Step by step
The question
Question: in a sample of 100, 45 say yes: 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.
45 divided by 100 is 0.45.
q prime is the rest: 1 minus 0.45 is 0.55.
Careful!
p prime is the fraction 0.45, not the count 45.
Both counts, 45 and 55, 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.
0.45 times 0.55 is 0.2475.
Divided by 100: 0.002475.
The square root of that is about 0.0498.
Times 1.96: about 0.0975.
4 The interval
Take the error bound away from the middle: 0.45 minus 0.0975 is 0.3525.
And add it: 0.45 plus 0.0975 is 0.5475.
The 95 percent confidence interval is (0.3525, 0.5475).
5 Say it in words
In words: we are 95 percent confident that the true proportion is between 35.25% and 54.75%.
6 Check
Let's check: the middle of the interval should be 0.45.
0.3525 plus 0.5475, divided by 2, is 0.45.
Half the width should be the error bound: 0.5475 minus 0.3525, divided by 2, is 0.0975.
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.