Confidence Intervals For Proportions and
Choosing the Sample Size
A Large Sample Confidence Interval for a Population Proportion
Recall that a confidence interval for a population mean is given by
We can make a similar construction for a confidence interval for a population
proportion. Instead of x, we can use
p and instead of s, we use
, hence, we can write the confidence interval for a large sample
1000 randomly selected Americans were asked if they believed the minimum wage should be raised. 600 said yes. Construct a 95% confidence interval for the proportion of Americans who believe that the minimum wage should be raised.
p = 600/1000 = .6 zc = 1.96 and n = 1000
Hence we can conclude that between 57 and 63 percent of all Americans agree with the proposal. In other words, with a margin of error of .03 , 60% agree.
Calculating n for Estimating a Mean
Suppose that you were interested in the average number of units that students take at a two year college to get an AA degree. Suppose you wanted to find a 95% confidence interval with a margin of error of .5 for m knowing s = 10. How many people should we ask?
Solving for n in
Margin of Error = E = zc s/
E = zcs
Squaring both sides, we get
We use the formula:
A Subaru dealer wants to find out the age of their customers (for advertising purposes). They want the margin of error to be 3 years old. If they want a 90% confidence interval, how many people do they need to know about?
E = 3, zc = 1.65
but there is no way of finding sigma exactly. They use the following reasoning: most car customers are between 16 and 68 years old hence the range is
Range = 68 - 16 = 52
The range covers about four standard deviations hence one standard deviation is about
s @ 52/4 = 13
We can now calculate n:
Hence the dealer should survey at least 52 people.
Finding n to Estimate a Proportion
Suppose that you are in charge to see if dropping a computer will damage it. You want to find the proportion of computers that break. If you want a 90% confidence interval for this proportion, with a margin of error of 4%, How many computers should you drop?
The formula states that
Squaring both sides, we get that
zc2 p(1 - p)
Multiplying by n, we get
nE2 = zc2[p(1 - p)]
This is the formula for finding n.
Since we do not know p, we use .5 ( A conservative estimate)
We round 425.4 up for greater accuracy
We will need to drop at least 426 computers. This could get expensive.