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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.RE.54
Chapter 7, Problem 7.RE.54

In Exercises 51–54, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance.


Left-tailed test, n=6, α=0.05

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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 = 6, so df = 6 - 1 = 5.
Identify the level of significance (α) for the test. Here, α = 0.05, which represents the probability of rejecting the null hypothesis when it is true.
Since this is a left-tailed test, locate the critical value for the chi-square distribution corresponding to df = 5 and α = 0.05. Use a chi-square distribution table or statistical software to find this value.
Define the rejection region for the left-tailed test. The rejection region consists of all chi-square values less than the critical value obtained in the previous step.
Summarize the critical value and rejection region. State that if the test statistic falls within the rejection region, the null hypothesis will be rejected at the 0.05 significance level.

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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 particularly useful in analyzing contingency tables and goodness-of-fit problems.
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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 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 significantly different from what is expected under the null hypothesis.
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Step 4: State Conclusion
Related Practice
Textbook Question

In Exercises 27 and 28, (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.


A substance abuse counselor claims that the mean annual drug overdose death rate for the 50 states is at least 25 deaths per 100,000 people. In a random sample of 30 states, the mean annual drug overdose rate is 22.48 per 100,000 people. Assume the population standard deviation is 10.69 deaths per 100,000. At α=0.01, is there enough evidence to reject the claim?

Textbook Question

In Exercises 29–34, find the critical value(s) and rejection region(s) for the type of t-test with level of significance α and sample size n.


Left-tailed test, α=0.05, n=48

Textbook Question

In Exercises 7–10, explain how you should interpret a decision that rejects the null hypothesis.


A nonprofit consumer organization says that the standard deviation of the starting prices of its top-rated vehicles for a recent year is no more than \$2900.

Textbook Question

n Exercises 1–6, the statement represents a claim. Write its complement and state which is H0 and which is Ha.


μ ≤ 375

Textbook Question

In Exercises 55–58, test the claim about the population variance or standard deviation at the level of significance . Assume the population is normally distributed.


Claim: σ^2 > 2; α=0.10. Sample statistics: s^2 = 2.95, n=18

Textbook Question

In Exercises 7–10, (c) explain whether the hypothesis test is left-tailed, right-tailed, or two-tailed.


An energy bar maker claims that the mean number of grams of carbohydrates in one bar is less than 25.