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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 7.5.9
Chapter 7, Problem 7.5.9

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.


Left-tailed test, n=7,α=0.01

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Step 1: Understand the problem. This is a chi-square test, specifically a left-tailed test. The goal is to find the critical value(s) and rejection region(s) based on the sample size (n=7) and the level of significance (α=0.01).
Step 2: Recall the formula for degrees of freedom in a chi-square test. Degrees of freedom (df) are calculated as df = n - 1, where n is the sample size. For this problem, df = 7 - 1 = 6.
Step 3: Use the chi-square distribution table or a statistical software to find the critical value for a left-tailed test with df = 6 and α = 0.01. In a left-tailed test, the critical value corresponds to the area to the left of the curve equal to α.
Step 4: Define the rejection region. For a left-tailed test, the rejection region is the range of chi-square values less than the critical value obtained in Step 3. This region represents where the null hypothesis would be rejected.
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 these values in the context of hypothesis testing.

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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 frequencies expected under the null hypothesis. This test is commonly used in hypothesis testing to assess goodness-of-fit or independence in contingency tables.
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Intro to Least Squares Regression

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 left-tailed chi-square test, the critical value indicates the point below which the test statistic must fall to reject the null hypothesis.
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Rejection Region

The rejection region is the set 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 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 its rejection.
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Step 4: State Conclusion
Related Practice
Textbook Question

Hypothesis Testing Using a P-Value In Exercises 33–38,

         

a. identify the claim and state and .

b. find the standardized test statistic z.

c. find the corresponding P-value.

d. decide whether to reject or fail to reject the null hypothesis.

e. interpret the decision in the context of the original claim.


MCAT Scores A random sample of 100 medical school applicants at a university has a mean total score of 505 on the MCAT. According to a report, the mean total score for the school’s applicants is more than 503. Assume the population standard deviation is 10.6. At alpha=0.01, is there enough evidence to support the report’s claim?

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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.


Attendance An amusement park claims that the mean daily attendance at the park is at least 20,000 people.

Textbook Question

How do the requirements for a chi-square test for a variance or standard deviation differ from a z-test or a t-test for a mean?

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.


Annual Salary An employment information service claims the mean annual salary for senior level statisticians is more than \(124,000. The annual salaries (in dollars) for a random sample of 12 senior level statisticians are shown in the table at the left. At α=0.01, is there enough evidence to support the claim that the mean salary is more than \)124,000?


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≠52,200; α=0.05. Sample statistics: x_bar=53,220, s=2700, n=34

Textbook Question

In Exercises 15–22, test the claim about the population variance or standard deviation at the level of significance Assume the population is normally distributed.

Claim: σ^2=63, α=0.01 . Sample statistics: s^2=58, n=29