Stating the Null and Alternative Hypotheses In Exercises 25–30, write the claim as a mathematical statement. State the null and alternative hypotheses, and identify which represents the claim.


College Debt According to a recent survey, 14% of adults currently carry student loan debt.

Verified step by step guidance

1. Identify the parameter of interest: The problem is about the proportion of adults who currently carry student loan debt. Let the parameter be denoted as p, where p represents the true proportion of adults with student loan debt.
2. Translate the claim into a mathematical statement: The claim is that 14% of adults currently carry student loan debt. Mathematically, this is written as p = 0.14.
3. State the null hypothesis (H₀): The null hypothesis is a statement of no change or no effect. It is typically written as H₀: p = 0.14, which aligns with the claim in this case.
4. State the alternative hypothesis (H₁): The alternative hypothesis is the complement of the null hypothesis and depends on the context of the problem. If the problem does not specify a direction (greater than or less than), we use a two-tailed test. Thus, H₁: p ≠ 0.14.
5. Identify which hypothesis represents the claim: Since the claim is that 14% of adults carry student loan debt, and this matches the null hypothesis (H₀: p = 0.14), the null hypothesis represents the claim.

Key Concepts

Null Hypothesis (H0)

The null hypothesis is a statement that indicates no effect or no difference, serving as a default position that assumes any observed effect is due to sampling variability. In this context, it would assert that the proportion of adults with student loan debt is equal to 14%, symbolically represented as H0: p = 0.14.

Alternative Hypothesis (H1)

The alternative hypothesis is a statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. For this scenario, it would claim that the proportion of adults with student loan debt is not equal to 14%, represented as H1: p ≠ 0.14, indicating a potential change in the debt situation.

Claim

In hypothesis testing, the claim is the assertion that is being tested, often aligned with the alternative hypothesis. In this case, the claim is that the proportion of adults carrying student loan debt differs from 14%, which is encapsulated in the alternative hypothesis, H1.