Normal mean 170, sd 7: value with 2.5% below it Step by Step

Step by step

The question

Question: X is normal, with mean 170 and SD 7: which value has 2.5% below it?

μ = 170, σ = 7: x with 2.5% below it?

1 Shade the area

The bell, with the mean 170 in the middle.

x is the cut-off we want.

2.5 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

2.5 percent is 0.025.

It is already on the left.

area to the left = 0.025

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 0.0250 inside the table.

0.0250 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 7 is −13.72.

170 minus 13.72 is 156.28.

x = 170 + (−1.96) × 7 = 170 − 13.72 = 156.28

5 Check

Let's check: the z of 156.28 is 156.28 minus 170, over 7.

That is −1.96.

z = 156.28 − 1707 = −1.96

The table at −1.96 gives 0.025: about 0.025 to the left.

So it is right!

table at z = −1.96: 0.025 ≈ 0.025

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