Stating Hypotheses In Exercises 11–16, the statement represents a claim. Write its complement and state which is H0 and which is Ha.


μ < 128

Verified step by step guidance

1. Step 1: Understand the problem. The claim provided is μ < 128, which is a statement about the population mean (μ). In hypothesis testing, we need to define two hypotheses: the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis typically represents the status quo or no effect, while the alternative hypothesis represents the claim or effect being tested.
2. Step 2: Write the complement of the claim. The complement of μ < 128 is μ ≥ 128. This is because the complement includes all values that are not less than 128.
3. Step 3: Assign the hypotheses. The null hypothesis (H0) is usually the complement of the claim, as it represents the default assumption. Therefore, H0: μ ≥ 128. The alternative hypothesis (Ha) represents the claim being tested, so Ha: μ < 128.
4. Step 4: Verify the direction of the test. Since the claim is μ < 128, this is a one-tailed test, specifically a left-tailed test, because we are testing for values less than 128.
5. Step 5: Summarize the hypotheses. The null hypothesis is H0: μ ≥ 128, and the alternative hypothesis is Ha: μ < 128. These hypotheses will be used in the hypothesis testing process to determine whether there is sufficient evidence to support the claim.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which represents the claim being tested. The goal is to determine whether there is enough evidence to reject H0 in favor of Ha.

Null and Alternative Hypotheses

The null hypothesis (H0) is a statement that indicates no significant difference or effect exists, serving as a default position. In contrast, the alternative hypothesis (Ha) represents the claim or effect that the researcher aims to support. For the given statement μ < 128, H0 would typically be μ ≥ 128, while Ha would be μ < 128, indicating a claim that the population mean is less than 128.

Complements of Hypotheses

The complement of a hypothesis refers to the opposite scenario of the original claim. In hypothesis testing, if the alternative hypothesis states a specific condition (e.g., μ < 128), its complement would encompass all other possibilities (e.g., μ ≥ 128). Understanding complements is crucial for correctly formulating H0 and Ha, as they must cover all potential outcomes in the context of the hypothesis being tested.