Sample size: 95% confidence, sd 2.5, error bound 0.5 Step by Step

Step by step

The question

Question: the SD of all the numbers is 2.5: how big a sample do we need to be 95% confident with an error bound of 0.5?

n? 95% confidence, σ = 2.5, EBM = 0.5

The error bound is how far the sample mean may be from the true mean.

EBM: how far the sample mean may be from the true mean

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 is z* times σ over the square root of n.

Solved for n, it gives this formula.

from EBM = z* × σ ÷ √n, solved for n

n is z* squared times σ squared, over the error bound squared.

n = z*² × σ²EBM²

3 Put the numbers in

1.96 squared is 3.8416.

2.5 squared is 6.25.

0.5 squared is 0.25.

3.8416 times 6.25 is 24.01.

Divided by 0.25: 96.04.

n = 1.96² × 2.5²0.5² = 3.8416 × 6.250.25 = 24.010.25 = 96.04

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

96.04 rounds up to 97.

n = 97

Careful!

Rounding down to 96 gives an error bound a bit too big.

Always round up.

Careful: not n = 96 ✗

5 Check

Let's check: with 97 people the error bound is about 0.4975.

That is not more than 0.5.

Good.

n = 97: EBM ≈ 0.4975 ≤ 0.5

With one person less, 96, it is about 0.5001: too big.

So 97 is the smallest that works.

So it is right!

n = 96: EBM ≈ 0.5001 > 0.5

Let's look back.

First: find z*.

Then: the formula.

Then: put the numbers in.

Then: round up.

Last: check.

Answer