Step by step
The question
Question: X is normal, with mean 20 and SD 3: what is the probability that X is between 18 and 22?
μ is the mean.
σ is the standard deviation, the SD.
1 Find z
z counts how many SDs x is from the mean: x minus the mean, over the SD.
18 minus 20 is −2.
Divided by 3, that is about −0.67.
And for 22: 22 minus 20 is 2.
Divided by 3, that is about 0.67.
2 Draw and shade
The bell: the mean, 20, in the middle.
18 is left of it.
22 is right of it.
We want X between 18 and 22.
We shade that part.
3 Read the table
The z-table gives the area to the left of z.
For z = −0.67: 0.2514.
For z = 0.67: 0.7486.
4 The area we want
Careful!
Both table numbers start at the far left.
So we take away the smaller one.
Between them: the bigger area minus the smaller one.
0.7486 minus 0.2514 is 0.4972.
5 Check
Let's check: the three parts of the bell.
Left of z = −0.67: 0.2514.
Right of z = 0.67: 1 minus 0.7486, 0.2514.
They add up to 1, the whole bell.
So it is right!
Let's look back.
First: find z.
Then: draw and shade.
Then: read the table.
Then: the area we want.
Last: check.