Step by step
The question
Question: how big a sample do we need to find a proportion, 95% confident, with an error bound of 0.03?
1 Find z*
95 percent of the bell is in the middle.
That leaves 5 percent: 2.5 percent in each end.
z* marks the cut between the middle and the ends.
Left of z* are the middle and one end: 0.95 plus 0.025 is 0.9750.
Now we look for 0.9750 inside the table.
0.9750 is in the table: row 1.9, column .06.
So z = 1.96.
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
1.96 squared is 3.8416.
Times 0.5 times 0.5: 0.9604.
0.03 squared is 0.0009.
Dividing gives about 1067.11.
4 Round up
A sample size must be a whole number.
We always round up: a smaller sample would miss the goal.
1067.11 rounds up to 1068.
Careful!
Rounding down to 1067 gives an error bound a bit too big.
Always round up.
5 Check
Let's check: with 1068 people the error bound is about 0.03.
That is not more than 0.03.
Good.
With one person less, 1067, it is about 0.03: too big.
So 1068 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.