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Ch. 8 - Hypothesis Testing with Two Samples
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. 8 - Hypothesis Testing with Two SamplesProblem 8.Ch6-9.9
Chapter 8, Problem 8.Ch6-9.9

In Exercises 7–10, the statement represents a claim. Write its complement and state which is Ho and which is Ha.


σ=0.63

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Understand the problem: The given statement is a claim about the population standard deviation (σ). The claim is that σ = 0.63. We need to write the complement of this claim and identify which statement is the null hypothesis (Ho) and which is the alternative hypothesis (Ha).
Recall the definitions: The null hypothesis (Ho) is a statement of no change, no effect, or equality. It is the hypothesis that we assume to be true unless there is strong evidence against it. The alternative hypothesis (Ha) is the statement that contradicts the null hypothesis and represents the claim we are testing.
Write the null hypothesis (Ho): Since the claim is that σ = 0.63, the null hypothesis will be Ho: σ = 0.63. This represents the assumption that the population standard deviation is equal to 0.63.
Write the alternative hypothesis (Ha): The complement of the claim is that σ ≠ 0.63. This represents the possibility that the population standard deviation is not equal to 0.63. Therefore, the alternative hypothesis is Ha: σ ≠ 0.63.
Summarize the hypotheses: The null hypothesis (Ho) is σ = 0.63, and the alternative hypothesis (Ha) is σ ≠ 0.63. These hypotheses will be used in hypothesis testing to determine whether there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.

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Key Concepts

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

Null Hypothesis (Ho)

The null hypothesis, denoted as Ho, is a statement that indicates no effect or no difference in a statistical test. It serves as the default assumption that any observed effect is due to sampling variability. In this case, Ho would assert that the population standard deviation σ is equal to 0.63.
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Step 1: Write Hypotheses Example 1

Alternative Hypothesis (Ha)

The alternative hypothesis, represented as Ha, is a statement that contradicts the null hypothesis. It suggests that there is an effect or a difference. For the given claim, Ha would state that the population standard deviation σ is not equal to 0.63, indicating a deviation from the null hypothesis.
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Step 1: Write Hypotheses

Complement of a Hypothesis

The complement of a hypothesis refers to the opposite statement of the original claim. In hypothesis testing, if Ho states that σ equals a specific value, the complement would assert that σ does not equal that value. This is crucial for determining the scope of the test and understanding the relationship between Ho and Ha.
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Complementary Events
Related Practice
Textbook Question

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

b. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.

The mean score on a reading assessment test for 49 randomly selected male high school students was 279. Assume the population standard deviation is 41. The mean score on the same test for 50 randomly selected female high school students was 292. Assume the population standard deviation is 39. At α=0.05, can you support the claim that the mean score on the reading assessment test for male high school students is less than the mean score for female high school students? (Adapted from National Center for Education Statistics)

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Textbook Question

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

f. Interpret the decision in the context of the original claim.

The mean score on a reading assessment test for 49 randomly selected male high school students was 279. Assume the population standard deviation is 41. The mean score on the same test for 50 randomly selected female high school students was 292. Assume the population standard deviation is 39. At α=0.05, can you support the claim that the mean score on the reading assessment test for male high school students is less than the mean score for female high school students? (Adapted from National Center for Education Statistics)

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Textbook Question

[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)


48,69446,85642,91261,67271,11254,861


69,45471,84159,75169,61254,28452,166


66,36048,16465,27235,25061,12765,397


58,92558,91659,01753,07045,19969,941


69,49257,08553,82952,69268,29853,792



Construct a 95% confidence interval for the population mean annual earnings for locksmiths.

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Textbook Question

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.


a. Identify the claim and state Ho and Ha


b. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.


c. Find the critical value(s) and identify the rejection region(s).


d. Find the appropriate standardized test statistic.


e. Decide whether to reject or fail to reject the null hypothesis.


f. Interpret the decision in the context of the original claim.



[APPLET] The table shows the credit scores for 12 randomly selected adults who are considered high-risk borrowers before and two years after they attend a personal finance seminar. At α=0.01, is there enough evidence to support the claim that the personal finance seminar helps adults increase their credit scores? Assume the populations are normally distributed.

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Textbook Question

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \$31. Assume the population is normally distributed. (Adapted from Expedia)


Construct a 99% confidence interval for the population variance.

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Textbook Question

[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)


48,69446,85642,91261,67271,11254,861


69,45471,84159,75169,61254,28452,166


66,36048,16465,27235,25061,12765,397


58,92558,91659,01753,07045,19969,941


69,49257,08553,82952,69268,29853,792



A researcher claims that the mean annual earnings for locksmiths is \$55,000. At α=0.05, can you reject the researcher’s claim? Interpret the decision in the context of the original claim.

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