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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.1
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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1
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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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

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 17 and 18, (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 real estate agent claims that there is no difference between the mean household incomes of two neighborhoods. The mean income of 12 randomly selected households from the first neighborhood is \$52,750 with a standard deviation of \$2900. In the second neighborhood, 10 randomly selected households have a mean income of \$51,200 with a standard deviation of \$2225. At α=0.01, can you reject the real estate agent’s claim? Assume the population variances are equal.

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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



86
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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=0.01


Population statistics: σ1= 52 and σ2= 68


Sample statistics: x̅1 = 5595, n1 = 156, and x̅2= 5575, n2= 216

42
views
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
views
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=0.015, s1=0.011, n1= 8 and x̅2=0.019, s2=0.004, n2= 6

68
views