A pollster wants a 95% confidence interval for a proportion with a… 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 0.04?

n? 95% confidence, EBP = 0.04

There are four choices.

First we solve it, the normal way.

Then we pick.

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

The error bound of a proportion is z* times the square root of p prime times q prime over n.

We solve it for n.

EBP = z* × √p′ × q′n

To remove the square root, we square both sides.

EBP² = z*² × p′ × q′n

n is on the bottom.

We multiply both sides by n, and divide both sides by EBP squared: 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: p prime times q prime is biggest then, 0.5 times 0.5 is 0.25, while 0.4 times 0.6 is only 0.24.

The biggest n 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.

0.04 squared is 0.0016.

Dividing gives about 600.25.

n = 3.8416 × 0.5 × 0.50.04² = 0.96040.0016 ≈ 600.25

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

600.25 rounds up to 601.

n = 601

Careful!

Rounding down to 600 gives an error bound a bit too big: the error bound divides by the root of n, so a smaller n makes it bigger.

Always round up.

The right way: n = 601.

Careful: not n = 600 ✗

right: n = 601 ✓

5 Check

Let's check: with 601 people the error bound is about 0.04.

That is not more than 0.04.

Good.

n = 601: EBP ≈ 0.04 ≤ 0.04

With one person less, 600, it is about 0.04: too big.

So 601 is the smallest that works.

So it is right!

n = 600: EBP ≈ 0.04 > 0.04

6 Pick the choice

Our answer is n = 601.

That is choice C.

C) 601

7 Why not the others

A is 600.

Wrong: rounded down: always round up.

That is the Careful mistake from before.

A) 600 ✗ rounded down: always round up

B is 25.

Wrong: forgot to square.

B) 25 ✗ forgot to square

D is 423.

Wrong: used 1.645: that is for 90%.

D) 423 ✗ used 1.645: that is for 90%

Let's look back.

First: find z*.

Then: the formula.

Then: put the numbers in.

Then: round up.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer