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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 7.4.9b
Chapter 7, Problem 7.4.9b

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


b. t = 0
Graph of a t-distribution showing t0 = -2.086 with shaded area to the left, indicating rejection region for hypothesis testing.

Verified step by step guidance
1
Step 1: Understand the context of the problem. The standardized test statistic t is used in hypothesis testing to determine whether to reject the null hypothesis. The decision depends on the critical t-value and the significance level (α).
Step 2: Analyze the graph provided. The graph shows a t-distribution with a critical t-value of t₀ = -2.086, which marks the rejection region on the left tail of the distribution. The shaded area represents the rejection region.
Step 3: Compare the given test statistic t = 0 to the critical t-value. Since t = 0 lies at the center of the distribution and does not fall within the rejection region (shaded area), it does not meet the criteria for rejecting the null hypothesis.
Step 4: Explain the reasoning. The null hypothesis is rejected only if the test statistic falls within the rejection region, which is determined by the critical t-value and the significance level. In this case, t = 0 is not extreme enough to reject the null hypothesis.
Step 5: Conclude the analysis. Based on the comparison and the graph, the test statistic t = 0 does not allow you to reject the null hypothesis.

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Key Concepts

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

Standardized Test Statistic (t)

A standardized test statistic, such as t, is a value derived from sample data that measures how far the sample mean is from the null hypothesis mean, expressed in terms of standard errors. It helps determine whether to reject the null hypothesis by comparing the calculated t value to critical values from the t-distribution.
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Step 2: Calculate Test Statistic

Null Hypothesis (H0)

The null hypothesis (H0) is a statement that there is no effect or no difference, serving as a default position in hypothesis testing. It is tested against an alternative hypothesis (H1) and is typically rejected if the evidence from the data is strong enough, often determined by the significance level and the calculated test statistic.
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Step 1: Write Hypotheses

Rejection Region

The rejection region is the area in the tails of the distribution where, if the test statistic falls within this area, the null hypothesis is rejected. In the context of a t-distribution, this region is determined by the significance level (alpha) and is critical for making decisions about the null hypothesis based on the calculated t value.
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Step 4: State Conclusion
Related Practice
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?


Rent A recent study claims that at least 20% of renters are behind on rent payments in New Jersey. 

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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?


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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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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Textbook Question

Writing Hypotheses: Internet Provider An Internet provider is trying to gain advertising deals and claims that the mean time a customer spends online per day is greater than 28 minutes. You are asked to test this claim. How would you write the null and alternative hypotheses when


b. you represent a competing advertiser and want to reject the claim?

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Textbook Question

Graphical Analysis In Exercises 13 and 14, state whether each standardized test statistic X^2 allows you to reject the null hypothesis. Explain.


b. X^2=23.309

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