Normal mean 50, sd 10: value with 5% above it Step by Step

Step by step

The question

Question: X is normal, with mean 50 and SD 10: which value has 5% above it?

μ = 50, σ = 10: x with 5% above it?

1 Shade the area

The bell, with the mean 50 in the middle.

x is the cut-off we want.

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

5 percent is 0.05.

It is on the right.

So the area to the left is 1 minus 0.05: 0.95.

area to the left = 1 − 0.05 = 0.95

Careful!

0.05 is on the right.

The table needs the area to the left, 0.95.

Careful: not 0.05 in the table ✗

3 Find it inside the table

Now we look for 0.9500 inside the table.

0.9500 is halfway between 0.9495 and 0.9505.

So z is halfway between 1.64 and 1.65: 1.645.

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

1.645 times 10 is 16.45.

50 plus 16.45 is 66.45.

x = 50 + 1.645 × 10 = 50 + 16.45 = 66.45

5 Check

Let's check: the z of 66.45 is 66.45 minus 50, over 10.

That is 1.645.

z = 66.45 − 5010 = 1.645

The table at 1.65 gives 0.9505: about 0.95 to the left.

So it is right!

table at z = 1.65: 0.9505 ≈ 0.95

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