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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.9d
Chapter 7, Problem 7.RE.9d

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.

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Understand the null hypothesis (H₀) and the alternative hypothesis (H₁). In this case, the null hypothesis (H₀) is that the standard deviation of the starting prices of the top-rated vehicles is no more than \$2900 (σ ≤ 2900). The alternative hypothesis (H₁) is that the standard deviation is greater than \$2900 (σ > 2900).
Identify the type of hypothesis test to be used. Since the problem involves the standard deviation, a chi-square test for variance or standard deviation is appropriate. The test statistic for this test is based on the chi-square distribution.
Calculate the test statistic using the formula: χ² = ((n - 1) * s²) / σ₀², where n is the sample size, s² is the sample variance, and σ₀ is the hypothesized standard deviation (2900 in this case).
Compare the calculated test statistic to the critical value from the chi-square distribution table at the chosen significance level (e.g., α = 0.05). Alternatively, calculate the p-value and compare it to the significance level.
If the test statistic exceeds the critical value or if the p-value is less than the significance level, reject the null hypothesis. This means there is sufficient evidence to conclude that the standard deviation of the starting prices is greater than \$2900. Interpret this decision in the context of the problem: the nonprofit consumer organization’s claim that the standard deviation is no more than \$2900 is not supported by the data.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Null Hypothesis

The null hypothesis is a statement that there is no effect or no difference, serving as a default position in statistical testing. In this context, it posits that the standard deviation of starting prices for the vehicles is $2900 or less. Rejecting the null hypothesis suggests that there is sufficient evidence to believe that the standard deviation exceeds this value.
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Statistical Significance

Statistical significance refers to the likelihood that a result or relationship is caused by something other than mere chance. When rejecting the null hypothesis, it indicates that the observed data is unlikely to occur if the null hypothesis were true, often assessed using a p-value. A low p-value (typically less than 0.05) suggests strong evidence against the null hypothesis.
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Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this scenario, it quantifies how much the starting prices of the vehicles deviate from the average price. A higher standard deviation indicates greater variability in prices, which is critical for understanding the financial implications for consumers and the organization.
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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

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

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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, (a) state the null and alternative hypotheses and identify which represents the claim.

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.

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Textbook Question

In Exercises 7–10, describe type I and type II errors for a hypothesis test of the claim.


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

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Textbook Question

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

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