Normal: P(45 < X < 60), mean 50, sd 10 Step by Step

Step by step

The question

Question: X is normal, with mean 50 and SD 10: what is the probability that X is between 45 and 60?

P(45 < X < 60): μ = 50, σ = 10

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 10, that is −0.5.

z = 45 − 5010 = −510 = −0.5

And for 60: 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.

45 is left of it.

60 is right of it.

We want X between 45 and 60.

We shade that part.

3 Read the table

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

For z = −0.5: 0.3085.

For z = 1: 0.8413.

4 The area we want

Careful!

Both table numbers start at the far left.

So we take away the smaller one.

Careful: between: big area − small area

Between them: the bigger area minus the smaller one.

0.8413 minus 0.3085 is 0.5328.

P(45 < X < 60) = 0.8413 − 0.3085 = 0.5328

5 Check

Let's check: the three parts of the bell.

Left of z = −0.5: 0.3085.

Right of z = 1: 1 minus 0.8413, 0.1587.

left 0.3085 + middle 0.5328 + right 0.1587

They add up to 1, the whole bell.

So it is right!

= 1.0000

Let's look back.

First: find z.

Then: draw and shade.

Then: read the table.

Then: the area we want.

Last: check.

Answer