Normal mean 170, sd 8: probability that x is less than 150 Step by Step

Step by step

The question

Question: X is normal, with mean 170 and SD 8: what is the probability that X is less than 150?

P(X < 150): μ = 170, σ = 8

μ is the mean.

σ is the standard deviation, the SD.

μ = the mean, σ = the SD

1 Find z

z counts how many SDs x is from the mean: x minus the mean, over the SD.

z = x − μσ

150 minus 170 is −20.

Divided by 8, that is −2.5.

z = 150 − 1708 = −208 = −2.5

2 Draw and shade

The bell: the mean, 170, in the middle.

150 is left of it.

We want X less than 150.

We shade that part.

3 Read the table

The z-table gives the area to the left of z.

Row −2.5, column .00.

For z = −2.5: the row −2.5 and the column .00 meet at 0.0062.

4 The area we want

Careful!

The z-table always gives the area to the left of z.

Here that is what we want.

Careful: the table gives the area to the LEFT

Less than: the area to the left.

That is just the table number, 0.0062.

P(X < 150) = 0.0062

5 Check

Let's check another way.

The bell is the same on both sides.

So the area left of −2.5 is the area right of 2.5.

left of −2.5 = right of 2.5

1 minus the table at 2.5: 1 minus 0.9938 is 0.0062.

The same!

So it is right!

1 − table at 2.5: 1 − 0.9938 = 0.0062

Let's look back.

First: find z.

Then: draw and shade.

Then: read the table.

Then: the area we want.

Last: check.

Answer