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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.55
Chapter 7, Problem 7.RE.55

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

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Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: σ² ≤ 2, and the alternative hypothesis is H₁: σ² > 2. This is a right-tailed test since the claim is that the population variance is greater than 2.
Step 2: Determine the test statistic formula for a chi-square test for variance. The formula is χ² = ((n - 1) * s²) / σ₀², where n is the sample size, s² is the sample variance, and σ₀² is the hypothesized population variance.
Step 3: Substitute the given values into the formula. Here, n = 18, s² = 2.95, and σ₀² = 2. Compute χ² = ((18 - 1) * 2.95) / 2.
Step 4: Determine the critical value for the chi-square distribution at α = 0.10 with degrees of freedom df = n - 1 = 17. Use a chi-square distribution table or statistical software to find the critical value for a right-tailed test.
Step 5: Compare the calculated χ² value to the critical value. If χ² > critical value, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Conclude whether there is sufficient evidence to support the claim that the population variance is greater than 2.

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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 a null hypothesis (H0) and an alternative hypothesis (H1). In this case, the null hypothesis would state that the population variance is less than or equal to 2, while the alternative hypothesis claims that it is greater than 2. The outcome of the test determines whether to reject or fail to reject the null hypothesis.
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Significance Level (α)

The significance level, denoted as α, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. In this scenario, α is set at 0.10, meaning there is a 10% risk of concluding that the population variance is greater than 2 when it is not. This threshold helps determine the critical value for the test statistic, guiding the decision-making process.
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Chi-Square Distribution

The Chi-Square distribution is a statistical distribution commonly used in hypothesis testing for variance and standard deviation. It is particularly relevant when the population is normally distributed, as is the case here. The test statistic for variance is calculated using the sample variance and the sample size, and it follows a Chi-Square distribution with degrees of freedom equal to n-1. This distribution helps determine the critical value needed to assess the hypothesis.
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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?

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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

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Textbook Question

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


p < 0.205

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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=15

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Textbook Question

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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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.

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