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.


Paying for College According to a recent survey, 54% of today’s college students used student loans to pay for college.

Verified step by step guidance

1. Identify the claim: The problem states that 54% of today's college students used student loans to pay for college. This is the claim we will test.
2. Define the population proportion: Let p represent the proportion of college students who use student loans to pay for college.
3. Write the claim as a mathematical statement: The claim is that p = 0.54.
4. State the null hypothesis (H₀): The null hypothesis always includes equality. Here, H₀: p = 0.54.
5. State the alternative hypothesis (H₁): The alternative hypothesis depends on the context. If we are testing whether the proportion is different from 54%, then H₁: p ≠ 0.54. If we are testing whether it is greater or less than 54%, the alternative hypothesis would be H₁: p > 0.54 or H₁: p < 0.54, respectively.

Key Concepts

Null Hypothesis

The null hypothesis (H0) is a statement that indicates no effect or no difference in a statistical test. It serves as a default position that assumes any observed effect is due to sampling variability. In this context, it would state that the proportion of college students using loans is equal to 54%, suggesting that there is no significant change from the reported figure.

Alternative Hypothesis

The alternative hypothesis (H1) is a statement that contradicts the null hypothesis, indicating that there is an effect or a difference. It represents the claim that researchers aim to support. In this case, it would assert that the proportion of college students using loans is not equal to 54%, suggesting a significant change in the usage of student loans.

Statistical Significance

Statistical significance refers to the likelihood that a relationship observed in data is not due to chance. It is often determined through p-values in hypothesis testing. If the results show statistical significance, it implies that the alternative hypothesis may be accepted, indicating a meaningful difference in the proportion of students using loans compared to the stated 54%.