Normal mean 100, sd 10: top 2.5% Step by Step

Step by step

The question

Question: X is normal, with mean 100 and SD 10: which value has 2.5% above it?

μ = 100, σ = 10: x with 2.5% above it?

1 Shade the area

The bell, with the mean 100 in the middle.

x is the cut-off we want.

2.5 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

2.5 percent is 0.025.

It is on the right.

So the area to the left is 1 minus 0.025: 0.975.

area to the left = 1 − 0.025 = 0.975

Careful!

0.025 is on the right.

The table needs the area to the left, 0.975.

Careful: not 0.025 in the table ✗

3 Find it inside the table

Now we look for 0.9750 inside the table.

0.9750 is in the table: row 1.9, column .06.

So z = 1.96.

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

1.96 times 10 is 19.6.

100 plus 19.6 is 119.6.

x = 100 + 1.96 × 10 = 100 + 19.6 = 119.6

5 Check

Let's check: the z of 119.6 is 119.6 minus 100, over 10.

That is 1.96.

z = 119.6 − 10010 = 1.96

The table at 1.96 gives 0.975: about 0.975 to the left.

So it is right!

table at z = 1.96: 0.975 ≈ 0.975

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