Step by step
The question
Question: in a sample of 500, 250 agree: 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.
250 divided by 500 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 250.
Both counts, 250 and 250, 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.
0.5 times 0.5 is 0.25.
Divided by 500: 0.0005.
The square root of that is about 0.0224.
Times 1.96: about 0.0438.
4 The interval
Take the error bound away from the middle: 0.5 minus 0.0438 is 0.4562.
And add it: 0.5 plus 0.0438 is 0.5438.
The 95 percent confidence interval is (0.4562, 0.5438).
5 Say it in words
In words: we are 95 percent confident that the true proportion is between 45.62% and 54.38%.
6 Check
Let's check: the middle of the interval should be 0.5.
0.4562 plus 0.5438, divided by 2, is 0.5.
Half the width should be the error bound: 0.5438 minus 0.4562, divided by 2, is 0.0438.
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.