Step by step
The question
Question: a sample of 36 has mean 68, and the SD of all the numbers is 3: find the 90% confidence interval for the true mean.
x̄, the sample mean, is 68: the middle of our interval.
σ, the SD of all the numbers, is known: 3.
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*
90 percent of the bell is in the middle.
That leaves 10 percent: 5 percent in each end.
z* marks the cut between the middle and the ends.
Left of z* are the middle and one end: 0.9 plus 0.05 is 0.9500.
Now we look for 0.9500 inside the table.
0.9500 is halfway between 0.9495 and 0.9505.
So z is halfway between 1.64 and 1.65: 1.645.
2 Error bound
The error bound: z* times σ over the square root of n.
The square root of 36 is 6.
3/6 is 0.5.
Times 1.645: 0.8225.
Careful!
We divide by the square root of 36, not by 36.
3 The interval
Take the error bound away from the middle: 68 minus 0.8225 is 67.1775.
And add it: 68 plus 0.8225 is 68.8225.
The 90 percent confidence interval is (67.1775, 68.8225).
4 Say it in words
In words: we are 90 percent confident that the true mean is between 67.1775 and 68.8225.
5 Check
Let's check: the middle of the interval should be 68.
67.1775 plus 68.8225, divided by 2, is 68.
Half the width should be the error bound: 68.8225 minus 67.1775, divided by 2, is 0.8225.
So it is right!
Let's look back.
First: find z*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.