Textbook Question
Stating Hypotheses In Exercises 11–16, the statement represents a claim. Write its complement and state which is H0 and which is Ha.
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9. Hypothesis Testing for One Sample
Steps in Hypothesis Testing
2:06 minutesProblem 7.1.19
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
a. 
b. 
c. 
d. 
Verified step by step guidance1
Step 1: Understand the alternative hypothesis (Ha). The alternative hypothesis states that the population mean (μ) is not equal to 3, which is written as Ha: μ ≠ 3. This indicates a two-tailed test where we are testing for any deviation from μ = 3.
Step 2: Match the alternative hypothesis with its graph. The graph corresponding to Ha: μ ≠ 3 will show rejection regions on both sides of μ = 3, indicating that values significantly less than or greater than 3 are part of the rejection region. Based on the provided images, the graph with rejection regions on both sides of μ = 3 is the second image.
Step 3: State the null hypothesis (H0). The null hypothesis is the complement of the alternative hypothesis and states that the population mean is equal to 3, written as H0: μ = 3.
Step 4: Sketch the graph for the null hypothesis. The graph for H0: μ = 3 will have no rejection regions, and the entire area will be centered around μ = 3, indicating that we assume μ = 3 unless evidence suggests otherwise.
Step 5: Review the graphs and ensure proper interpretation. Confirm that the graph for Ha: μ ≠ 3 has rejection regions on both sides of μ = 3, while the graph for H0: μ = 3 has no rejection regions and is centered around μ = 3.
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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, and it serves as the default assumption in hypothesis testing. In this case, it would state that the population mean (μ) is equal to 3. The null 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 not equal to 3, which implies that it could be either less than or greater than 3. This hypothesis is what researchers aim to support through statistical testing.
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Graphical Representation of Hypotheses
Graphical representations of hypotheses, such as the one provided, visually depict the regions of acceptance and rejection for the null hypothesis. In this case, the graph shows a critical region on either side of the mean value of 3, indicating where the null hypothesis would be rejected if the sample mean falls outside this range. Understanding these graphs is crucial for interpreting the results of hypothesis tests.
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