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?
μ is the mean.
σ is the standard deviation, 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.
45 minus 50 is −5.
Divided by 10, that is −0.5.
And for 60: 60 minus 50 is 10.
Divided by 10, that is 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.
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.
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.
Between them: the bigger area minus the smaller one.
0.8413 minus 0.3085 is 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.
They add up to 1, the whole bell.
So it is right!
6 Pick the choice
Our answer is P(45 < X < 60) = 0.5328.
That is choice D.
7 Why not the others
A is 1.1498.
Wrong: added the two areas: a probability is never above 1.
B is 0.8413.
Wrong: that is P(X < 60) only: take away P(X < 45).
C is 0.3085.
Wrong: that is P(X < 45).
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.