A survey says 40% with a margin of error of 4%. Is 45% a plausible… Step by Step

Step by step

The question

Question: A survey says 40% with a margin of error of 4%.

Is 45% a plausible value?

1 What the margin means

40% comes from a sample: only some of the people.

The true value for everyone can be a bit different.

The margin of error, 4%, says how far it can be.

40% ± 4%

± means plus or minus: we go 4% down and 4% up.

The true value is likely in between.

± = plus or minus

true value: likely within 4% of 40%

2 Lowest and highest

The lowest likely value: 40% minus 4% is 36%.

lowest = 40% − 4% = 36%

The highest: 40% plus 4% is 44%.

highest = 40% + 4% = 44%

3 Draw it

A number line from 34 to 48: a bit of room on each side.

The likely values go from 36% to 44%.

We colour them green.

40%, the sample's answer, is right in the middle.

Now 45%, the value in the question.

We mark it in red.

4 Inside or outside

45% is above 44%: outside the green part.

= 45% > 44%

Careful!

Close is not enough.

It must be inside the margin of error.

Careful: not 45% is close to 40% → plausible ✗

So no: 45% is not a plausible value.

= no, 45% is not plausible

5 Check

Let's check another way: how far is 45% from 40%?

45% minus 40% is 5%.

That is more than the margin, 4%.

So it is right!

45% − 40% = 5%, 5% > 4%

Let's look back.

First: what the margin means.

Then: lowest and highest.

Then: draw it.

Then: inside or outside.

Last: check.

Answer