Step by step
The question
Question: a sample of 50 has mean 12.5, and the SD of all the numbers is 3: find the 95% confidence interval for the true mean.
x̄, the sample mean, is 12.5: the middle of our interval.
σ, the SD of all the numbers, is known: 3.
And n, the sample size, is 50.
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.
3 over the square root of 50 is about 0.4243.
Times 1.96: about 0.8316.
Careful!
We divide by the square root of 50, not by 50.
3 The interval
Take the error bound away from the middle: 12.5 minus 0.8316 is 11.6684.
And add it: 12.5 plus 0.8316 is 13.3316.
The 95 percent confidence interval is (11.6684, 13.3316).
4 Say it in words
In words: we are 95 percent confident that the true mean is between 11.6684 and 13.3316.
5 Check
Let's check: the middle of the interval should be 12.5.
11.6684 plus 13.3316, divided by 2, is 12.5.
Half the width should be the error bound: 13.3316 minus 11.6684, divided by 2, is 0.8316.
So it is right!
Let's look back.
First: find z*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.