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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 12.2.11c
Chapter 12, Problem 12.2.11c

Transformations of Data Example 1 illustrated the use of two-way ANOVA to analyze the sample data in Table 12-3. How are the results affected in each of the following cases?


c. The format of the table is transposed so that the row and column factors are interchanged.

Verified step by step guidance
1
Understand the concept of two-way ANOVA: Two-way ANOVA is used to analyze the effect of two independent categorical variables (factors) on a dependent continuous variable. It also examines the interaction between these two factors.
Recognize the structure of the data table: In the original format, the rows represent one factor (Factor A) and the columns represent another factor (Factor B). The cells contain the dependent variable values.
Consider the transposed format: When the table is transposed, the roles of the row and column factors are swapped. Factor A becomes the column factor, and Factor B becomes the row factor. The dependent variable values remain unchanged.
Analyze the impact on the results: The statistical results of the two-way ANOVA (such as F-statistics, p-values, and interaction effects) are not affected by transposing the table. This is because the analysis is based on the relationship between the factors and the dependent variable, not the physical arrangement of the table.
Conclude that the interpretation of the factors changes: While the numerical results remain the same, the labels and interpretation of the factors are reversed. For example, if Factor A originally represented 'Treatment Type' and Factor B represented 'Location,' after transposing, 'Location' would be the column factor and 'Treatment Type' would be the row factor.

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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 (Analysis of Variance) 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. This technique is particularly useful when analyzing complex datasets with multiple factors.
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Factor Interaction

Factor interaction occurs 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, understanding interaction is crucial as it can reveal whether the combined influence of factors is greater or less than their individual effects. This can significantly affect the interpretation of results.
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Data Transposition

Data transposition involves switching the rows and columns of a data table, which can change the way factors are represented in an analysis. In the context of two-way ANOVA, transposing the table may affect the interpretation of the factors and their interactions, but the underlying data remains the same. It is essential to understand how this change can influence the analysis and results.
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Related Practice
Textbook Question

Bonferroni Test Shown below are weights (kg) of poplar trees obtained from trees planted in a rich and moist region. The trees were given different treatments identified in the table below. The data are from a study conducted by researchers at Pennsylvania State University and were provided by Minitab, Inc. Also shown are partial results from using the Bonferroni test with the sample data.

c. Use the Bonferroni test procedure with a 0.05 significance level to test for a significant difference between the mean amount of the irrigation treatment group and the group treated with both fertilizer and irrigation. Identify the test statistic and either the P-value or critical values. What do the results indicate?

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Textbook Question

In Exercises 1–4, use the following listed measured amounts of chest compression (mm) from car crash tests (from Data Set 35 “Car Data” in Appendix B). Also shown are the SPSS results from analysis of variance. Assume that we plan to use a 0.05 significance level to test the claim that the different car sizes have the same mean amount of chest compression.



Anova


b. If the objective is to test the claim that the four car sizes have the same mean chest compression, why is the method referred to as analysis of variance?

Textbook Question

Transformations of Data Example 1 illustrated the use of two-way ANOVA to analyze the sample data in Table 12-3. How are the results affected in each of the following cases?


b. Each sample value is multiplied by the same nonzero constant.

Textbook Question

In Exercises 1–5, refer to the following list of numbers of years that deceased U.S. presidents, popes, and British monarchs lived after their inauguration, election, or coronation, respectively. (As of this writing, the last president is George H. W. Bush, the last pope is John Paul II, and the last British monarch is George VI.) Assume that the data are samples from larger populations.


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Exploring the Data Include appropriate units in all answers.


d. Are there any obvious outliers?

Textbook Question

c. Shown below is an interaction graph constructed from the data in Exercise 1. What does the graph suggest?

Textbook Question

Transformations of Data Example 1 illustrated the use of two-way ANOVA to analyze the sample data in Table 12-3. How are the results affected in each of the following cases?


d. The first sample value in the first cell is changed so that it becomes an outlier.