Step by step
The question
Question: X is normal, with mean 100 and SD 15: what is the probability that X is between 85 and 115?
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.
85 minus 100 is −15.
Divided by 15, that is −1.
And for 115: 115 minus 100 is 15.
Divided by 15, that is 1.
2 Draw and shade
The bell: the mean, 100, in the middle.
85 is left of it.
115 is right of it.
We want X between 85 and 115.
We shade that part.
3 Read the table
The z-table gives the area to the left of z.
For z = −1: 0.1587.
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.
Between them: the bigger area minus the smaller one.
0.8413 minus 0.1587 is 0.6826.
5 Check
Let's check: the three parts of the bell.
Left of z = −1: 0.1587.
Right of z = 1: 1 minus 0.8413, 0.1587.
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.