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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 7.5.11
Chapter 7, Problem 7.5.11

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.


Two-tailed test, n=81,α=0.10

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1
Determine the degrees of freedom (df) for the chi-square test. The formula for degrees of freedom is df = n - 1, where n is the sample size. In this case, n = 81, so df = 81 - 1.
Identify the level of significance (α) for the test. For a two-tailed test, the significance level is split equally between the two tails of the chi-square distribution. Thus, each tail will have an area of α/2 = 0.10/2.
Use a chi-square distribution table or statistical software to find the critical values corresponding to the upper and lower tails of the distribution. Look up the chi-square values for df = 80 and cumulative probabilities of 1 - α/2 (upper tail) and α/2 (lower tail).
Define the rejection regions based on the critical values. The rejection region for a two-tailed test includes chi-square values less than the lower critical value and greater than the upper critical value.
Summarize the critical values and rejection regions. Clearly state the critical values and the intervals that define the rejection regions for the test.

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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.
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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 chi-square test, critical values can be found using chi-square distribution tables based on the degrees of freedom and the specified α.
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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 two-tailed test, this region is split between both tails of the distribution. For a chi-square test with a significance level of α, the rejection regions are determined by the critical values, indicating where the test statistic must fall to reject the null hypothesis.
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Step 4: State Conclusion
Related Practice
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.


[APPLET] Gross Domestic Product A politician estimates that the mean gross domestic product (GDP) per country in a recent year is greater than \$400 billion. You want to test this estimate. To do so, you determine the GDPs of 42 randomly selected countries for that year. The results (in billions of dollars) are shown in the table at the left. Assume the population standard deviation is \$2099 billion. At alpha=0.06, can you support the politician’s estimate?


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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.


σ ≠ 5

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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.


p < 0.45

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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≥8000; α=0.01. Sample statistics: x_bar=77,000, s=450, n=25

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Textbook Question

Graphical Analysis In Exercises 9–12, match the P-value or z-statistic with the graph that represents the corresponding area. Explain your reasoning.


z = -2.37


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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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