Normal mean 60, sd 8: bottom 25% Step by Step

Step by step

The question

Question: X is normal, with mean 60 and SD 8: which value has 25% below it?

μ = 60, σ = 8: x with 25% below it?

A value with 25 percent below it is also called the 25th percentile.

25% below it = the 25th percentile

1 Shade the area

The bell, with the mean 60 in the middle.

x is the cut-off we want.

25 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

25 percent is 0.25.

It is already on the left.

area to the left = 0.25

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 0.2500 inside the table.

The closest number is 0.2514: row −0.6, column .07.

So z is about −0.67.

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

−0.67 times 8 is −5.36.

60 minus 5.36 is 54.64.

x = 60 + (−0.67) × 8 = 60 − 5.36 = 54.64

5 Check

Let's check: the z of 54.64 is 54.64 minus 60, over 8.

That is −0.67.

z = 54.64 − 608 = −0.67

The table at −0.67 gives 0.2514: about 0.25 to the left.

So it is right!

table at z = −0.67: 0.2514 ≈ 0.25

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