Sample size for a proportion: 90% confidence, error bound 0.03 Step by Step

Step by step

The question

Question: how big a sample do we need to find a proportion, 90% confident, with an error bound of 0.03?

n? 90% confidence, EBP = 0.03

1 Find z*

90 percent of the bell is in the middle.

That leaves 10 percent: 5 percent in each end.

90% in the middle: 10% left over, 5% in each end

z* marks the cut between the middle and the ends.

Left of z* are the middle and one end: 0.9 plus 0.05 is 0.9500.

left of z* = 0.9 + 0.05 = 0.9500

Now we look for 0.9500 inside the table.

0.9500 is halfway between 0.9495 and 0.9505.

So z is halfway between 1.64 and 1.65: 1.645.

z* = 1.645

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

1.645 squared is about 2.706.

Times 0.5 times 0.5: 0.6765.

0.03 squared is 0.0009.

Dividing gives about 751.67.

n = 2.706 × 0.5 × 0.50.03² = 0.67650.0009 ≈ 751.67

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

751.67 rounds up to 752.

n = 752

Careful!

Rounding down to 751 gives an error bound a bit too big.

Always round up.

Careful: not n = 751 ✗

5 Check

Let's check: with 752 people the error bound is about 0.03.

That is not more than 0.03.

Good.

n = 752: EBP ≈ 0.03 ≤ 0.03

With one person less, 751, it is about 0.03: too big.

So 752 is the smallest that works.

So it is right!

n = 751: EBP ≈ 0.03 > 0.03

Let's look back.

First: find z*.

Then: the formula.

Then: put the numbers in.

Then: round up.

Last: check.

Answer