Ch. 7 - Hypothesis Testing with One Sample

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 7 - Hypothesis Testing with One Sample

Problem 7.1.12

Chapter 7, Problem 7.1.12

Stating Hypotheses In Exercises 11–16, the statement represents a claim. Write its complement and state which is H0 and which is Ha.

μ < 128

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Step 1: Understand the problem. The claim provided is μ < 128, which is a statement about the population mean (μ). In hypothesis testing, we need to define two hypotheses: the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis typically represents the status quo or no effect, while the alternative hypothesis represents the claim or effect being tested.

Step 2: Write the complement of the claim. The complement of μ < 128 is μ ≥ 128. This is because the complement includes all values that are not less than 128.

Step 3: Assign the hypotheses. The null hypothesis (H0) is usually the complement of the claim, as it represents the default assumption. Therefore, H0: μ ≥ 128. The alternative hypothesis (Ha) represents the claim being tested, so Ha: μ < 128.

Step 4: Verify the direction of the test. Since the claim is μ < 128, this is a one-tailed test, specifically a left-tailed test, because we are testing for values less than 128.

Step 5: Summarize the hypotheses. The null hypothesis is H0: μ ≥ 128, and the alternative hypothesis is Ha: μ < 128. These hypotheses will be used in the hypothesis testing process 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.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which represents the claim being tested. The goal is to determine whether there is enough evidence to reject H0 in favor of Ha.

Null and Alternative Hypotheses

The null hypothesis (H0) is a statement that indicates no significant difference or effect exists, serving as a default position. In contrast, the alternative hypothesis (Ha) represents the claim or effect that the researcher aims to support. For the given statement μ < 128, H0 would typically be μ ≥ 128, while Ha would be μ < 128, indicating a claim that the population mean is less than 128.

Complements of Hypotheses

The complement of a hypothesis refers to the opposite scenario of the original claim. In hypothesis testing, if the alternative hypothesis states a specific condition (e.g., μ < 128), its complement would encompass all other possibilities (e.g., μ ≥ 128). Understanding complements is crucial for correctly formulating H0 and Ha, as they must cover all potential outcomes in the context of the hypothesis being tested.

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06:21

Step 1: Write Hypotheses

Related Practice

Textbook Question

In Exercises 13–18, test the claim about the population mean μ at the level of significance α. Assume the population is normally distributed.

Claim: μ=4915; α=0.01. Sample statistics: x_bar=5017, s=5613, n=51

111

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Textbook Question

Finding a P-Value In Exercises 13–18, find the P-value for the hypothesis test with the standardized test statistic z. Decide whether to reject H0 for the level of significance alpha.

Left-tailed test

z= 1.95

alpha=0.08

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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.

Changing Jobs A researcher claims that 40% of U.S. adults would consider changing jobs. In a random sample of 50 U.S. adults, 25 say they would consider changing jobs. At α=0.10, is there enough evidence to reject the researcher’s claim?

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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.

p = 0.21

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Textbook Question

Vaccinations In 2021, a reporter claims that at least 55% of U.S. adults feel that COVID-19 vaccinations should be required for high school students to attend school in the fall. In a random sample of 200 U.S. adults, 56% feel that COVID-19 vaccinations should be required for high school students to attend school in the fall. At α=0.10, is there enough evidence to reject the reporter’s claim?

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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.

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Stating Hypotheses In Exercises 11–16, the statement represents a claim. Write its complement and state which is H0 and which is Ha.μ < 128

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Key Concepts

Hypothesis Testing

Null and Alternative Hypotheses

Complements of Hypotheses

Stating Hypotheses In Exercises 11–16, the statement represents a claim. Write its complement and state which is H0 and which is Ha.

μ < 128