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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 7.1.42
Chapter 7, Problem 7.1.42

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.


High School Graduation Rate A high school claims that its mean graduation rate is more than 97%.

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Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ) in both words and symbols. The null hypothesis (H₀) represents the claim that the mean graduation rate is 97% or less, while the alternative hypothesis (Hₐ) represents the claim that the mean graduation rate is more than 97%. In symbols: H₀: μ ≤ 97% and Hₐ: μ > 97%.
Step 2: Identify the type of hypothesis test. Since the alternative hypothesis (Hₐ) states that the mean graduation rate is greater than 97%, this is a right-tailed test. A right-tailed test is used when the alternative hypothesis involves a 'greater than' inequality.
Step 3: Explain the reasoning for the test type. The direction of the inequality in the alternative hypothesis (Hₐ: μ > 97%) determines the tail of the test. A 'greater than' inequality corresponds to the right tail of the normal distribution.
Step 4: Sketch the normal sampling distribution. Draw a bell-shaped curve representing the normal distribution. Mark the hypothesized mean (97%) on the horizontal axis. Shade the area to the right of the test statistic, as this represents the P-value for a right-tailed test.
Step 5: Interpret the P-value area. The shaded area under the curve to the right of the test statistic represents the probability of observing a sample mean as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true. This area will be used to determine whether to reject or fail to reject the null hypothesis.

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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 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, assuming 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

Getting at the Concept Explain why a level of significance of α=0 is not used.

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


Textbook Question

True or False? In Exercises 5–10, determine whether the statement is true or false. If it is false, rewrite it as a true statement.


The level of significance is the maximum probability you allow for rejecting a null hypothesis when it is actually true.

Textbook Question

Hypothesis Testing Using Rejection Regions In Exercises 19–26, (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 t, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the population is normally distributed.


Credit Card Debt A credit reporting agency claims that the mean credit card debt in Colorado is greater than \$5540 per borrower. You want to test this claim. You find that a random sample of 30 borrowers has a mean credit card debt of \$5594 per person and a standard deviation of \$597 per person. At , can you support the claim α=0.05?

Textbook Question

Explain the difference between the z-test for μ using a P-value and the z-test for μ using rejection region(s).

Textbook Question

Does failing to reject the null hypothesis mean that the null hypothesis is true? Explain.