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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 7.3.3
Chapter 7, Problem 7.3.3

In Exercises 3–8, find the critical value(s) and rejection region(s) for the type of t-test with level of significance alpha and sample size n.


Left-tailed test, α=0.10, n=20

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Determine the degrees of freedom (df) for the t-test. The formula for degrees of freedom is df = n - 1, where n is the sample size. In this case, df = 20 - 1.
Identify the level of significance (α) for the test. Here, α = 0.10, which represents the probability of rejecting the null hypothesis when it is true.
Since this is a left-tailed test, the critical value corresponds to the t-score where the cumulative probability to the left of the critical value equals α. Use a t-distribution table or statistical software to find the t-score for df = 19 and α = 0.10.
Define the rejection region. For a left-tailed test, the rejection region is t < critical value, where the critical value is the t-score found in the previous step.
Summarize the critical value and rejection region. Clearly state the critical value and describe the rejection region in terms of the t-statistic (e.g., 'Reject the null hypothesis if t < critical value').

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Key Concepts

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

Critical Value

A critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis. It is determined based on the significance level (alpha) and the type of test being conducted. For a left-tailed test, the critical value corresponds to the point where the cumulative probability equals alpha, indicating the threshold for rejection.
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Critical Values: t-Distribution

Rejection Region

The rejection region is the range of values for the test statistic that leads to the rejection of the null hypothesis. In a left-tailed test, this region is located to the left of the critical value. If the calculated test statistic falls within this region, it suggests that the sample provides sufficient evidence to reject the null hypothesis at the specified significance level.
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Guided course
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Step 4: State Conclusion

T-Test

A t-test is a statistical test used to determine if there is a significant difference between the means of two groups, especially when the sample size is small and the population standard deviation is unknown. The t-test accounts for sample size and variability, making it suitable for hypothesis testing in various scenarios, including one-sample, independent two-sample, and paired sample tests.
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Related Practice
Textbook Question

Hypothesis Testing Using Rejection Region(s) In Exercises 39–44, (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.


[APPLET] Fluorescent Lamps A compact fluorescent lamp (CFL) bulb manufacturer guarantees that the mean life of a CFL bulb is at least 10,000 hours. You want to test this guarantee. To do so, you record the lives of a random sample of 32 CFL bulbs. The results (in hours) are listed. Assume the population standard deviation is 1850 hours. At alpha=0.11, do you have enough evidence to reject the manufacturer’s claim?


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 ≥0.48, α=0.08. Sample statistics: p_hat = 0.40, n=90

Textbook Question

Hypothesis Testing Using Rejection Regions In Exercises 19–26, (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 t, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the population is normally distributed.


Lead Levels As part of your work for an environmental awareness group, you want to test a claim that the mean amount of lead in the air in U.S. cities is less than 0.032 microgram per cubic meter. You find that the mean amount of lead in the air for a random sample of 56 U.S. cities is 0.021 microgram per cubic meter and the standard deviation is 0.034 microgram per cubic meter. At α=0.01, can you support the claim?

Textbook Question

True or False? In Exercises 5–10, determine whether the statement is true or false. If it is false, rewrite it as a true statement.


To support a claim, state it so that it becomes the null hypothesis.

Textbook Question

What are the two types of hypotheses used in a hypothesis test? How are they related?

Textbook Question

[APPLET] A weight loss program claims that program participants have a mean weight loss of at least 10.5 pounds after 1 month. The weight losses after 1 month (in pounds) of a random sample of 40 program participants are listed below. At α=0.01, is there enough evidence to reject the program’s claim?