A poll finds that 54% of voters like the plan, with a margin of… Step by Step

Step by step

The question

Question: A poll finds that 54% of voters like the plan, with a margin of error of 5%.

Can we be sure that more than half of all voters like the plan?

1 What the margin means

54% comes from a sample: only some of the people.

The true value for everyone can be a bit different.

The margin of error, 5%, says how far it can be.

54% ± 5%

± means plus or minus: we go 5% down and 5% up.

The true value is likely in between.

± = plus or minus

true value: likely within 5% of 54%

2 Lowest and highest

The lowest likely value: 54% minus 5% is 49%.

lowest = 54% − 5% = 49%

The highest: 54% plus 5% is 59%.

highest = 54% + 5% = 59%

3 Draw it

A number line from 46 to 62: a bit of room on each side.

The likely values go from 49% to 59%.

We colour them green.

54%, the sample's answer, is right in the middle.

Half is 50%.

We mark it in red.

4 Inside or outside

The lowest likely value, 49%, is below half.

= 49% < 50%

So the true value could be 49%: not more than half.

49% is plausible too, and it is not more than half

Careful!

54% is only the sample's answer.

Everyone together could give less than half.

Careful: not 54% > 50% → sure ✗

So no: we can't be sure it is more than half.

= no, we can't be sure

5 Check

Let's check another way: how far is 50% from 54%?

54% minus 50% is 4%.

That is within the margin, 5%.

Half itself is plausible.

So it is right!

54% − 50% = 4%, 4% ≤ 5%

Let's look back.

First: what the margin means.

Then: lowest and highest.

Then: draw it.

Then: inside or outside.

Last: check.

Answer