Interpreting a Decision In Exercises 43–48, determine whether the claim represents the null hypothesis or the alternative hypothesis. If a hypothesis test is performed, how should you interpret a decision that
         
a. rejects the null hypothesis?


A recent study claims that at least 20% of renters are behind on rent payments in New Jersey. 

Verified step by step guidance

1. Step 1: Understand the problem. The claim states that 'at least 20% of renters are behind on rent payments in New Jersey.' This means the claim is about a population proportion (p) being at least 0.20. In hypothesis testing, the null hypothesis (H₀) typically represents the status quo or the claim being tested, while the alternative hypothesis (Hₐ) represents the complement or challenge to the null hypothesis.
2. Step 2: Define the null hypothesis (H₀). Since the claim is 'at least 20%,' this translates to H₀: p ≥ 0.20. The null hypothesis assumes that the proportion of renters behind on rent payments is at least 20%.
3. Step 3: Define the alternative hypothesis (Hₐ). The alternative hypothesis is the opposite of the null hypothesis. In this case, Hₐ: p < 0.20. This means the proportion of renters behind on rent payments is less than 20%.
4. Step 4: Interpret a decision to reject the null hypothesis. If the null hypothesis (H₀: p ≥ 0.20) is rejected, it means there is sufficient evidence to support the alternative hypothesis (Hₐ: p < 0.20). This suggests that the proportion of renters behind on rent payments is likely less than 20%.
5. Step 5: Interpret a decision to fail to reject the null hypothesis. If the null hypothesis is not rejected, it means there is not enough evidence to support the alternative hypothesis. In this case, we would conclude that the data does not provide sufficient evidence to suggest the proportion of renters behind on rent payments is less than 20%, and the claim that at least 20% are behind on rent payments cannot be ruled out.

Key Concepts

Null Hypothesis

The null hypothesis (H0) is a statement that indicates no effect or no difference, serving as a default position in hypothesis testing. In this context, it would assert that the proportion of renters behind on payments is less than 20%. Researchers aim to test this hypothesis against an alternative hypothesis to determine if there is enough evidence to reject it.

Alternative Hypothesis

The alternative hypothesis (H1) represents a statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. In this case, it posits that at least 20% of renters are behind on rent payments. This hypothesis is what researchers seek to support through statistical evidence when conducting a hypothesis test.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves calculating a test statistic and comparing it to a critical value to determine whether to reject the null hypothesis. A decision to reject the null hypothesis indicates that the sample provides sufficient evidence to support the alternative hypothesis.