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
         
b. fails to reject the null hypothesis?


Rent 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 is a statistical hypothesis testing problem where we need to determine whether the claim represents the null hypothesis (H₀) or the alternative hypothesis (Hₐ).
2. Step 2: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis typically represents the status quo or the claim being tested. In this case, H₀: p ≥ 0.20, where p is the proportion of renters behind on rent payments. The alternative hypothesis (Hₐ) would be Hₐ: p < 0.20, as it challenges the claim.
3. Step 3: Interpret the decision to 'fail to reject the null hypothesis.' Failing to reject H₀ means that there is not enough statistical evidence to support the alternative hypothesis (Hₐ). In this context, it suggests that the data does not provide sufficient evidence to conclude that less than 20% of renters are behind on rent payments.
4. Step 4: Emphasize the importance of the decision. Failing to reject H₀ does not mean that H₀ is true; it simply means that the sample data does not provide strong enough evidence to reject it. The claim that at least 20% of renters are behind on rent payments remains plausible based on the data.
5. Step 5: Highlight the role of significance level (α). The decision to fail to reject H₀ depends on the chosen significance level (e.g., 0.05). If the p-value from the hypothesis test is greater than α, we fail to reject H₀. This step ensures that the interpretation aligns with the statistical framework used in the test.

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 gather evidence to either reject or fail to reject this hypothesis based on sample data.

Alternative Hypothesis

The alternative hypothesis (H1) represents the claim 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 in New Jersey are behind on rent payments. This hypothesis is what researchers seek to support through statistical testing.

Failing to Reject the Null Hypothesis

Failing to reject the null hypothesis means that the evidence from the sample data is not strong enough to support the alternative hypothesis. It does not prove that the null hypothesis is true; rather, it indicates that there is insufficient evidence to conclude that at least 20% of renters are behind on payments. This decision can lead to further investigation or a reassessment of the data.