Textbook Question
In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.
Sample 1: The retail prices of 20 motorcycles
Sample 2: The retail prices of 20 minivans
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.R.1
In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.
Sample 1: The heights of 37 children
Sample 2: The heights of the same 37 children after 1 year
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Identify the two samples provided in the problem: Sample 1 consists of the heights of 37 children, and Sample 2 consists of the heights of the same 37 children after 1 year.
Understand the concept of independent and dependent samples: Independent samples are those where the observations in one sample do not influence or 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 on the same subjects at different times.
Examine the relationship between the two samples: Since Sample 2 involves the same 37 children as in Sample 1, but measured after 1 year, the two samples are inherently related. Each observation in Sample 1 corresponds directly to an observation in Sample 2 for the same child.
Conclude that the two samples are dependent because the measurements in Sample 2 are paired with the measurements in Sample 1 for the same individuals (the same 37 children).
Justify the classification: The dependency arises because the heights in Sample 2 are influenced by the same individuals whose heights were measured in Sample 1, making the samples paired and not independent.
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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 groups that are not related or paired in any way. In statistical analysis, this means that the data collected from one sample does not influence or affect the data from another sample. For example, if you were to measure the heights of two different groups of children, those groups would be considered independent.
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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 more than once, such as before and after a treatment. In the given question, the heights of the same 37 children measured after one year represent dependent samples because the second measurement is directly related to the first.
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Justification in Statistical Analysis
Justification in statistical analysis involves providing reasoning or evidence to support the classification of samples as independent or dependent. This includes explaining the relationship between the samples and how the data collection method influences the results. In this case, the justification is based on the fact that the same children are measured at two different times, indicating a dependent relationship.
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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.
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)
Textbook Question
In Exercises 9 and 10, (a) identify the claim and state Ho 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. Assume the samples are random and independent, and the populations are normally distributed.
A researcher claims that the mean sodium content of sandwiches at Restaurant A is less than the mean sodium content of sandwiches at Restaurant B. The mean sodium content of 22 randomly selected sandwiches at Restaurant A is 670 milligrams. Assume the population standard deviation is 20 milligrams. The mean sodium content of 28 randomly selected sandwiches at Restaurant B is 690 milligrams. Assume the population standard deviation is 30 milligrams. At α=0.05, is there enough evidence to support the claim?
Textbook Question
In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.
Claim: μ1>μ2; α=0.05
Population statistics: σ1= 0.30 and σ2= 0.23
Sample statistics: x̅1 = 1.28, n1 = 96, and x̅2= 1.34, n2= 85
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
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. (Adapted from National Center for Education Statistics)
Textbook Question
In Exercises 29 and 30, (a) identify the claim and state Ho 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. Assume the samples are random and independent.
A medical research team conducted a study to test the effect of a drug used to treat a type of inflammation. In the study, 68 subjects took the drug and 68 subjects took a placebo. The results are shown below. At α=0.05, can you reject the claim that the proportion of subjects who had at least 24 weeks of accrued remission is the same for the two groups? (Source: The New England Journal of Medicine)