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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 7.1.34
Chapter 7, Problem 7.1.34

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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Understand the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis (H₀) is that the percentage of U.S. gamers who are women is 50%. The alternative hypothesis (H₁) is that the percentage of U.S. gamers who are women is not 50%.
Define a Type I error. A Type I error occurs when the null hypothesis (H₀) is rejected even though it is true. In this case, it would mean concluding that the percentage of U.S. gamers who are women is not 50%, when in reality it is 50%.
Define a Type II error. A Type II error occurs when the null hypothesis (H₀) is not rejected even though the alternative hypothesis (H₁) is true. In this case, it would mean failing to conclude that the percentage of U.S. gamers who are women is not 50%, when in reality it is not 50%.
Consider the implications of each error. A Type I error might lead to incorrect changes in policies or assumptions about the gaming demographic, while a Type II error might result in missed opportunities to address actual differences in the gaming demographic.
Ensure the researcher understands the importance of selecting an appropriate significance level (α) for the hypothesis test. The significance level determines the probability of making a Type I error and should be chosen based on the context and consequences of the study.

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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. In this case, the null hypothesis would state that the percentage of U.S. gamers that are women is 50%, while the alternative would claim it is not.
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Step 1: Write Hypotheses

Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected when it is actually true. In the context of the given claim, this would mean concluding that the percentage of female gamers is not 50% when, in fact, it is. This type of error is also known as a 'false positive' and is denoted by the significance level (alpha), which is the probability of making this error.
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Type II Error

A Type II error happens when the null hypothesis is not rejected when it is false. In this scenario, it would mean failing to recognize that the percentage of female gamers is not 50% when it actually is. This error is referred to as a 'false negative' and is denoted by beta (β), representing the probability of making this error. Understanding both types of errors is crucial for evaluating the reliability of hypothesis tests.
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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

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.


Chess A local chess club claims that the length of time to play a game has a standard deviation of more than 12 minutes.

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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

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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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

In Exercises 7–12, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance α.


Right-tailed test, n=10,α=0.10

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