Step by step
The question
Question: in a sample of 1000, 200 bike to work: find the 95% 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.
200 divided by 1000 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 200.
Both counts, 200 and 800, are at least 5.
So the normal bell works here.
2 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.
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 1000: 0.00016.
The square root of that is about 0.0126.
Times 1.96: about 0.0248.
4 The interval
Take the error bound away from the middle: 0.2 minus 0.0248 is 0.1752.
And add it: 0.2 plus 0.0248 is 0.2248.
The 95 percent confidence interval is (0.1752, 0.2248).
5 Say it in words
In words: we are 95 percent confident that the true proportion is between 17.52% and 22.48%.
6 Check
Let's check: the middle of the interval should be 0.2.
0.1752 plus 0.2248, divided by 2, is 0.2.
Half the width should be the error bound: 0.2248 minus 0.1752, divided by 2, is 0.0248.
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.