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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.R.3
Chapter 8, Problem 8.R.3

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

Verified step by step guidance
1
Understand the concept of independent and dependent samples: Independent samples are those where the observations in one sample do not influence or relate to the observations in the other sample. Dependent samples, on the other hand, involve paired or related observations, such as before-and-after measurements or matched pairs.
Examine the nature of the two samples provided: Sample 1 consists of the retail prices of 20 motorcycles, and Sample 2 consists of the retail prices of 20 minivans. Consider whether there is any inherent relationship or pairing between the observations in these two samples.
Ask whether the samples are paired or related: For example, are the motorcycles and minivans paired in any way, such as being sold by the same dealership or being part of a specific comparison study? If there is no such pairing or relationship, the samples are likely independent.
Determine if the samples influence each other: If the retail price of a motorcycle does not affect the retail price of a minivan, and vice versa, this is another indication that the samples are independent.
Conclude the classification: Based on the analysis, if the samples are unrelated and there is no pairing or dependency between the observations, classify them as independent samples. If there is a clear relationship or pairing, classify them as dependent 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 influenced by each other. In statistical analysis, this means that the selection or outcome of one sample does not affect the other. For example, if we compare the prices of motorcycles and minivans, the price of one does not impact the price of the other, making them independent.
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Dependent Samples

Dependent samples, also known as paired samples, occur when the samples are related or matched in some way. This could involve measurements taken from the same subjects at different times or under different conditions. In contrast to independent samples, the outcome of one sample can influence the other, which is not the case in the given question.
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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 they were collected. In the context of the question, one must analyze whether the motorcycle prices and minivan prices are influenced by each other or are separate entities.
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Related Practice
Textbook Question

In Exercises 11–16, 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=0.10. Assume (σ1)^2 ≠ (σ2)^2


Sample statistics: x̅1= 520, s1= 25, n1= 7 and x̅2= 500, s2= 55, n2= 6

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Textbook Question

In Exercises 19–22, test the claim about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed.


Claim: μd≠0; α=0.05.


Sample statistics: d̄=17.5, sd=4.05, n=37



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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?

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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

82
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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)


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Textbook Question

In Exercises 19–22, test the claim about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed.


Claim: μd<0; α=0.10.


Sample statistics: d̄=3.2, sd=5.68, n=25

60
views