Hypothesis Tests for Population Proportions (College Board AP® Statistics)

One-sample z-test for population proportion

What is a z-test for the population proportion?

• A z-test for the population proportion is used to test whether the population proportion, , has changed

  • A random sample of individuals from the population with a sample proportion of is used to try to prove the case

What are the hypotheses for a z-test for the population proportion?

• The null hypothesis, , is the assumption that the population proportion has not changed

  • e.g. The population proportion has the fixed value

    • It is assumed to be correct, unless evidence proves otherwise

    • Note that if then both success and failure are equally likely (there is no preference)

• The alternative hypothesis, , is how you think the population proportion has changed

  • e.g. or or

What are the conditions for a z-test for the population proportion?

• When performing a z-test 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:

    • 

    • 

    • where is the population proportion in the null hypothesis,

How do I calculate the standardized test statistic?

• You need a measure of how far the sample proportion is from the population proportion

  • This is the standardized test statistic

• The standardized test statistic for sample proportion is a z-score given by:

  • 

  • where is the sample proportion, is the population proportion under the null hypothesis, and is the standard error of the sample proportion

    • and where is the sample size

• The standardized test statistic shows how many standard errors the sample proportion is from the population proportion

How do I calculate the p-value?

• The p-value is the probability of obtaining a test statistic as extreme, or more extreme, than the one observed in the sample, assuming the null hypothesis is true

• Use the standard normal distribution, , to calculate the probability of being in the extreme region (tail) that extends from the z-score given by

  • You can use either the z-tables or a calculator to find this probability

• For a two-tail test, remember to work out the total probability across both tails

  • You can double the p-value from a one-tail test

How do I conclude a hypothesis test?

• Conclusions to a hypothesis test need to show two things:

  • a decision about the null hypothesis

  • an interpretation of this decision in the context of the question

• To make the decision, compare the p-value to the significance level

  • If then the null hypothesis should be rejected

  • If then the null hypothesis should not be rejected

• In a two-tailed test, remember to double the p-value and compare this to

What are the steps for performing a z-test for the population proportion on a calculator?

• When using a calculator to conduct a z-test for a population proportion, you must still write down all steps of the hypothesis testing process:

  • State the null and alternative hypotheses and clearly define your parameter

  • Describe the test being used and show that the situation meets the conditions required

  • Calculate the standardized test statistic (z-score)

  • Calculate the p-value using your calculator

  • Compare the p-value to the significance level

  • Write down the conclusion to the test and interpret it in the context of the problem