Normal mean 100, sd 15: top 10% Step by Step

Step by step

The question

Question: X is normal, with mean 100 and SD 15: which value has 10% above it?

μ = 100, σ = 15: x with 10% above it?

1 Shade the area

The bell, with the mean 100 in the middle.

x is the cut-off we want.

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.

z-table: 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.

area to the left = 1 − 0.1 = 0.9

Careful!

0.1 is on the right.

The table needs the area to the left, 0.9.

Careful: not 0.1 in the table ✗

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 15 is 19.2.

100 plus 19.2 is 119.2.

x = 100 + 1.28 × 15 = 100 + 19.2 = 119.2

5 Check

Let's check: the z of 119.2 is 119.2 minus 100, over 15.

That is 1.28.

z = 119.2 − 10015 = 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