n Exercises 1–6, the statement represents a claim. Write its complement and state which is H0 and which is Ha.


σ > 1.9

Verified step by step guidance

1. Understand the problem: The claim is about the population standard deviation (σ) being greater than 1.9. We need to write the complement of this claim and identify which hypothesis is the null hypothesis (H₀) and which is the alternative hypothesis (Hₐ).
2. Recall the definitions: The null hypothesis (H₀) is a statement of no effect or equality, and it is the hypothesis we assume to be true unless there is strong evidence against it. The alternative hypothesis (Hₐ) is the claim we are testing and is typically the statement of inequality or difference.
3. Write the complement of the claim: The complement of 'σ > 1.9' is 'σ ≤ 1.9'. This represents all values of the standard deviation that are less than or equal to 1.9.
4. Assign the hypotheses: The null hypothesis (H₀) is the complement of the claim, so H₀: σ ≤ 1.9. The alternative hypothesis (Hₐ) is the claim itself, so Hₐ: σ > 1.9.
5. Summarize the hypotheses: H₀: σ ≤ 1.9 (null hypothesis), Hₐ: σ > 1.9 (alternative hypothesis). These hypotheses will be used in hypothesis testing to determine whether there is sufficient evidence to support the claim.

Key Concepts

Null Hypothesis (H0)

The null hypothesis (H0) is a statement that indicates no effect or no difference, serving as a default position in statistical testing. In this context, it represents the claim that the population standard deviation (σ) is less than or equal to 1.9, suggesting that any observed effect is due to sampling variability rather than a true effect.

Alternative Hypothesis (Ha)

The alternative hypothesis (Ha) is a statement that contradicts the null hypothesis, indicating the presence of an effect or difference. For the given claim σ > 1.9, Ha asserts that the population standard deviation is greater than 1.9, suggesting a significant deviation from the null hypothesis that warrants further investigation.

Complement of a Statement

The complement of a statement refers to the opposite condition of that statement. In hypothesis testing, if the claim is σ > 1.9, the complement would be σ ≤ 1.9. Understanding complements is crucial for defining the null and alternative hypotheses, as they represent all possible outcomes in the context of the statistical test.