Normal mean 50, sd 10: value with 90% below it Step by Step

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The question

Question: X is normal, with mean 50 and SD 10: which value has 90% below it?

μ = 50, σ = 10: x with 90% below it?

A value with 90 percent below it is also called the 90th percentile.

90% below it = the 90th percentile

1 Shade the area

The bell, with the mean 50 in the middle.

x is the cut-off we want.

90 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

90 percent is 0.9.

It is already on the left.

area to the left = 0.9

Careful!

More than half the bell is on the left.

So x is above the mean, and z is positive.

Careful: more than half on the left: z is positive

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 10 is 12.8.

50 plus 12.8 is 62.8.

x = 50 + 1.28 × 10 = 50 + 12.8 = 62.8

5 Check

Let's check: the z of 62.8 is 62.8 minus 50, over 10.

That is 1.28.

z = 62.8 − 5010 = 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