Step by step
The question
Question: X is normal, with mean 100 and SD 15: what is the probability that X is more than 130?
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.
130 minus 100 is 30.
Divided by 15, that is 2.
2 Draw and shade
The bell: the mean, 100, in the middle.
130 is right of it.
We want X more than 130.
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.
Row 2.0, column .00.
For z = 2: the row 2.0 and the column .00 meet at 0.9772.
4 The area we want
Careful!
0.9772 is the area to the left.
We want the right side.
The whole bell is 1.
The right side is 1 minus the left: 1 minus 0.9772 is 0.0228.
5 Check
Let's check another way.
The bell is the same on both sides.
So the area right of 2 is the area left of −2.
The table at −2 gives 0.0228.
The same!
So it is right!
6 Pick the choice
Our answer is P(X > 130) = 0.0228.
That is choice D.
7 Why not the others
A is 0.9772.
Wrong: that is P(X < 130).
That is the Careful mistake from before.
B is 0.0456.
Wrong: that is both tails: only the upper tail counts.
C is 0.1587.
Wrong: used z = 1: z = 30/15 = 2.
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.