Wiki Page, Math 13 - Perez Sandbox : Module 21 - Hypothesis Test for a Population Proportion (1b of 3)

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

Now let's look at the hypothesis test for a population proportion in more detail. As we already know, the first step of a hypothesis test is to determine the hypotheses. In the previous section, our hypotheses were in words. In this section, we use symbols. As before, we will use the symbol p for the population proportion.

EXAMPLE Health insurance coverage

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According to the Government Accountability Office, 80% of all college students ages 18 to 23 had health insurance coverage in 2006. (This is the population proportion in 2006.) The Patient Protection and Affordable Care Act passed in 2010 allowed young people under age 26 to stay on their parents' health insurance policy.

Has the proportion of college students ages 18 to 23 who have health insurance increased since 2006? We will conduct a hypothesis test to find out.

A survey of 800 randomly selected college students ages 18 to 23 indicated that 83% of them had health insurance coverage.

: (No change; the proportion of college students ages 18 to 23 who have health insurance is still 80%.)

: (The proportion of college students ages 18 to 23 who have health insurance is now greater than 80%.)

Note here that our hypothesis are not about the college population aged 18-23 in 2006. We do not need to state a hypothesis about the 2006 population because we already know the population proportion is 0.80. Rather, the hypotheses are about this year's population of college students aged 18-23.

The results of this year's survey (83% with health insurance this year) do not affect our hypotheses. We will use the sample results to determine whether to reject the null hypothesis in favor of the alternative hypothesis. More on that later.

EXAMPLE Internet access

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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 survey 500 randomly selected children ages 8 to 18 and find that 430 of them have Internet access at home.

The research question helps us form our hypotheses:

: (No change; the proportion of children with Internet access at home this year has not changed.)

: (The proportion of children with Internet access at home this year has changed since 2009.)

Note again that our hypothesis are not about the population of children in 2009. We do not need to state a hypothesis about the 2009 population because we already know the population proportion is 0.84. Rather, the hypotheses are about this year's population of children.

The results of this year's survey (430 out of 500 = 86% with Internet access this year) do not affect our hypotheses. We will use the sample results to determine whether to reject the null hypothesis in favor of the alternative hypothesis. More on that later.

EXAMPLE Jury selection

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Jefferson Parish is a suburb of New Orleans, Louisiana. Its population is about 23% African American.

Is there evidence that African Americans are underrepresented on juries in murder trials in Jefferson Parish?

According to a New York Times article (June 4, 2007), there were 18 murder trials in Jefferson Parish between 1986 and 2007 in which the ethnicity of the jurors was known. Ten of the juries had no black jurors, 7 juries had 1 black juror, and 1 jury had 2 black jurors. The research question helps us to form our hypotheses:

: p = 0.23 (No difference; the proportion of African Americans on juries in murder trials is the same as the proportion of African Americans in the population.)

: p < 0.23 (The proportion of African Americans on juries in murder trials is less than the proportion of African Americans in the population.)

Summary of Hypotheses

As a reminder, the null hypothesis is always a statement of equality. The alternative hypothesis is always a statement of inequality, using <, >, or ≠. So hypotheses take the form:

:

: or or

We use to represent the proportion from the null hypothesis.

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