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


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Verified step by step guidance

1. Step 1: Understand the alternative hypothesis (Ha). The given alternative hypothesis is Ha: μ > 3, which means the population mean μ is greater than 3. This corresponds to a one-tailed test where the region of interest is to the right of 3.
2. Step 2: Match the alternative hypothesis with its graph. The graph that represents Ha: μ > 3 will have a shaded region starting at 3 and extending to the right, indicating values greater than 3. Based on the provided images, this matches the third graph.
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, which means the population mean μ is less than or equal to 3.
4. Step 4: Sketch the graph for the null hypothesis. The graph for H0: μ ≤ 3 will have a shaded region starting at 3 and extending to the left, indicating values less than or equal to 3. This matches the fourth graph provided.
5. Step 5: Verify the graphical representation. Ensure that the graphs correctly represent the hypotheses: the third graph for Ha: μ > 3 and the fourth graph for H0: μ ≤ 3. This visual representation helps in understanding the direction of the test and the regions of interest.

Key Concepts

Null Hypothesis (H0)

The null hypothesis is a statement that there is no effect or no difference, serving as a default position in hypothesis testing. In this case, it would state that the population mean (μ) is less than or equal to 3. It is typically denoted as H0: μ ≤ 3, and 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, suggesting that there is an effect or a difference. Here, Ha: μ > 3 indicates that the population mean is greater than 3. This hypothesis is what researchers aim to support through statistical testing, often leading to a one-tailed test in this context.

Graphical Representation of Hypotheses

Graphical representations of hypotheses help visualize the relationship between the null and alternative hypotheses. In this case, the graph would show a number line where the null hypothesis (H0) is represented by a shaded area up to 3, while the alternative hypothesis (Ha) would extend beyond this point, indicating values greater than 3. This visual aid is crucial for understanding the rejection region in hypothesis testing.