μ = 60, σ = 8: x with 25% below it?
A value with 25 percent below it is also called the 25th percentile.
25% below it = the 25th percentile
1 Shade the area
The bell, with the mean 60 in the middle.
x is the cut-off we want.
25 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.
z-table: area to the left of z
25 percent is 0.25.
It is already on the left.
area to the left = 0.25
Careful!
Less than half the bell is on the left.
So x is below the mean, and z is negative.
Careful: less than half on the left: z is negative
3 Find it inside the table
Now we look for 0.2500 inside the table.
The closest number is 0.2514: 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.
x = μ + z × σ
−0.67 times 8 is −5.36.
60 minus 5.36 is 54.64.
x = 60 + (−0.67) × 8 = 60 − 5.36 = 54.64
5 Check
Let's check: the z of 54.64 is 54.64 minus 60, over 8.
That is −0.67.
z = 54.64 − 608 = −0.67
The table at −0.67 gives 0.2514: about 0.25 to the left.
So it is right!
table at z = −0.67: 0.2514 ≈ 0.25
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.
Answer x = 60 + (−0.67) × 8 = 60 − 5.36 = 54.64
▶ Watch it solved
The question
Question: X is normal, with mean 100 and SD 15: which value has 99% below it?
μ = 100, σ = 15: x with 99% below it?
A value with 99 percent below it is also called the 99th percentile.
99% below it = the 99th percentile
1 Shade the area
The bell, with the mean 100 in the middle.
x is the cut-off we want.
99 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.
z-table: area to the left of z
99 percent is 0.99.
It is already on the left.
area to the left = 0.99
Careful!
More than half the bell is on the left.
So x is above the mean, and z is positive.
Careful: more than half on the left: z is positive
3 Find it inside the table
Now we look for 0.9900 inside the table.
The closest number is 0.9901: row 2.3, column .03.
So z is about 2.33.
4 Back to x
Start at the mean, then go z SDs: x is the mean plus z times the SD.
x = μ + z × σ
2.33 times 15 is 34.95.
100 plus 34.95 is 134.95.
x = 100 + 2.33 × 15 = 100 + 34.95 = 134.95
5 Check
Let's check: the z of 134.95 is 134.95 minus 100, over 15.
That is 2.33.
z = 134.95 − 10015 = 2.33
The table at 2.33 gives 0.9901: about 0.99 to the left.
So it is right!
table at z = 2.33: 0.9901 ≈ 0.99
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.
Answer x = 100 + 2.33 × 15 = 100 + 34.95 = 134.95
▶ Watch it solved
