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Ch. 7 - Hypothesis Testing with One Sample
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 7, Problem 7.1.20

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.

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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.
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.
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.
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.
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.

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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 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.
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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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Step 1: Write Hypotheses

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.
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Step 1: Write Hypotheses
Related Practice
Textbook Question

Identifying the Nature of a Hypothesis Test In Exercises 37–42, state and in words and in symbols. Then determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. Explain your reasoning. Sketch a normal sampling distribution and shade the area for the P-value.


Lung Cancer A report claims that lung cancer accounts for 25% of all cancer diagnoses.

Textbook Question

Hypothesis Testing Using Rejection Regions In Exercises 7–12, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.


Nursing A patient care manager claims that more than half of all nurses feel they became better professionals during the coronavirus pandemic. In a random sample of 300 nurses, 174 say they became better professionals during the pandemic. At α=0.01, is there enough evidence to support the manager’s claim?

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Textbook Question

Hypothesis Testing Using a P-Value In Exercises 13–16, (a) identify the claim and state H0 and Ha, (b) use technology to find the P-value, (c) decide whether to reject or fail to reject the null hypothesis, and (d) interpret the decision in the context of the original claim.


Stray Cats An animal advocate claims that 25% of U.S. households have taken in a stray cat. In a random sample of 500 U.S. households, 105 say they have taken in a stray cat. At α=0.05, is there enough evidence to reject the advocate’s claim?

Textbook Question

Identifying the Nature of a Hypothesis Test In Exercises 37–42, state and in words and in symbols. Then determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. Explain your reasoning. Sketch a normal sampling distribution and shade the area for the P-value.


Golf A golf analyst claims that the standard deviation of the 18-hole scores for a golfer is less than 2.1 strokes.

Textbook Question

In Exercise 1, you rejected the claim that p=0.53. But this claim was true. What type of error is this?

Textbook Question

Interpreting a P-Value In Exercises 3–8, the P-value for a hypothesis test is shown. Use the P-value to decide whether to reject H0 when the level of significance is (a)α=0.01, (b) α=0.05 , and (c) α=0.10.


P = 0.0838