Normal: P(X < 45), mean 50, sd 5 Step by Step

Step by step

The question

Question: X is normal, with mean 50 and SD 5: what is the probability that X is less than 45?

P(X < 45): μ = 50, 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

45 minus 50 is −5.

Divided by 5, that is −1.

z = 45 − 505 = −55 = −1

2 Draw and shade

The bell: the mean, 50, in the middle.

45 is left of it.

We want X less than 45.

We shade that part.

3 Read the table

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

Row −1.0, column .00.

For z = −1: the row −1.0 and the column .00 meet at 0.1587.

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, 0.1587.

P(X < 45) = 0.1587

5 Check

Let's check another way.

The bell is the same on both sides.

So the area left of −1 is the area right of 1.

left of −1 = right of 1

1 minus the table at 1: 1 minus 0.8413 is 0.1587.

The same!

So it is right!

1 − table at 1: 1 − 0.8413 = 0.1587

Let's look back.

First: find z.

Then: draw and shade.

Then: read the table.

Then: the area we want.

Last: check.

Answer