Sample size for a proportion: 95% confidence, error bound 0.02 Step by Step

Step by step

The question

Question: how big a sample do we need to find a proportion, 95% confident, with an error bound of 0.02?

n? 95% confidence, EBP = 0.02

1 Find z*

95 percent of the bell is in the middle.

That leaves 5 percent: 2.5 percent in each end.

95% in the middle: 5% left over, 2.5% 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.

left of z* = 0.95 + 0.025 = 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.

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.

n = z*² × p′ × q′EBP²

We have no guess for p prime.

So we use 0.5: it gives the biggest n, so it is safe.

no guess for p′: use 0.5 (the biggest n, safe)

p prime is 0.5, so q prime is 1 minus 0.5: 0.5.

p′ = 0.5, q′ = 1 − 0.5 = 0.5

3 Put the numbers in

1.96 squared is 3.8416.

Times 0.5 times 0.5: 0.9604.

0.02 squared is 0.0004.

Dividing gives about 2401.

n = 3.8416 × 0.5 × 0.50.02² = 0.96040.0004 ≈ 2401

4 Round up

A sample size must be a whole number.

We always round up: a smaller sample would miss the goal.

always round UP: a bigger sample is safe

2401 is already whole: 2401.

n = 2401

Careful!

If n had decimals, we would always round up.

Careful: a decimal n always rounds UP

5 Check

Let's check: with 2401 people the error bound is about 0.02.

That is not more than 0.02.

Good.

n = 2401: EBP ≈ 0.02 ≤ 0.02

With one person less, 2400, it is about 0.02: too big.

So 2401 is the smallest that works.

So it is right!

n = 2400: EBP ≈ 0.02 > 0.02

Let's look back.

First: find z*.

Then: the formula.

Then: put the numbers in.

Then: round up.

Last: check.

Answer