Normal mean 64, sd 3: bottom 15% Step by Step

Step by step

The question

Question: X is normal, with mean 64 and SD 3: which value has 15% below it?

μ = 64, σ = 3: x with 15% below it?

A value with 15 percent below it is also called the 15th percentile.

15% below it = the 15th percentile

1 Shade the area

The bell, with the mean 64 in the middle.

x is the cut-off we want.

15 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

15 percent is 0.15.

It is already on the left.

area to the left = 0.15

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 0.1500 inside the table.

The closest number is 0.1492: row −1.0, column .04.

So z is about −1.04.

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

−1.04 times 3 is −3.12.

64 minus 3.12 is 60.88.

x = 64 + (−1.04) × 3 = 64 − 3.12 = 60.88

5 Check

Let's check: the z of 60.88 is 60.88 minus 64, over 3.

That is −1.04.

z = 60.88 − 643 = −1.04

The table at −1.04 gives 0.1492: about 0.15 to the left.

So it is right!

table at z = −1.04: 0.1492 ≈ 0.15

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