AP Statistics 3.5 Setting Up a Test for a Population Proportion Study Notes - New Syllabus

3.5.B.1 Null and Alternative Hypotheses for a Population Proportion

Every hypothesis test begins by stating two competing hypotheses about the population proportion.

• The null hypothesis, denoted by \(H_0\), is the statement that is assumed to be true unless the sample data provide convincing evidence otherwise. It represents the status quo, the current belief, or the claim being tested.
• The alternative hypothesis, denoted by \(H_a\), is the statement for which evidence is being collected. It represents the researcher’s claim or the belief that differs from the null hypothesis.

During a hypothesis test, the sample data are used to determine whether there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.

Roles of the Two Hypotheses

Example 1

A company claims that 75% of its customers are satisfied with its service.

A researcher believes the actual satisfaction rate is different from 75%.

Interpretation:

• The company’s claim represents the null hypothesis.
• The researcher’s belief represents the alternative hypothesis.

Example 2

A school believes that 60% of students participate in extracurricular activities.

A principal wants to investigate whether participation has increased.

Interpretation:

• The current belief (60%) is the null hypothesis.
• The claim that participation has increased is the alternative hypothesis.

Important AP Exam Notes

• The null hypothesis represents the status quo or current belief.
• The alternative hypothesis represents the claim being investigated.
• Evidence is collected to determine whether there is convincing evidence against the null hypothesis.
• Hypothesis tests always compare the sample data with what would be expected if the null hypothesis were true.
• Decisions are made about the null hypothesis, not about proving the alternative hypothesis.

Common AP Exam Mistakes

3.5.B.2 Writing Null and Alternative Hypotheses

When writing hypotheses for a one-sample z-test for a population proportion, the null hypothesis always contains an equality.

The equality may be written as \(=\), \(\le\), or \(\ge\). However, in AP Statistics, the null hypothesis is always tested at the boundary of equality.

The alternative hypothesis expresses the researcher’s claim and determines whether the test is one-sided or two-sided.

• If the alternative hypothesis uses \(<\) or \(>\), the test is one-sided.
• If the alternative hypothesis uses \(\neq\), the test is two-sided.

Forms of Hypotheses

Writing the Null Hypothesis

Although the original claim may use \(\ge\) or \(\le\), the null hypothesis is always written using the equality value.

For example:

• “At least 70%” → \(H_0:p=0.70\)
• “No more than 45%” → \(H_0:p=0.45\)
• “Exactly 60%” → \(H_0:p=0.60\)

Example 1

A restaurant claims that 80% of customers are satisfied with their service.

A customer believes that the true satisfaction rate is less than 80%.

Hypotheses:

\(H_0:p=0.80\)

\(H_a:p<0.80\)

Type of Test: Left-tailed (one-sided)

Example 2

A school reports that 65% of students participate in extracurricular activities.

A researcher wants to determine whether the true proportion is different from 65%.

Hypotheses:

\(H_0:p=0.65\)

\(H_a:p\neq0.65\)

Type of Test: Two-sided

Important AP Exam Notes

• The null hypothesis always contains the equality.
• The alternative hypothesis never contains an equality symbol.
• The wording of the research question determines the alternative hypothesis.
• \(<\) or \(>\) in the alternative hypothesis indicates a one-sided test.
• \(\neq\) in the alternative hypothesis indicates a two-sided test.
• Always write hypotheses using the population proportion \(p\), never the sample proportion \(\hat{p}\).

Common AP Exam Mistakes

3.5.B.3 Writing Hypotheses for a One-Sample z-Test for a Population Proportion

For a one-sample z-test for a population proportion, the hypotheses are written using the population proportion \(p\) and the hypothesized value \(p_0\).

The value \(p_0\) is the claimed or hypothesized population proportion that is assumed to be true under the null hypothesis.

• The null hypothesis always states that the population proportion is equal to the hypothesized value.
• The alternative hypothesis depends on the research question and determines whether the test is left-tailed, right-tailed, or two-tailed.

General Forms of the Hypotheses

Choosing the Correct Alternative Hypothesis

Example 1: Left-Tailed Test

A manufacturer claims that 90% of its products pass inspection.

An inspector believes the true passing rate is less than 90%.

Hypotheses:

\(H_0:p=0.90\)

\(H_a:p<0.90\)

Example 2: Right-Tailed Test

A fitness program claims that 40% of members exercise at least five days per week.

A researcher believes the true proportion is greater than 40%.

Hypotheses:

\(H_0:p=0.40\)

\(H_a:p>0.40\)

Example 3: Two-Tailed Test

A university reports that 55% of students live on campus.

A researcher wants to determine whether the true proportion is different from 55%.

Hypotheses:

\(H_0:p=0.55\)

\(H_a:p\neq0.55\)

Important AP Exam Notes

• The null hypothesis is always written as \(H_0:p=p_0\).
• \(p_0\) is the hypothesized population proportion.
• The alternative hypothesis determines whether the test is left-tailed, right-tailed, or two-tailed.
• Write hypotheses using the population proportion \(p\), not the sample proportion \(\hat{p}\).
• Choose the alternative hypothesis based on the wording of the research question or claim.

Common AP Exam Mistakes