Step by step
The question
Question: in a sample of 500, 421 have a phone: 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.
421 divided by 500 is 0.842.
q prime is the rest: 1 minus 0.842 is 0.158.
Careful!
p prime is the fraction 0.842, not the count 421.
Both counts, 421 and 79, 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.
0.842 times 0.158 is 0.133.
Divided by 500: about 0.00026607.
The square root of that is about 0.0163.
Times 1.96: about 0.032.
4 The interval
Take the error bound away from the middle: 0.842 minus 0.032 is 0.81.
And add it: 0.842 plus 0.032 is 0.874.
The 95 percent confidence interval is (0.81, 0.874).
5 Say it in words
In words: we are 95 percent confident that the true proportion is between 81% and 87.4%.
6 Check
Let's check: the middle of the interval should be 0.842.
0.81 plus 0.874, divided by 2, is 0.842.
Half the width should be the error bound: 0.874 minus 0.81, divided by 2, is 0.032.
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.