Textbook Question
In Exercises 7–10, the statement represents a claim. Write its complement and state which is Ho and which is Ha.
μ≠2.28
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Ch. 8 - Hypothesis Testing with Two Samples
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 8, Problem 8.1.8
Independent and Dependent Samples In Exercises 5–8, classify the two samples as independent or dependent and justify your answer.
Sample 1: The commute times of 10 workers when they use their own vehicles
Sample 2: The commute times of the same 10 workers when they use public transportation
Verified step by step guidance1
Identify the two samples provided in the problem: Sample 1 consists of the commute times of 10 workers using their own vehicles, and Sample 2 consists of the commute times of the same 10 workers using public transportation.
Understand the distinction between independent and dependent samples. Independent samples are those where the observations in one sample are not related to the observations in the other sample. Dependent samples, on the other hand, are paired or related in some way, such as measurements taken from the same individuals under different conditions.
Examine the relationship between the two samples. In this case, the same 10 workers are being observed under two different conditions: using their own vehicles and using public transportation. This creates a natural pairing between the observations in Sample 1 and Sample 2.
Conclude that the two samples are dependent because the commute times in Sample 1 and Sample 2 are linked to the same individuals, making the observations paired.
Justify the classification by noting that the comparison involves the same group of workers under two different commuting conditions, which inherently creates a dependency between the samples.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Independent Samples
Independent samples refer to two or more groups that are not related or paired in any way. In statistical analysis, this means that the selection or outcome of one sample does not influence the other. For example, if we were comparing the test scores of two different classes, the scores from one class would not affect the scores from the other.
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Independence Test
Dependent Samples
Dependent samples, also known as paired samples, involve groups that are related or matched in some way. This typically occurs when the same subjects are measured under different conditions, such as before and after a treatment. In the given question, the same workers' commute times are measured under two different transportation methods, making the samples dependent.
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Justification in Statistical Analysis
Justification in statistical analysis involves providing a rationale for classifying samples as independent or dependent. This includes explaining the relationship between the samples and how they were collected. In the context of the question, the justification would highlight that the same individuals are being observed in both samples, thus establishing a dependent relationship.
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Related Practice
Textbook Question
A pediatrician claims that the mean birth weight of a single-birth baby is greater than the mean birth weight of a baby that has a twin. The mean birth weight of a random sample of 85 single-birth babies is 3086 grams. Assume the population standard deviation is 563 grams. The mean birth weight of a random sample of 68 babies that have a twin is 2263 grams. Assume the population standard deviation is 624 grams. At α=0.10, can you support the pediatrician’s claim? Interpret the decision in the context of the original claim.
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Textbook Question
Describe another way you can perform a hypothesis test for the difference between the means of two populations using independent samples with and known that does not use rejection regions.
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Textbook Question
Getting at the Concept Explain why the null hypothesis Ho: μ1=μ2 is equivalent to the null hypothesis .Ho: μ1-μ2=0
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Textbook Question
Seat Belt Use In a survey of 1000 drivers from the West, 934 wear a seat belt. In a survey of 1000 drivers from the Northeast, 909 wear a seat belt. At α=0.05, can you support the claim that the proportion of drivers who wear seat belts is greater in the West than in the Northeast? (Adapted from National Highway Traffic Safety Administration)
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Textbook Question
Constructing Confidence Intervals for μ1-μ2. When the sampling distribution for x̅1-x̅2 is approximated by a t-distribution and the populations have equal variances, you can construct a confidence interval for μ1-μ2, as shown below.
Construct the indicated confidence interval for μ1-μ2 . Assume the populations are approximately normal with equal variances.
10K Race
To compare the mean ages of male and female participants in a 10K race, you randomly select several ages from both sexes. The results are shown below. Construct a 95% confidence interval for the difference in mean ages of male and female participants in the race. (Adapted from Great Race)
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