Ch. 7 - Hypothesis Testing with One Sample

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 7 - Hypothesis Testing with One Sample

Problem 7.1.56

Chapter 7, Problem 7.1.56

Writing A null hypothesis is rejected with a level of significance of 0.10. Is it also rejected at a level of significance of 0.05? Explain.

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Understand the concept of the level of significance: The level of significance (denoted as α) is the probability of rejecting the null hypothesis when it is actually true. A smaller α represents a stricter criterion for rejecting the null hypothesis.

Recognize the relationship between α = 0.10 and α = 0.05: A level of significance of 0.05 is more stringent than 0.10. This means that if a null hypothesis is rejected at α = 0.10, it does not necessarily mean it will be rejected at α = 0.05.

Recall the p-value approach: The p-value is the probability of observing the test statistic or something more extreme under the null hypothesis. If the p-value is less than or equal to the level of significance, the null hypothesis is rejected.

Compare the p-value to both levels of significance: If the p-value is less than or equal to 0.10 but greater than 0.05, the null hypothesis will be rejected at α = 0.10 but not at α = 0.05. If the p-value is less than or equal to 0.05, the null hypothesis will be rejected at both levels of significance.

Conclude based on the p-value: To determine whether the null hypothesis is rejected at α = 0.05, you need to know the p-value. If the p-value is not provided, you cannot definitively conclude whether the null hypothesis is rejected at the stricter level of significance.

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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, and it serves as the default assumption in hypothesis testing. Researchers aim to gather evidence to either reject or fail to reject this hypothesis based on sample data. In this context, rejecting the null hypothesis at a significance level indicates that the observed data is unlikely under the assumption that the null hypothesis is true.

Level of Significance

The level of significance, often denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. Common levels of significance are 0.05 and 0.10, with lower values indicating stricter criteria for rejecting the null hypothesis.

P-Value

The p-value is a statistical measure that helps determine the strength of the evidence against the null hypothesis. It quantifies the probability of observing the sample data, or something more extreme, if the null hypothesis is true. If the p-value is less than the chosen level of significance, the null hypothesis is rejected. Therefore, if a null hypothesis is rejected at 0.10, it may or may not be rejected at 0.05, depending on the p-value.

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Step 3: Get P-Value

Related Practice

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 > 0.70, α=0.04. Sample statistics: p_hat = 0.64, n=225

Textbook Question

Hypothesis Testing Using Rejection Regions In Exercises 23–30, (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 X^2, (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.

Salaries The annual salaries (in dollars) of 15 randomly chosen senior level graphic design specialists are shown in the table at the left. At α=0.05, is there enough evidence to support the claim that the standard deviation of the annual salaries is different from \$13,056?

Textbook Question

In Exercises 29–32, test the claim about the population mean at the level of significance α. Assume the population is normally distributed.

Claim: ; μ ≤ 22,500; α = 0.01; α = 1200

Sample statistics: x_bar = 23,500, n = 45

Textbook Question

Graphical Analysis In Exercises 9–12, match the P-value or z-statistic with the graph that represents the corresponding area. Explain your reasoning.

z = -0.51