CK-12 Probability and Statistics - Advanced

7.4. Confidence Intervals http://www.ck12.org

proportion. This is shown in the diagram below:

Therefore, if a single sample proportion is within 1. 96

√

p( 1 −p)

n

of the population proportion, then the interval

pˆ− 1. 96

√

p( 1 −p)

n

to ˆp+ 1. 96

√

p( 1 −p)

n

will capture the population proportion. This will happen for 95% of

all possible samples. If you look at the above formulas, you should notice that the population proportion(p)and
the sample proportion(pˆ)are both used to calculate the confidence interval. However, in real-life situations, the
population proportion is seldom known. Therefore,(p)is most often replaced with(pˆ)in the formulas above so that
they now become:

pˆ− 1. 96

√

pˆ( 1 −pˆ)

n

and ˆp+ 1. 96

√

pˆ( 1 −pˆ)

n

or in a more standard formp±z

√

pˆ( 1 −pˆ)

n

There are two restrictions

that apply to this formula: 1)n p≥5 and 2)n( 1 −p)≥5.

As before, the margin of error isz

√

pˆ( 1 −pˆ)

n

and the confidence interval is ˆp±the margin of error.

Example:

A large grocery store has been recording data regarding the number of shoppers that use savings coupons at their
outlet. Last year it was reported that 77% of all shoppers used coupons, and these results were considered accurate
within 2.9%, 19 times out of 20.

a) Are you dealing with a 90%,95% or 99% confidence level?

b) What is the margin of error?

c) Calculate the confidence interval.

d) Explain the meaning of the confidence interval.

Solution:

a) The statement 19 times out of 20 indicates that you are dealing with a 95% confidence interval.

b) The results were accurate within 2.9%, so the margin of error is 2.9%.

c) The confidence interval is simply ˆp±the margin of error.