Step by step
The question
Question: the SD of all the numbers is 6: how big a sample do we need to be 95% confident with an error bound of 1.5?
The error bound is how far the sample mean may be from the true mean.
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
The error bound is z* times σ over the square root of n.
Solved for n, it gives this formula.
n is z* squared times σ squared, over the error bound squared.
3 Put the numbers in
1.96 squared is 3.8416.
6 squared is 36.
1.5 squared is 2.25.
3.8416 times 36 is 138.2976.
Divided by 2.25: 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.
Always round up.
5 Check
Let's check: with 62 people the error bound is about 1.4935.
That is not more than 1.5.
Good.
With one person less, 61, it is about 1.5057: too big.
So 62 is the smallest that works.
So it is right!
Let's look back.
First: find z*.
Then: the formula.
Then: put the numbers in.
Then: round up.
Last: check.