Wiki Page, Math 13 - Perez Sandbox : Module 21 - Hypothesis Test for a Population Proportion (3a of 3)

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2026-01-17T21:15:33-08:00

Interpreting the P-Value

Previously, we worked on interpreting the P-value for a one-tailed hypothesis test. Now we will work on interpreting the P-value for a two-tailed test.

EXAMPLE Health insurance coverage

Recall this example from a previous page. According to the Kaiser Family Foundation, 84% of U.S. children ages 8 to 18 had Internet access at home as of August 2009. Researchers wonder if this percentage has changed since then. They test the following hypotheses: : and : , where p is the proportion of children ages 8 to 18 with Internet access at home now.

They survey 500 randomly selected children (ages 8 to 18) and find that 430 of them have Internet access at home; therefore, the observed sample proportion is 86% (430 out of 500.)

The standard error is 0.0164. The P-value is 0.22.

What does the P-value tell us?

Let's start by thinking about the sampling distribution to make sure we understand what 0.22 describes.

The 0.22 is the area in the two tails combined (the white area). What does this area describe? Area is probability, so 0.22 is a probability.

Now think about the X-values, i.e. the sample proportions that correspond to this probability. These sample proportions are less than 0.82 and greater than 0.86. In other words, the sample proportions deviate more than 0.02 from the center of the sampling distribution in both directions.

What is the center? The center is the hypothesized population proportion from the null hypothesis.

Now let's put all of these observations together:

In random sampling from a population of children in which 84% have Internet access, there is a 22% probability that a sample of 500 children deviates more than 2% from the population proportion.

Here other ways to interpret this P-value:

Sample proportions with more than 0.02 error will occur about 22% of the time by chance alone in random sampling from a population with p = 0.84 and n = 500.

If the null hypothesis is true (84% of children have Internet access at home), then estimates from random samples of 500 children will have on average 1.64% error (standard error is 1.64%.) Data we collected from our sample had 2% error. In this context, error this large or larger will occur about 22% of the time in random sampling.

EXAMPLE Wrong ways to interpret the P-value

Sometimes it is helpful to look at wrong answers. Here we share some common misinterpretations of the P-value and explain why each one is incorrect.

There is a 22% chance that the proportion of children with Internet access is equal to 0.84.

This statement says in essence that there is a 22% chance that the null hypothesis is correct. This is wrong because probability always involves repeated random events, such as random sampling or random assignment or flipping a coin. The null hypothesis being right or wrong is not a random event. The null hypothesis is either right or it is wrong; it cannot have a 22% chance of being right or wrong.

For a similar reason the following interpretation of the P-value is also incorrect: There is a 22% chance that the proportion of children with internet access is not equal to 0.84.

If the null hypothesis is true (84% of children have Internet access at home), then estimates from random samples of 500 children will not equal 84% about 22% of the time in random sampling.

This an incorrect interpretation of the P-value because it says in essence that 22% of the time sample proportions will not equal 84% (the center of the sampling distribution.) This is clearly wrong because it does not take into account the evidence that we actually gathered, which is the observed sample proportion of 86%. The P-value is a measure of the strength of the evidence against the null, so it must include the observed sample proportion.

If the null hypothesis is true (84% of children have Internet access at home), then random samples of 500 children will have an error 22% of the time.

This interpretation has a problem similar to the previous statement. It does not mention the evidence we actually gathered, which is the observed sample proportion of 86%. In addition, all random samples with sample proportions not equal to 84% have error because error is the amount the sample proportion deviates from the hypothesized population proportion of 84%. Therefore, it is incorrect to say that random samples have error only 22% of the time.

Summary

The P-value is a conditional probability where the condition is that the null hypothesis is assumed to be true.

If the population has the proportion stated in the null hypothesis, the sample proportions from random samples will vary with an average error given by the standard error. The P-value describes the likelihood of getting sample proportions with more error than observed in the data collected from our sample.

Another way to say this: The P-value is the probability that sample results are more extreme than the data if the null hypothesis is true. The phrase “as extreme as or more extreme than” means farther from the center of the sampling distribution in the direction of the alternative hypothesis for a one-tailed test and in both directions for a two-tailed test.

More generally, we view the P-value a description of the strength of the evidence against the null hypothesis and in support of the alternative hypothesis. But the P-value is a probability statement about random samples coming from the population described by the null hypothesis; it is not a probability statement about the whether the null hypothesis is true.

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