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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 7.3.8
Chapter 7, Problem 7.3.8

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=0.10, n=38

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Step 1: Understand the problem. This is a left-tailed t-test, meaning the rejection region is located in the left tail of the t-distribution. The level of significance (α) is given as 0.10, and the sample size (n) is 38.
Step 2: Determine the degrees of freedom (df). For a t-test, degrees of freedom are calculated as df = n - 1. Use the formula: df=n-1.
Step 3: Look up the critical value in the t-distribution table. Use the degrees of freedom (df = 38 - 1 = 37) and the level of significance (α = 0.10) for a left-tailed test. The critical value corresponds to the t-score where the cumulative probability equals α.
Step 4: Define the rejection region. For a left-tailed test, the rejection region is all t-values less than the critical value. Express this as: t<critical value.
Step 5: Verify the setup. Ensure the degrees of freedom and α are correctly applied, and confirm the critical value and rejection region are consistent with the left-tailed test.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Critical Value

A critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis. It is determined based on the significance level (alpha) and the type of test being conducted. For a left-tailed test, the critical value corresponds to the point where the cumulative probability equals alpha, indicating the threshold for rejecting the null hypothesis.
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Critical Values: t-Distribution

Rejection Region

The rejection region is the range of values for the test statistic that leads to the rejection of the null hypothesis. In a left-tailed test, this region is located to the left of the critical value. If the calculated test statistic falls within this region, it suggests that the sample provides sufficient evidence to reject the null hypothesis at the specified significance level.
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Guided course
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Step 4: State Conclusion

T-Test

A t-test is a statistical test used to determine if there is a significant difference between the means of two groups, or between a sample mean and a known value. It is particularly useful when the sample size is small (typically n < 30) and the population standard deviation is unknown. The type of t-test (one-sample, independent, or paired) depends on the data structure and research question.
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Related Practice
Textbook Question

Finding Critical Values and Rejection Regions In Exercises 23–28, find the critical value(s) and rejection region(s) for the type of z-test with level of significance α. Include a graph with your answer.


Right-tailed test, α = 0.08

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Textbook Question

use the figure at the left, which suggests what adults think about protecting the environment.


[Image]


Are People Concerned About Protecting the Environment? You interview a random sample of 100 adults. The results of the survey show that 58% of the adults said they live in ways that help protect the environment some of the time. At α=0.05, can you reject the claim that at least 64% of adults make an effort to live in ways that help protect the environment some of the time?

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Textbook Question

Explain how to test a population proportion p.

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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.


Two-tailed test, α=0.05, n=27

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Textbook Question

Hypothesis Testing Using Rejection Region(s) In Exercises 39–44, (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.


Light Bulbs A light bulb manufacturer guarantees that the mean life of a certain type of light bulb is at least 750 hours. A random sample of 25 light bulbs has a mean life of 745 hours. Assume the population is normally distributed and the population standard deviation is 60 hours. At alpha= 0.02, do you have enough evidence to reject the manufacturer’s claim?

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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.


Video Game Systems A researcher claims that the percentage of U.S. gamers that are women is not 50%.

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