Sample size for a proportion: 95% confidence, error bound 0.04, p′ =… Step by Step

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The question

Question: how big a sample do we need to find a proportion, 95% confident, with an error bound of 0.04, if we guess p prime is 0.2?

n? 95% confidence, EBP = 0.04, p′ = 0.2

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

p prime is 0.2, so q prime is 1 minus 0.2: 0.8.

p′ = 0.2, q′ = 1 − 0.2 = 0.8

3 Put the numbers in

1.96 squared is 3.8416.

Times 0.2 times 0.8: 0.614656.

0.04 squared is 0.0016.

Dividing gives about 384.16.

n = 3.8416 × 0.2 × 0.80.04² = 0.6146560.0016 ≈ 384.16

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

384.16 rounds up to 385.

n = 385

Careful!

Rounding down to 384 gives an error bound a bit too big.

Always round up.

Careful: not n = 384 ✗

5 Check

Let's check: with 385 people the error bound is about 0.04.

That is not more than 0.04.

Good.

n = 385: EBP ≈ 0.04 ≤ 0.04

With one person less, 384, it is about 0.04: too big.

So 385 is the smallest that works.

So it is right!

n = 384: EBP ≈ 0.04 > 0.04

Let's look back.

First: find z*.

Then: the formula.

Then: put the numbers in.

Then: round up.

Last: check.

Answer