Normal mean 75, sd 10: bottom 5% Step by Step

Step by step

The question

Question: X is normal, with mean 75 and SD 10: which value has 5% below it?

μ = 75, σ = 10: x with 5% below it?

A value with 5 percent below it is also called the 5th percentile.

5% below it = the 5th percentile

1 Shade the area

The bell, with the mean 75 in the middle.

x is the cut-off we want.

5 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

5 percent is 0.05.

It is already on the left.

area to the left = 0.05

Careful!

Less than half the bell is on the left.

So x is below the mean, and z is negative.

Careful: less than half on the left: z is negative

3 Find it inside the table

Now we look for 0.0500 inside the table.

0.0500 is halfway between 0.0495 and 0.0505.

So z is halfway between −1.65 and −1.64: −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.

75 minus 16.45 is 58.55.

x = 75 + (−1.645) × 10 = 75 − 16.45 = 58.55

5 Check

Let's check: the z of 58.55 is 58.55 minus 75, over 10.

That is −1.645.

z = 58.55 − 7510 = −1.645

The table at −1.65 gives 0.0495: about 0.05 to the left.

So it is right!

table at z = −1.65: 0.0495 ≈ 0.05

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