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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 7.3.11a
Chapter 7, Problem 7.3.11a

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


a. t = -1.755


Graph of a t-distribution showing critical values at -1.725 and 1.725, with shaded areas indicating rejection regions.

Verified step by step guidance
1
Identify the critical values from the graph, which are given as \(-1.725\) and \(1.725\). These values mark the boundaries of the rejection regions in the tails of the t-distribution.
Determine the rejection regions: any t-value less than \(-1.725\) or greater than \(1.725\) falls into the rejection region, meaning the null hypothesis would be rejected in those cases.
Compare the given test statistic \(t = -1.755\) to the critical values. Since \(-1.755\) is less than \(-1.725\), it lies in the left rejection region.
Conclude that because the test statistic falls in the rejection region, there is sufficient evidence to reject the null hypothesis at the given significance level.
Remember that this conclusion depends on the significance level associated with the critical values, which is typically set before the test (commonly 0.05 for two-tailed tests).

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Key Concepts

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

Null Hypothesis and Alternative Hypothesis

The null hypothesis (H0) is a statement of no effect or no difference, which we test against the alternative hypothesis (Ha). In hypothesis testing, we use sample data to decide whether to reject H0 in favor of Ha, based on the evidence provided by the test statistic.
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06:21
Step 1: Write Hypotheses

Critical Values and Rejection Regions

Critical values define the boundaries of rejection regions in a hypothesis test. If the test statistic falls into these regions (beyond the critical values), we reject the null hypothesis. The image shows critical values at ±1.725, marking the cutoff points for rejecting H0 at a given significance level.
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05:50
Critical Values: t-Distribution

Standardized Test Statistic (t-value)

The t-value measures how many standard errors the sample statistic is from the null hypothesis value. Comparing the calculated t-value to critical values helps determine if the observed data is statistically significant. Here, t = -1.755 lies beyond the critical value -1.725, indicating rejection of H0.
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Step 2: Calculate Test Statistic
Related Practice
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


a. you represent the manufacturer and want to support the claim?

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.


a. X^2=2.091

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

         

a. rejects 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.

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


Textbook Question

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.

Claim: p ≠0.15, α=0.05. Sample statistics: p_hat = 0.12, n=500

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

         

a. rejects the null hypothesis?


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