Step by step
The question
Question: a sample of 25 has mean 30, and the SD of all the numbers is 5: find the 99% confidence interval for the true mean.
x̄, the sample mean, is 30: the middle of our interval.
σ, the SD of all the numbers, is known: 5.
And n, the sample size, is 25.
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 25 is 5.
5/5 is 1.
Times 2.576: 2.576.
Careful!
We divide by the square root of 25, not by 25.
3 The interval
Take the error bound away from the middle: 30 minus 2.576 is 27.424.
And add it: 30 plus 2.576 is 32.576.
The 99 percent confidence interval is (27.424, 32.576).
4 Say it in words
In words: we are 99 percent confident that the true mean is between 27.424 and 32.576.
5 Check
Let's check: the middle of the interval should be 30.
27.424 plus 32.576, divided by 2, is 30.
Half the width should be the error bound: 32.576 minus 27.424, divided by 2, is 2.576.
So it is right!
Let's look back.
First: find z*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.