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Ch. 12 - Analysis of Variance
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 12 - Analysis of VarianceProblem 12.1.2
Chapter 12, Problem 12.1.2

Two-Way Anova If we have a goal of using the data given in Exercise 1 to (1) determine whether the femur side (left, right) has an effect on the crash force measurements and (2) to determine whether the vehicle size has an effect on the crash force measurements, should we use one-way analysis of variance for the two individual tests? Why or why not?

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Step 1: Understand the problem. The goal is to determine whether two factors—femur side (left, right) and vehicle size—have an effect on crash force measurements. This involves analyzing the interaction between these two factors and their individual effects.
Step 2: Recognize the limitations of one-way ANOVA. One-way ANOVA is designed to test the effect of a single factor on a dependent variable. It does not account for interactions between multiple factors, which is crucial in this case since we are dealing with two factors (femur side and vehicle size).
Step 3: Introduce Two-Way ANOVA. Two-way ANOVA is the appropriate statistical test here because it allows us to analyze the effects of two independent factors simultaneously on the dependent variable (crash force measurements). It also tests for interaction effects between the two factors.
Step 4: Explain why interaction effects matter. Interaction effects occur when the impact of one factor on the dependent variable depends on the level of the other factor. For example, the effect of femur side on crash force measurements might vary depending on the vehicle size. Two-way ANOVA can detect such interactions, whereas one-way ANOVA cannot.
Step 5: Conclude the reasoning. Since the problem involves two factors and potentially their interaction, one-way ANOVA is insufficient. Two-way ANOVA should be used to properly analyze the data and address both questions about femur side and vehicle size effects on crash force measurements.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Two-Way ANOVA

Two-Way ANOVA is a statistical method used to determine the effect of two independent categorical variables on a continuous dependent variable. It allows researchers to assess not only the individual impact of each factor but also any interaction effects between them. In this case, the two factors are femur side (left, right) and vehicle size, which can influence crash force measurements.
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One-Way ANOVA

One-Way ANOVA is a statistical test used to compare the means of three or more independent groups based on one categorical independent variable. It assesses whether there are statistically significant differences between the group means. In the context of the question, using one-way ANOVA for each factor separately would ignore potential interactions between the femur side and vehicle size.
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Interaction Effects

Interaction effects occur when the effect of one independent variable on the dependent variable differs depending on the level of another independent variable. In the context of Two-Way ANOVA, it is crucial to examine these interactions to understand how the combination of factors influences the outcome. Ignoring interaction effects by using one-way ANOVA could lead to misleading conclusions about the data.
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Related Practice
Textbook Question

In Exercises 5–16, use analysis of variance for the indicated test.


Triathlon Times Jeff Parent is a statistics instructor who participates in triathlons. Listed below are times (in minutes and seconds) he recorded while riding a bicycle for five stages through each mile of a 3-mile loop. Use a 0.05 significance level to test the claim that it takes the same time to ride each of the miles. Does one of the miles appear to have a hill?

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Textbook Question

Weights from ANSUR I and ANSUR II The following table lists weights (kg) of randomly selected U.S. Army personnel obtained from the ANSUR I study conducted in 1988 and the ANSUR II study conducted in 2012. If we use the data with two-way analysis of variance and a 0.05 significance level, we get the accompanying display. What do you conclude?

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Textbook Question

Distance Between Pupils The following table lists distances (mm) between pupils of randomly selected U.S. Army personnel collected as part of the ANSUR II study. Results from two-way analysis of variance are also shown. Use the displayed results and use a 0.05 significance level. What do you conclude? Are the results as you would expect?

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Textbook Question

Two-Way Anova The measurements of crash test forces on the femur in Table 12-3 from Example 1 are reproduced below with fabricated measurement data (in red) used for the left femur in a small car. What characteristic of the data suggests that the appropriate method of analysis is two-way analysis of variance? That is, what is “two-way” about the data entered in this table?

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Textbook Question

Balanced Design Does the table given in Exercise 1 constitute a balanced design? Why or why not?

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