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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.RE.2
Chapter 8, Problem 8.RE.2

In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.


Sample 1: The weights of 45 oranges
Sample 2: The weights of 40 grapefruits

Verified step by step guidance
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Step 1: 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, involve paired or related observations, such as before-and-after measurements or matched pairs.
Step 2: Analyze the given samples. Sample 1 consists of the weights of 45 oranges, and Sample 2 consists of the weights of 40 grapefruits. Consider whether there is any inherent relationship or pairing between the weights of the oranges and the weights of the grapefruits.
Step 3: Determine if the samples are paired or related. Since the weights of the oranges and grapefruits are measured independently and there is no indication that the oranges and grapefruits are matched or paired in any way, the samples are likely independent.
Step 4: Justify the classification. The samples are independent because the weights of the oranges do not influence or depend on the weights of the grapefruits, and there is no pairing or relationship between the two groups.
Step 5: Conclude the classification. Based on the analysis, classify the two samples as independent and provide reasoning that supports this conclusion.

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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 you measure the weights of oranges and grapefruits from different sources, the results from one do not impact the results from the other.
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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 can happen when the same subjects are measured under different conditions or at different times. For instance, if you measured the weights of the same oranges before and after a specific treatment, those samples would be dependent.
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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, such as whether they come from different populations or if they are linked through a common factor. Clear justification is essential for ensuring the validity of the statistical methods applied.
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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=0.01. Assume (σ1)^2 = (σ2)^2


Sample statistics: x̅1= 44.5, s1= 5.85, n1= 17 and x̅2= 49.1, s2= 5.25, n2= 18

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Textbook Question

In Exercises 17 and 18, (c) find the standardized test statistic t, 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 1–4, classify the two samples as independent or dependent and justify your answer.


Sample 1: The fuel efficiencies of 12 cars

Sample 2: The fuel efficiencies of the same 12 cars using an alternative fuel




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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= 14 and σ2= 15


Sample statistics: x̅1 = 87, n1 = 410, and x̅2= 85, n2= 340

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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=0.01. Assume (σ1)^2 = (σ2)^2


Sample statistics: x̅1= 61, s1= 3.3, n1= 5 and x̅2= 55.1, s2= 1.2, n2= 7

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Textbook Question

In Exercises 17 and 18, (a) identify the claim and state Ho and Ha, 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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