Step by step
The question
Question: the SD of all the numbers is 12: how big a sample do we need to be 95% confident with an error bound of 3?
The error bound is how far the sample mean may be from the true mean.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find z*
95 percent of the bell is in the middle.
That leaves 5 percent: 2.5 percent in each end.
z* marks the cut between the middle and the ends.
Left of z* are the middle and one end: 0.95 plus 0.025 is 0.9750.
Now we look for 0.9750 inside the table.
0.9750 is in the table: row 1.9, column .06.
So z = 1.96.
2 The formula
We plan the sample before we collect any data, so we need n.
The error bound is z* times σ over the square root of n.
We solve it for n.
The square root of n is on the bottom.
We multiply both sides by it, and divide both sides by EBM.
To remove the square root, we square both sides.
Each part squared: n is z* squared times σ squared, over the error bound squared.
3 Put the numbers in
1.96 squared is 3.8416.
12 squared is 144.
3 squared is 9.
3.8416 times 144 is 553.1904.
Divided by 9: about 61.47.
4 Round up
A sample size must be a whole number.
We always round up: a smaller sample would miss the goal.
61.47 rounds up to 62.
Careful!
Rounding down to 61 gives an error bound a bit too big: the error bound divides by the root of n, so a smaller n makes it bigger.
Always round up.
The right way: n = 62.
5 Check
Let's check: with 62 people the error bound is about 2.987.
That is not more than 3.
Good.
With one person less, 61, it is about 3.0114: too big.
So 62 is the smallest that works.
So it is right!
6 Pick the choice
Our answer is n = 62.
That is choice B.
7 Why not the others
A is 8.
Wrong: forgot to square: (1.96 × 12/3)².
C is 61.
Wrong: rounded down: always round up.
That is the Careful mistake from before.
D is 16.
Wrong: used (12/3)² without 1.96.
Let's look back.
First: find z*.
Then: the formula.
Then: put the numbers in.
Then: round up.
Then: check.
Then: pick the choice.
Last: why not the others.