Step by step
The question
Question: how big a sample do we need to find a proportion, 99% confident, with an error bound of 0.05?
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
From the error bound of a proportion, solved for n: z* squared times p prime times q prime, over the error bound squared.
We have no guess for p prime.
So we use 0.5: it gives the biggest n, so it is safe.
p prime is 0.5, so q prime is 1 minus 0.5: 0.5.
3 Put the numbers in
2.576 squared is about 6.6358.
Times 0.5 times 0.5: 1.65895.
0.05 squared is 0.0025.
Dividing gives about 663.58.
4 Round up
A sample size must be a whole number.
We always round up: a smaller sample would miss the goal.
663.58 rounds up to 664.
Careful!
Rounding down to 663 gives an error bound a bit too big.
Always round up.
5 Check
Let's check: with 664 people the error bound is about 0.05.
That is not more than 0.05.
Good.
With one person less, 663, it is about 0.05: too big.
So 664 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.