Graphical Analysis In Exercises 17–20, match the alternative hypothesis with its graph. Then state the null hypothesis and sketch its graph.


Ha: μ < 3

Verified step by step guidance

1. Step 1: Understand the alternative hypothesis (Ha: μ < 3). This indicates that the mean μ is less than 3, which corresponds to a left-tailed test. The graph should show shading to the left of 3.
2. Step 2: Match the alternative hypothesis with the correct graph. Among the provided graphs, the one that shows shading to the left of 3 is graph b.
3. Step 3: State the null hypothesis (H0). The null hypothesis is the complement of the alternative hypothesis. For Ha: μ < 3, the null hypothesis is H0: μ ≥ 3.
4. Step 4: Sketch the graph for the null hypothesis. The graph for H0: μ ≥ 3 should show shading to the right of 3, including the point at 3.
5. Step 5: Verify the interpretation of the graphs and hypotheses. Ensure that the alternative hypothesis graph (b) and the null hypothesis graph are consistent with the problem statement.

Key Concepts

Null Hypothesis (H0)

The null hypothesis is a statement that there is no effect or no difference, and it serves as the default assumption in hypothesis testing. In this case, it would state that the population mean (μ) is equal to or greater than 3 (H0: μ ≥ 3). This hypothesis is tested against the alternative hypothesis to determine if there is enough evidence to reject it.

Alternative Hypothesis (Ha)

The alternative hypothesis represents a statement that contradicts the null hypothesis, indicating the presence of an effect or a difference. Here, Ha: μ < 3 suggests that the population mean is less than 3. This hypothesis is what researchers aim to support through statistical testing, often leading to a rejection of the null hypothesis if sufficient evidence is found.

Graphical Representation of Hypotheses

Graphical representations of hypotheses, such as number lines, help visualize the null and alternative hypotheses. In this context, the graph for Ha: μ < 3 would show a shaded area to the left of 3, indicating the values where the alternative hypothesis holds true. Understanding how to sketch these graphs is crucial for interpreting the results of hypothesis tests.