Sample size: 90% confidence, sd 5, error bound 1 Step by Step

Step by step

The question

Question: the SD of all the numbers is 5: how big a sample do we need to be 90% confident with an error bound of 1?

n? 90% confidence, σ = 5, EBM = 1

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*

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

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

1.645 squared is about 2.706.

5 squared is 25.

1 squared is 1.

2.706 times 25 is 67.65.

Divided by 1: 67.65.

n = 1.645² × 5²1² = 2.706 × 251 = 67.651 = 67.65

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

67.65 rounds up to 68.

n = 68

Careful!

Rounding down to 67 gives an error bound a bit too big.

Always round up.

Careful: not n = 67 ✗

5 Check

Let's check: with 68 people the error bound is about 0.9974.

That is not more than 1.

Good.

n = 68: EBM ≈ 0.9974 ≤ 1

With one person less, 67, it is about 1.0048: too big.

So 68 is the smallest that works.

So it is right!

n = 67: EBM ≈ 1.0048 > 1

Let's look back.

First: find z*.

Then: the formula.

Then: put the numbers in.

Then: round up.

Last: check.

Answer