Textbook Question
True or False? In Exercises 5–10, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
To support a claim, state it so that it becomes the null hypothesis.
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9. Hypothesis Testing for One Sample
Steps in Hypothesis Testing
1:49 minutesProblem 7.1.17
Textbook Question
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 guidance1
Step 1: Understand the alternative hypothesis (Ha: μ > 3). This indicates that the mean μ is greater than 3, so the graph should represent values of μ greater than 3 with shading or emphasis to the right of 3.
Step 2: Analyze the provided graphs. Look for the graph where the shading or emphasis starts at 3 and extends to the right, representing μ > 3. In this case, the correct graph is option 'a'.
Step 3: State the null hypothesis (H0). The null hypothesis is the opposite of the alternative hypothesis, so H0: μ ≤ 3. This means the mean μ is less than or equal to 3.
Step 4: Sketch the graph for the null hypothesis. The graph should represent values of μ less than or equal to 3, with shading or emphasis to the left of 3.
Step 5: Verify the match between the alternative hypothesis and its graph, and ensure the null hypothesis graph is correctly sketched to represent the opposite scenario.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Null Hypothesis (H0)
The null hypothesis is a statement that there is no effect or no difference, serving as a default position that indicates no change from a known value. In this case, it would state that the population mean (μ) is less than or equal to 3 (H0: μ ≤ 3). This hypothesis is tested against the alternative hypothesis to determine if there is enough evidence to reject it.
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Step 1: Write Hypotheses
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.
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Graphical Representation of Hypotheses
Graphical representations of hypotheses help visualize the relationship between the null and alternative hypotheses. In this case, the graph would illustrate the critical region for the alternative hypothesis (Ha: μ > 3) on the right side of the number line, indicating where we would reject the null hypothesis. Understanding how to sketch these graphs is crucial for interpreting statistical results.
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