Step by step
The question
Question: X is normal, with mean 70 and SD 6: what is the probability that X is more than 80?
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.
Every normal bell has the same shape when we count in SDs.
So one table, the z-table, works for every bell.
80 minus 70 is 10.
Divided by 6, that is about 1.67.
2 Draw and shade
The bell: the mean, 70, in the middle.
80 is right of it.
We want X more than 80.
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 1.6, column .07.
For z = 1.67: the row 1.6 and the column .07 meet at 0.9525.
4 The area we want
Careful!
0.9525 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 0.9525 is 0.0475.
5 Check
Let's check another way.
The bell is the same on both sides.
So the area right of 1.67 is the area left of −1.67.
The table at −1.67 gives 0.0475.
The same!
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.