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


μ = 82

Verified step by step guidance

1. Step 1: Understand the problem. The given claim is μ = 82, where μ represents the population mean. This is a statement about the population parameter.
2. Step 2: Write the complement of the claim. The complement of μ = 82 is μ ≠ 82, as the complement represents all values of μ that are not equal to 82.
3. Step 3: Identify the null hypothesis (H0). The null hypothesis is the statement of equality, so H0: μ = 82.
4. Step 4: Identify the alternative hypothesis (Ha). The alternative hypothesis is the complement of the null hypothesis, so Ha: μ ≠ 82.
5. Step 5: Summarize the hypotheses. H0: μ = 82 (null hypothesis), Ha: μ ≠ 82 (alternative hypothesis). These hypotheses will be used in hypothesis testing to determine if there is enough evidence to reject H0 in favor of Ha.

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 case, H0 states that the population mean (μ) is equal to 82. It is the hypothesis that researchers aim to test against, and it is assumed true until evidence suggests otherwise.

Alternative Hypothesis (Ha)

The alternative hypothesis (Ha) represents a statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. For the given claim, Ha would state that the population mean (μ) is not equal to 82. This hypothesis is what researchers seek to support through statistical evidence.

Complement of a Hypothesis

The complement of a hypothesis refers to the opposite scenario of the original claim. In this context, if the claim is μ = 82, the complement would be μ ≠ 82. Understanding the complement is crucial for hypothesis testing, as it helps define the scope of the alternative hypothesis and the potential outcomes of the statistical test.