Sample size for a proportion: 90% confidence, error bound 0.02, p′ =… Step by Step

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The question

Question: how big a sample do we need to find a proportion, 90% confident, with an error bound of 0.02, if we guess p prime is 0.3?

n? 90% confidence, EBP = 0.02, p′ = 0.3

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

p prime is 0.3, so q prime is 1 minus 0.3: 0.7.

p′ = 0.3, q′ = 1 − 0.3 = 0.7

3 Put the numbers in

1.645 squared is about 2.706.

Times 0.3 times 0.7: 0.56826.

0.02 squared is 0.0004.

Dividing gives about 1420.65.

n = 2.706 × 0.3 × 0.70.02² = 0.568260.0004 ≈ 1420.65

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

1420.65 rounds up to 1421.

n = 1421

Careful!

Rounding down to 1420 gives an error bound a bit too big.

Always round up.

Careful: not n = 1420 ✗

5 Check

Let's check: with 1421 people the error bound is about 0.02.

That is not more than 0.02.

Good.

n = 1421: EBP ≈ 0.02 ≤ 0.02

With one person less, 1420, it is about 0.02: too big.

So 1421 is the smallest that works.

So it is right!

n = 1420: 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