Step by step
The question
Question: a sample of 36 has mean 7.5, and the SD of all the numbers is 1.2: find the 99% confidence interval for the true mean.
x̄, the sample mean, is 7.5: the middle of our interval.
σ, the SD of all the numbers, is known: 1.2.
And n, the sample size, is 36.
The interval goes from the sample mean minus the error bound, EBM, to the sample mean plus the error bound.
1 Find z*
99 percent of the bell is in the middle.
That leaves 1 percent: 0.5 percent in each end.
z* marks the cut between the middle and the ends.
Left of z* are the middle and one end: 0.99 plus 0.005 is 0.9950.
Now we look for 0.9950 inside the table.
0.9950 is halfway between 0.9949 and 0.9951.
So z is halfway between 2.57 and 2.58: 2.575.
With more digits it is 2.576.
That is the number books use.
2 Error bound
The error bound: z* times σ over the square root of n.
The square root of 36 is 6.
1.2/6 is 0.2.
Times 2.576: 0.5152.
Careful!
We divide by the square root of 36, not by 36.
3 The interval
Take the error bound away from the middle: 7.5 minus 0.5152 is 6.9848.
And add it: 7.5 plus 0.5152 is 8.0152.
The 99 percent confidence interval is (6.9848, 8.0152).
4 Say it in words
In words: we are 99 percent confident that the true mean is between 6.9848 and 8.0152.
5 Check
Let's check: the middle of the interval should be 7.5.
6.9848 plus 8.0152, divided by 2, is 7.5.
Half the width should be the error bound: 8.0152 minus 6.9848, divided by 2, is 0.5152.
So it is right!
Let's look back.
First: find z*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.