Normal: P(X < 112), mean 100, sd 8 Step by Step

Step by step

The question

Question: X is normal, with mean 100 and SD 8: what is the probability that X is less than 112?

P(X < 112): μ = 100, 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

112 minus 100 is 12.

Divided by 8, that is 1.5.

z = 112 − 1008 = 128 = 1.5

2 Draw and shade

The bell: the mean, 100, in the middle.

112 is right of it.

We want X less than 112.

We shade that part.

3 Read the table

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

Row 1.5, column .00.

For z = 1.5: the row 1.5 and the column .00 meet at 0.9332.

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, 0.9332.

P(X < 112) = 0.9332

5 Check

Let's check another way.

The bell is the same on both sides.

So the area left of 1.5 is the area right of −1.5.

left of 1.5 = right of −1.5

1 minus the table at −1.5: 1 minus 0.0668 is 0.9332.

The same!

So it is right!

1 − table at −1.5: 1 − 0.0668 = 0.9332

Let's look back.

First: find z.

Then: draw and shade.

Then: read the table.

Then: the area we want.

Last: check.

Answer