Normal: P(X < 34), mean 40, sd 5 Step by Step

Step by step

The question

Question: X is normal, with mean 40 and SD 5: what is the probability that X is less than 34?

P(X < 34): μ = 40, σ = 5

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

34 minus 40 is −6.

Divided by 5, that is −1.2.

z = 34 − 405 = −65 = −1.2

2 Draw and shade

The bell: the mean, 40, in the middle.

34 is left of it.

We want X less than 34.

We shade that part.

3 Read the table

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

Row −1.2, column .00.

For z = −1.2: the row −1.2 and the column .00 meet at 0.1151.

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, 0.1151.

P(X < 34) = 0.1151

5 Check

Let's check another way.

The bell is the same on both sides.

So the area left of −1.2 is the area right of 1.2.

left of −1.2 = right of 1.2

1 minus the table at 1.2: 1 minus 0.8849 is 0.1151.

The same!

So it is right!

1 − table at 1.2: 1 − 0.8849 = 0.1151

Let's look back.

First: find z.

Then: draw and shade.

Then: read the table.

Then: the area we want.

Last: check.

Answer