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.
μ ≤ 645
Ch. 7 - Hypothesis Testing with One Sample
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 7, Problem 7.5.8
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
Verified step by step guidance1
Step 1: Understand the problem. This is a chi-square test with a right-tailed test, sample size n=10, and level of significance α=0.10. The goal is to find the critical value(s) and rejection region(s).
Step 2: Recall the formula for degrees of freedom in a chi-square test: \( \text{df} = n - 1 \). Since \( n = 10 \), calculate \( \text{df} = 10 - 1 = 9 \).
Step 3: Use the chi-square distribution table or a statistical software to find the critical value corresponding to \( \text{df} = 9 \) and \( \alpha = 0.10 \) for a right-tailed test. The critical value is the point where the area to the right under the chi-square curve equals \( \alpha \).
Step 4: Define the rejection region. For a right-tailed test, the rejection region is the set of chi-square values greater than the critical value obtained in Step 3. This means any test statistic exceeding the critical value falls in the rejection region.
Step 5: Summarize the findings. The critical value and rejection region are determined based on the chi-square distribution table or software output. Ensure you understand how to interpret the table or software results to identify the critical value accurately.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chi-Square Test
The chi-square test is a statistical method used to determine if there is a significant association between categorical variables. It compares the observed frequencies in each category to the expected frequencies, which are calculated under the assumption of no association. This test is commonly used in hypothesis testing to evaluate goodness-of-fit or independence.
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Critical Value
A critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the chosen significance level (α) and the distribution of the test statistic. For a right-tailed chi-square test, the critical value indicates the point beyond which the test statistic is considered significant, leading to the rejection of the null hypothesis.
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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 right-tailed test, this region is located to the right of the critical value on the chi-square distribution. If the calculated test statistic falls within this region, it suggests that the observed data is unlikely under the null hypothesis, prompting researchers to consider alternative explanations.
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09:56Step 4: State Conclusion
Related Practice
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.
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?
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?
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%.
Textbook Question
How do the critical values for a two-tailed test change as alpha decreases?