A survey finds that 46% of people agree, with a margin of error of… Step by Step

Step by step

The question

Question: A survey finds that 46% of people agree, with a margin of error of 3%.

Can we be sure that more than half of all people agree?

1 What the margin means

46% comes from a sample: only some of the people.

The true value for everyone can be a bit different.

The margin of error, 3%, says how far it can be.

46% ± 3%

± means plus or minus: we go 3% down and 3% up.

The true value is likely in between.

± = plus or minus

true value: likely within 3% of 46%

2 Lowest and highest

The lowest likely value: 46% minus 3% is 43%.

lowest = 46% − 3% = 43%

The highest: 46% plus 3% is 49%.

highest = 46% + 3% = 49%

3 Draw it

A number line from 41 to 52: a bit of room on each side.

The likely values go from 43% to 49%.

We colour them green.

46%, the sample's answer, is right in the middle.

Half is 50%.

We mark it in red.

4 Inside or outside

Even the highest likely value, 49%, is not more than half.

= 49% ≤ 50%

Careful!

Every likely value is half or less.

Very likely, but still not 100% sure.

Careful: plausible = likely, not sure

So no: it is likely half or less.

= no, likely half or less

5 Check

Let's check another way: how far is 50% from 46%?

50% minus 46% is 4%.

That is more than the margin, 3%.

Half is not plausible.

So it is right!

50% − 46% = 4%, 4% > 3%

Let's look back.

First: what the margin means.

Then: lowest and highest.

Then: draw it.

Then: inside or outside.

Last: check.

Answer