Step by step
The question
Question: in a sample of 250, 70 cycle 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.
70 divided by 250 is 0.28.
q prime is the rest: 1 minus 0.28 is 0.72.
Careful!
p prime is the fraction 0.28, not the count 70.
Both counts, 70 and 180, 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.
0.28 times 0.72 is 0.2016.
Divided by 250: 0.0008064.
The square root of that is about 0.0284.
Times 1.96: about 0.0557.
4 The interval
Take the error bound away from the middle: 0.28 minus 0.0557 is 0.2243.
And add it: 0.28 plus 0.0557 is 0.3357.
The 95 percent confidence interval is (0.2243, 0.3357).
5 Say it in words
In words: we are 95 percent confident that the true proportion is between 22.43% and 33.57%.
6 Check
Let's check: the middle of the interval should be 0.28.
0.2243 plus 0.3357, divided by 2, is 0.28.
Half the width should be the error bound: 0.3357 minus 0.2243, divided by 2, is 0.0557.
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.