Step by step
The question
Question: in a sample of 400, 80 walk to school: find the 99% 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.
80 divided by 400 is 0.2.
q prime is the rest: 1 minus 0.2 is 0.8.
Careful!
p prime is the fraction 0.2, not the count 80.
Both counts, 80 and 320, are at least 5.
So the normal bell works here.
2 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.
3 Error bound
The error bound for a proportion: z* times the square root of p prime times q prime over n.
0.2 times 0.8 is 0.16.
Divided by 400: 0.0004.
The square root of that is about 0.02.
Times 2.576: about 0.0515.
4 The interval
Take the error bound away from the middle: 0.2 minus 0.0515 is 0.1485.
And add it: 0.2 plus 0.0515 is 0.2515.
The 99 percent confidence interval is (0.1485, 0.2515).
5 Say it in words
In words: we are 99 percent confident that the true proportion is between 14.85% and 25.15%.
6 Check
Let's check: the middle of the interval should be 0.2.
0.1485 plus 0.2515, divided by 2, is 0.2.
Half the width should be the error bound: 0.2515 minus 0.1485, divided by 2, is 0.0515.
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.