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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
All textbooksLarson 8th EditionCh. 7 - Hypothesis Testing with One SampleProblem 7.RE.5
Chapter 7, Problem 7.RE.5

n Exercises 1–6, the statement represents a claim. Write its complement and state which is H0 and which is Ha.


σ > 1.9

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Understand the problem: The claim is about the population standard deviation (σ) being greater than 1.9. We need to write the complement of this claim and identify which hypothesis is the null hypothesis (H₀) and which is the alternative hypothesis (Hₐ).
Recall the definitions: The null hypothesis (H₀) is a statement of no effect or equality, and it is the hypothesis we assume to be true unless there is strong evidence against it. The alternative hypothesis (Hₐ) is the claim we are testing and is typically the statement of inequality or difference.
Write the complement of the claim: The complement of 'σ > 1.9' is 'σ ≤ 1.9'. This represents all values of the standard deviation that are less than or equal to 1.9.
Assign the hypotheses: The null hypothesis (H₀) is the complement of the claim, so H₀: σ ≤ 1.9. The alternative hypothesis (Hₐ) is the claim itself, so Hₐ: σ > 1.9.
Summarize the hypotheses: H₀: σ ≤ 1.9 (null hypothesis), Hₐ: σ > 1.9 (alternative hypothesis). These hypotheses will be used in hypothesis testing to determine whether there is sufficient evidence to support the claim.

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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 (H0) is a statement that indicates no effect or no difference, serving as a default position in statistical testing. In this context, it represents the claim that the population standard deviation (σ) is less than or equal to 1.9, suggesting that any observed effect is due to sampling variability rather than a true effect.
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Step 1: Write Hypotheses

Alternative Hypothesis (Ha)

The alternative hypothesis (Ha) is a statement that contradicts the null hypothesis, indicating the presence of an effect or difference. For the given claim σ > 1.9, Ha asserts that the population standard deviation is greater than 1.9, suggesting a significant deviation from the null hypothesis that warrants further investigation.
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Step 1: Write Hypotheses

Complement of a Statement

The complement of a statement refers to the opposite condition of that statement. In hypothesis testing, if the claim is σ > 1.9, the complement would be σ ≤ 1.9. Understanding complements is crucial for defining the null and alternative hypotheses, as they represent all possible outcomes in the context of the statistical test.
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Related Practice
Textbook Question

In Exercises 7–10, (d) explain how you should interpret a decision that rejects the null hypothesis.


An energy bar maker claims that the mean number of grams of carbohydrates in one bar is less than 25.

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Textbook Question

A nutrition bar manufacturer claims that the standard deviation of the number of grams of carbohydrates in a bar is 1.11 grams. A random sample of 26 bars has a standard deviation of 1.19 grams. At α=0.05, is there enough evidence to reject the manufacturer’s claim? Assume the population is normally distributed.

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Textbook Question

In Exercises 7–10, explain how you should interpret a decision that fails to reject the null hypothesis.


A nonprofit consumer organization says that the standard deviation of the starting prices of its top-rated vehicles for a recent year is no more than \$2900.

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Textbook Question

In Exercises 11 and 12, find the P-value for the hypothesis test with the standardized test statistic z. Decide whether to reject H0 for the level of significance α.


Left-tailed test, z = -0.94, α = 0.05

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Textbook Question

You want your test to support a positive claim about your college, not just fail to reject one. Should you state your claim so that the null hypothesis contains the claim or the alternate hypothesis contains the claim? Explain.

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Textbook Question

A travel analyst says that the mean price of a meal for a family of 4 in a resort restaurant is at most \$100. A random sample of 33 meal prices for families of 4 has a mean of \$110 and a standard deviation of \$19. At α=0.01, is there enough evidence to reject the analyst’s claim?

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