X is normal with mean 50 and standard deviation 10. What is P(X

Step by step

The question

Question: X is normal, with mean 50 and SD 10: what is the probability that X is less than 60?

P(X < 60): μ = 50, σ = 10

μ is the mean.

σ is the standard deviation, the SD.

μ = the mean

σ = the SD

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Find z

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

Every normal bell has the same shape when we count in SDs.

So one table, the z-table, works for every bell.

z = x − μσ

60 minus 50 is 10.

Divided by 10, that is 1.

z = 60 − 5010 = 1010 = 1

2 Draw and shade

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

60 is right of it.

We want X less than 60.

We shade that part.

The whole bell has area 1, and the shaded area is the probability.

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 0.8413.

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, 0.8413.

P(X < 60) = 0.8413

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 0.1587 is 0.8413.

The same!

So it is right!

1 − table at −1: 1 − 0.1587 = 0.8413

6 Pick the choice

Our answer is P(X < 60) = 0.8413.

That is choice A.

A) 0.8413

7 Why not the others

B is 0.6826.

Wrong: that is within 1 SD on both sides.

B) 0.6826 ✗ that is within 1 SD on both sides

C is 0.1587.

Wrong: that is P(X > 60).

C) 0.1587 ✗ that is P(X > 60)

D is 0.9772.

Wrong: used z = 2.

D) 0.9772 ✗ used z = 2

Let's look back.

First: find z.

Then: draw and shade.

Then: read the table.

Then: the area we want.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer