Step by step
The question
Question: in a sample of 820, 410 agree: find the 90% confidence interval for the true proportion.
1 Name the numbers
p prime is the part of the sample that says yes: x divided by n.
410 divided by 820 is 0.5.
q prime is the rest: 1 minus 0.5 is 0.5.
Careful!
p prime is the fraction 0.5, not the count 410.
Both counts, 410 and 410, are at least 5.
So the normal bell works here.
2 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.
3 Error bound
The error bound for a proportion: z* times the square root of p prime times q prime over n.
0.5 times 0.5 is 0.25.
Divided by 820: about 0.00030488.
The square root of that is about 0.0175.
Times 1.645: about 0.0287.
4 The interval
Take the error bound away from the middle: 0.5 minus 0.0287 is 0.4713.
And add it: 0.5 plus 0.0287 is 0.5287.
The 90 percent confidence interval is (0.4713, 0.5287).
5 Say it in words
In words: we are 90 percent confident that the true proportion is between 47.13% and 52.87%.
6 Check
Let's check: the middle of the interval should be 0.5.
0.4713 plus 0.5287, divided by 2, is 0.5.
Half the width should be the error bound: 0.5287 minus 0.4713, divided by 2, is 0.0287.
So it is right!
Let's look back.
First: name the numbers.
Then: find z*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.