Step by step
The question
Question: X is normal, with mean 50 and SD 4: what is the probability that X is between 52 and 58?
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.
52 minus 50 is 2.
Divided by 4, that is 0.5.
And for 58: 58 minus 50 is 8.
Divided by 4, that is 2.
2 Draw and shade
The bell: the mean, 50, in the middle.
52 is right of it.
58 is right of it.
We want X between 52 and 58.
We shade that part.
3 Read the table
The z-table gives the area to the left of z.
For z = 0.5: 0.6915.
For z = 2: 0.9772.
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.
0.9772 minus 0.6915 is 0.2857.
5 Check
Let's check: the three parts of the bell.
Left of z = 0.5: 0.6915.
Right of z = 2: 1 minus 0.9772, 0.0228.
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.