Normal mean 30, sd 4: top 30% Step by Step

Step by step

The question

Question: X is normal, with mean 30 and SD 4: which value has 30% above it?

μ = 30, σ = 4: x with 30% above it?

1 Shade the area

The bell, with the mean 30 in the middle.

x is the cut-off we want.

30 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

30 percent is 0.3.

It is on the right.

So the area to the left is 1 minus 0.3: 0.7.

area to the left = 1 − 0.3 = 0.7

Careful!

0.3 is on the right.

The table needs the area to the left, 0.7.

Careful: not 0.3 in the table ✗

3 Find it inside the table

Now we look for 0.7000 inside the table.

The closest number is 0.6985: row 0.5, column .02.

So z is about 0.52.

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

0.52 times 4 is 2.08.

30 plus 2.08 is 32.08.

x = 30 + 0.52 × 4 = 30 + 2.08 = 32.08

5 Check

Let's check: the z of 32.08 is 32.08 minus 30, over 4.

That is 0.52.

z = 32.08 − 304 = 0.52

The table at 0.52 gives 0.6985: about 0.7 to the left.

So it is right!

table at z = 0.52: 0.6985 ≈ 0.7

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