Worked examples
The question
Question: A survey says 52% with a margin of error of 3%.
Is 50% a plausible value?
1 What the margin means
52% 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.
52% ± 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 52%
2 Lowest and highest
The lowest likely value: 52% minus 3% is 49%.
lowest = 52% − 3% = 49%
The highest: 52% plus 3% is 55%.
highest = 52% + 3% = 55%
3 Draw it
A number line from 47 to 57: a bit of room on each side.
The likely values go from 49% to 55%.
We colour them green.
52%, the sample's answer, is right in the middle.
Now 50%, the value in the question.
We mark it in red.
4 Inside or outside
50% is between 49% and 55%: inside the green part.
= 49% ≤ 50% ≤ 55%
Careful!
Plausible means likely.
It does not mean the true value is 50% for sure.
Careful: plausible = likely, not sure
So yes: 50% is a plausible value.
= yes, 50% is plausible
5 Check
Let's check another way: how far is 50% from 52%?
52% minus 50% is 2%.
That is within the margin, 3%.
So it is right!
52% − 50% = 2%, 2% ≤ 3%
Let's look back.
First: what the margin means.
Then: lowest and highest.
Then: draw it.
Then: inside or outside.
Last: check.
Answer = yes, 50% is plausible
▶ Watch it solved
The question
Question: The mean sleep in a sample is 6.2 hours with a margin of error of 0.4 hours.
Is 7 hours a plausible value for the mean?
1 What the margin means
6.2 hours comes from a sample: only some of the people.
The true value for everyone can be a bit different.
The margin of error, 0.4 hours, says how far it can be.
6.2 ± 0.4
± means plus or minus: we go 0.4 hours down and 0.4 hours up.
The true value is likely in between.
± = plus or minus
true value: likely within 0.4 of 6.2
2 Lowest and highest
The lowest likely value: 6.2 minus 0.4 is 5.8.
lowest = 6.2 − 0.4 = 5.8
The highest: 6.2 plus 0.4 is 6.6.
highest = 6.2 + 0.4 = 6.6
3 Draw it
A number line from 5.5 to 7.2: a bit of room on each side.
The likely values go from 5.8 to 6.6.
We colour them green.
6.2, the sample's answer, is right in the middle.
Now 7, the value in the question.
We mark it in red.
4 Inside or outside
7 is above 6.6: outside the green part.
= 7 > 6.6
Careful!
Close is not enough.
It must be inside the margin of error.
Careful: not 7 is close to 6.2 → plausible ✗
So no: 7 hours is not a plausible value.
= no, 7 is not plausible
5 Check
Let's check another way: how far is 7 from 6.2?
7 minus 6.2 is 0.8.
That is more than the margin, 0.4.
So it is right!
7 − 6.2 = 0.8, 0.8 > 0.4
Let's look back.
First: what the margin means.
Then: lowest and highest.
Then: draw it.
Then: inside or outside.
Last: check.
Answer = no, 7 is not plausible
▶ Watch it solved
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 = no, likely half or less
▶ Watch it solved
