There are many times when a proportion from a population is required. For example, the proportion of people who may vote for a particular candidate in an election or the percentage of people watching a particular TV channel.

To estimate π a proportion from a population, the proportion from a sample, p, can be used as a point estimate of π

As with estimating the mean, a range of values with some level of probability or confidence that the population parameters will lie within this range is a better way to estimate the population proportion.

This range of values is called a confidence interval for proportions.

Degree of Confidence

The degree of confidence is given as a percentage probability changed into a z-score, using the standard normal distribution tables. Check out where the z-scores are found in the tables for the following degrees of confidence.

Any degree of confidence can be found in this manner. e.g. For an 80% level of confidence look up 0.40 in the tables.

Confidence Intervals for Proportions

The confidence interval, based on the Central Limit Theorem, for the population proportion π is given by:

p ± z. Y12_Confidence_Intervals_for_Proportions_02.gif   

Where π is the population proportion. As π is often unknown, p the sample proportion can be used instead.

The value of z will depend upon the degree of confidence required. The z values for 90, 95 and 99% are shown above.

Y12_Confidence_Intervals_for_Proportions_02.gif is called the standard error of the proportion.

In opinion polls z.Y12_Confidence_Intervals_for_Proportions_02.gif is often called the margin of error.



In a survey before an election, a poll (sample) was taken of 300 potential voters. 120 said they would vote for Candidate A. Calculate a 95% confidence interval for the proportion of voters who would vote for Candidate A in the election.

From the poll p = 120 / 300 = 0.4.

The 95% confidence interval is:

p ± z. Y12_Confidence_Intervals_for_Proportions_03.gif

= 0.4 ± 1.96 Y12_Confidence_Intervals_for_Proportions_04.gif
=0.4 ± 0.0554

= (0.3446, 0.4554)

The 95% confidence interval for π is ( (0.3446, 0.4554) or between 34.46% and 45.54%

This is sometimes written 0.3446 < π < 0.4554

Note The margin of error for the poll is 0.0554 or 5.54%

Meaning of Confidence Intervals

The answer in the example above means that 95% of such confidence intervals will contain the population proportion.