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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 7.1.46b
Chapter 7, Problem 7.1.46b

Interpreting a Decision In Exercises 43–48, determine whether the claim represents the null hypothesis or the alternative hypothesis. If a hypothesis test is performed, how should you interpret a decision that
         
b. fails to reject the null hypothesis?


Gas Mileage An automotive manufacturer claims that the standard deviation for the gas mileage of one of the vehicles it manufactures is 3.9 miles per gallon.

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Step 1: Understand the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis (H₀) is a statement of no change, no effect, or a status quo. In this case, the null hypothesis would be that the standard deviation of the gas mileage is equal to 3.9 miles per gallon (σ = 3.9). The alternative hypothesis (Hₐ) would be that the standard deviation is not equal to 3.9 miles per gallon (σ ≠ 3.9).
Step 2: Recall the meaning of 'failing to reject the null hypothesis.' When a hypothesis test is performed, failing to reject the null hypothesis means that there is not enough statistical evidence to support the alternative hypothesis. This does not prove that the null hypothesis is true; it simply means that the data does not provide strong enough evidence to conclude otherwise.
Step 3: Interpret the decision in the context of the problem. If the null hypothesis is not rejected, it means that the claim made by the automotive manufacturer (that the standard deviation of the gas mileage is 3.9 miles per gallon) is consistent with the data. However, this does not confirm the claim; it only indicates that the data does not contradict it.
Step 4: Consider the implications of the test result. Failing to reject the null hypothesis suggests that the variability in gas mileage (as measured by the standard deviation) is not significantly different from the claimed value of 3.9 miles per gallon. This could mean that the manufacturer's claim is reasonable based on the available data.
Step 5: Emphasize the importance of sample size and significance level. The decision to fail to reject the null hypothesis could be influenced by factors such as the sample size and the chosen significance level (α). A small sample size or a high significance level might make it harder to detect a difference, even if one exists.

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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 (H0) is a statement that indicates no effect or no difference, serving as a default position in hypothesis testing. In this context, it would assert that the standard deviation of gas mileage is equal to 3.9 miles per gallon. Researchers aim to gather evidence against the null hypothesis to support an alternative hypothesis.
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Step 1: Write Hypotheses

Alternative Hypothesis

The alternative hypothesis (H1) is a statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. In this scenario, it would propose that the standard deviation of gas mileage is not equal to 3.9 miles per gallon. This hypothesis is what researchers hope to support through statistical testing.
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Step 1: Write Hypotheses

Failing to Reject the Null Hypothesis

Failing to reject the null hypothesis means that the evidence collected from the data is not strong enough to support the alternative hypothesis. This does not prove that the null hypothesis is true; rather, it indicates that there is insufficient evidence to conclude a significant difference or effect, such as the standard deviation of gas mileage being different from 3.9 miles per gallon.
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Step 1: Write Hypotheses
Related Practice
Textbook Question

Graphical Analysis In Exercises 9–12, state whether each standardized test statistic t allows you to reject the null hypothesis. Explain.


c. t = -2.096


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Textbook Question

Interpreting a Decision In Exercises 43–48, determine whether the claim represents the null hypothesis or the alternative hypothesis. If a hypothesis test is performed, how should you interpret a decision that

         

b. fails to reject the null hypothesis?


Marketing A fitness equipment company claims that its competitor’s home gym does not have a customer satisfaction rate of 99%.

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Textbook Question

Graphical Analysis In Exercises 9–12, state whether each standardized test statistic t allows you to reject the null hypothesis. Explain.


b. t = 0

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Textbook Question

Graphical Analysis In Exercises 9–12, state whether each standardized test statistic t allows you to reject the null hypothesis. Explain.


c. t = 1.7


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Textbook Question

Graphical Analysis In Exercises 9–12, state whether each standardized test statistic t allows you to reject the null hypothesis. Explain.


b. t = 1.42


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Textbook Question

Writing Hypotheses: Backpack Manufacturer A backpack manufacturer claims that the mean life of its competitor’s backpacks is less than 5 years. You are asked to perform a hypothesis test to test this claim. How would you write the null and alternative hypotheses when


b. you represent the competitor and want to reject the claim?

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