Step by step
The question
Question: a sample of 16 has mean 20, and the SD of all the numbers is 4: find the 95% confidence interval for the true mean.
x̄, the sample mean, is 20: the middle of our interval.
σ, the SD of all the numbers, is known: 4.
And n, the sample size, is 16.
The interval goes from the sample mean minus the error bound, EBM, to the sample mean plus the error bound.
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 Error bound
The error bound: z* times σ over the square root of n.
The square root of 16 is 4.
4/4 is 1.
Times 1.96: 1.96.
Careful!
We divide by the square root of 16, not by 16.
3 The interval
Take the error bound away from the middle: 20 minus 1.96 is 18.04.
And add it: 20 plus 1.96 is 21.96.
The 95 percent confidence interval is (18.04, 21.96).
4 Say it in words
In words: we are 95 percent confident that the true mean is between 18.04 and 21.96.
5 Check
Let's check: the middle of the interval should be 20.
18.04 plus 21.96, divided by 2, is 20.
Half the width should be the error bound: 21.96 minus 18.04, divided by 2, is 1.96.
So it is right!
Let's look back.
First: find z*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.