Normal mean 70, sd 4: bottom 10% Step by Step

Step by step

The question

Question: X is normal, with mean 70 and SD 4: which value has 10% below it?

μ = 70, σ = 4: x with 10% below it?

A value with 10 percent below it is also called the 10th percentile.

10% below it = the 10th percentile

1 Shade the area

The bell, with the mean 70 in the middle.

x is the cut-off we want.

10 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

10 percent is 0.1.

It is already on the left.

area to the left = 0.1

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 0.1000 inside the table.

The closest number is 0.1003: 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 4 is −5.12.

70 minus 5.12 is 64.88.

x = 70 + (−1.28) × 4 = 70 − 5.12 = 64.88

5 Check

Let's check: the z of 64.88 is 64.88 minus 70, over 4.

That is −1.28.

z = 64.88 − 704 = −1.28

The table at −1.28 gives 0.1003: about 0.1 to the left.

So it is right!

table at z = −1.28: 0.1003 ≈ 0.1

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