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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 7.1.41
Chapter 7, Problem 7.1.41

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.


Survey A polling organization reports that the number of responses to a survey mailed to 100,000 U.S. residents is not 100,000.

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Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁) in both words and symbols. The null hypothesis (H₀) typically represents the status quo or no change. In this case, H₀: 'The number of responses to the survey is equal to 100,000' (symbolically, H₀: μ = 100,000). The alternative hypothesis (H₁) represents the claim being tested. Here, H₁: 'The number of responses to the survey is not equal to 100,000' (symbolically, H₁: μ ≠ 100,000).
Step 2: Determine the type of hypothesis test. Since the alternative hypothesis (H₁) uses 'not equal to' (≠), this indicates a two-tailed test. A two-tailed test is used when we are interested in deviations in both directions (greater than or less than the hypothesized value).
Step 3: Sketch the normal sampling distribution. Draw a bell-shaped curve to represent the normal distribution. Mark the mean (μ = 100,000) at the center of the curve. Since this is a two-tailed test, shade the areas in both tails of the curve that correspond to the P-value. These shaded areas represent the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, under the assumption that H₀ is true.
Step 4: Explain the reasoning for the test type. The reasoning for a two-tailed test is based on the alternative hypothesis (H₁: μ ≠ 100,000). This hypothesis does not specify a direction (greater than or less than), so we are testing for any significant difference from the hypothesized value in either direction.
Step 5: To calculate the P-value, you would first compute the test statistic (e.g., z-score or t-score) using the sample data. Then, use statistical tables or software to find the probability associated with the test statistic in both tails of the distribution. The total P-value is the sum of the probabilities in both tails.

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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 (H1), which indicates the presence of an effect or difference. The goal is to determine whether there is enough evidence in the sample data to reject the null hypothesis in favor of the alternative.
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Step 1: Write Hypotheses

Types of Hypothesis Tests

Hypothesis tests can be classified as left-tailed, right-tailed, or two-tailed based on the direction of the alternative hypothesis. A left-tailed test is used when the alternative hypothesis states that a parameter is less than a certain value, while a right-tailed test is used when it states that the parameter is greater. A two-tailed test is appropriate when the alternative hypothesis indicates that the parameter is simply different from a certain value, without specifying a direction.
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Step 1: Write Hypotheses

P-value and Normal Distribution

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value under the assumption that the null hypothesis is true. In the context of a normal distribution, the P-value corresponds to the area under the curve in the tail(s) of the distribution, which is shaded to visually represent the likelihood of observing the sample data if the null hypothesis holds.
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Step 3: Get P-Value
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

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Textbook Question

Stating the Null and Alternative Hypotheses In Exercises 25–30, write the claim as a mathematical statement. State the null and alternative hypotheses, and identify which represents the claim.


Tablets A tablet manufacturer claims that the mean life of the battery for a certain model of tablet is more than 8 hours.

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Textbook Question

In Exercises 3–8, find the critical value(s) and rejection region(s) for the type of t-test with level of significance alpha and sample size n.


Left-tailed test, α=0.01, n=35

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Textbook Question

Identifying Type I and Type II Errors In Exercises 31–36, describe type I and type II errors for a hypothesis test of the indicated claim.


Repeat Customers A used textbook selling website claims that at least 60% of its new customers will return to buy their next textbook.

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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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