Step by step
The question
Question: the SD of all the numbers is 12: how big a sample do we need to be 99% confident with an error bound of 3?
The error bound is how far the sample mean may be from the true mean.
1 Find z*
99 percent of the bell is in the middle.
That leaves 1 percent: 0.5 percent in each end.
z* marks the cut between the middle and the ends.
Left of z* are the middle and one end: 0.99 plus 0.005 is 0.9950.
Now we look for 0.9950 inside the table.
0.9950 is halfway between 0.9949 and 0.9951.
So z is halfway between 2.57 and 2.58: 2.575.
With more digits it is 2.576.
That is the number books use.
2 The formula
The error bound is z* times σ over the square root of n.
Solved for n, it gives this formula.
n is z* squared times σ squared, over the error bound squared.
3 Put the numbers in
2.576 squared is about 6.6358.
12 squared is 144.
3 squared is 9.
6.6358 times 144 is 955.5552.
Divided by 9: about 106.17.
4 Round up
A sample size must be a whole number.
We always round up: a smaller sample would miss the goal.
106.17 rounds up to 107.
Careful!
Rounding down to 106 gives an error bound a bit too big.
Always round up.
5 Check
Let's check: with 107 people the error bound is about 2.9884.
That is not more than 3.
Good.
With one person less, 106, it is about 3.0024: too big.
So 107 is the smallest that works.
So it is right!
Let's look back.
First: find z*.
Then: the formula.
Then: put the numbers in.
Then: round up.
Last: check.