Confidence Intervals for Population Proportions (College Board AP® Statistics)

One-sample z-interval for population proportion

What is a confidence interval for the population proportion?

• A confidence interval for the population proportion is

  • a symmetric range of values centered about a sample proportion

  • designed to capture the actual value of the population proportion

• Different samples generate different confidence intervals

  • e.g. a sample proportion of 0.3 may have a confidence interval of (0.25, 0.35)

  • while another sample proportion of 0.3 may have a confidence interval of (0.29, 0.31)

How do I calculate a confidence interval for the population proportion?

• The confidence interval for the population proportion is given by

  • 

• Where:

  • The sample proportion, , is calculated from the sample

  • The critical value is the relevant z-value

    • The critical value depends on the confidence level C%

  • The standard error of the sample proportion is an estimate of the population standard deviation from the data

    • given by

    • where is the size of the sample

What are the conditions for a confidence interval for the population proportion?

• When calculating a z-interval for a population proportion, you must show that it meets the following conditions:

  • Items in the sample (or experiment) must satisfy the independence condition

    • by verifying that data is collected by random sampling

    • or random assignment (in an experiment)

    • and, if sampling without replacement, showing that the sample size is less than 10% of the population size

  • The sample size is large enough such that the sampling distribution of is approximately a normal distribution

    • by verifying that the expected number of successes, , and expected number of failures, , are both at least 10:

    • 

    • 

What is the margin of error?

• The margin of error is the half-width of the confidence interval

  • 

• The confidence interval is

  • 

• The total width of a confidence interval is

• You may be given an interval and asked to calculate its margin of error

  • or another value, such as

    • This involves forming and solving an equation

    • If no value of is known, use (as this gives the upper bound of )

How do I interpret a confidence interval for a population proportion?

• You must conclude calculations of a confidence interval by referring to the context

  • Start by saying 'we can be C% confident that the interval from [lower limit] to [upper limit]...'

    • using the limits from the confidence interval

  • then end with it capturing the population proportion in context

    • e.g. 'captures the actual proportion of students in the school who are left-handed'

How do I use confidence intervals to justify a claim about a population proportion?

• If a population proportion is claimed to be a specific value

  • check if that value lies in your confidence interval

• If it does, the sample data provides sufficient evidence that the population proportion is that value

  • If it does not, the sample data does not provide sufficient evidence that the population proportion is that value