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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 12.2.11a
Chapter 12, Problem 12.2.11a

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?


a. The same constant is added to each sample value.

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 variables on a dependent variable, as well as their interaction. It assumes that the data is normally distributed and that variances are equal across groups.
Recall the property of adding a constant to all data points: Adding a constant to all values in a dataset shifts the mean of the dataset by that constant but does not affect the variance or the relative differences between data points.
Analyze the effect on the two-way ANOVA results: Since the F-statistic in ANOVA is based on the ratio of variances (between-group variance to within-group variance), adding a constant to all sample values does not change the variances. Therefore, the F-statistic and p-values remain unaffected.
Consider the interpretation of the results: The statistical conclusions drawn from the two-way ANOVA, such as whether the factors or their interaction are significant, will remain the same because the underlying variances and relationships between groups are unchanged.
Summarize the impact: Adding a constant to each sample value does not affect the results of the two-way ANOVA because it does not alter the variances or the relationships between the groups being compared.

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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 helps in understanding if there are any significant interactions between the two factors and how they influence the outcome. This technique is particularly useful when analyzing data with multiple groups and can provide insights into both main effects and interaction effects.
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Data Transformation

Data transformation involves modifying the values in a dataset to meet the assumptions of statistical tests or to improve the interpretability of the data. Adding a constant to each sample value is a common transformation that shifts the entire dataset without affecting the relative differences between values. This type of transformation can influence the mean but does not change the variance or the results of ANOVA tests regarding group differences.
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Effect of Adding a Constant

When a constant is added to each value in a dataset, it uniformly shifts the data without altering the relationships between the groups. In the context of ANOVA, this means that while the overall means of the groups will change, the differences between group means remain the same. Consequently, the statistical significance of the ANOVA results is unaffected, as the test focuses on the variance among group means rather than their absolute values.
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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.

a. Use a 0.05 significance level to test the claim that the different treatments result in the same mean weight.

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Textbook Question

Interaction


a. What is an interaction between two factors?


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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.

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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.

b. What do the displayed Bonferroni SPSS results tell us?

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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


a. What characteristic of the data above indicates that we should use one-way analysis of variance?

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Textbook Question

Birth Weights Data Set 6 “Births” includes birth weights (g), hospitals, and the day of the week that mothers were admitted to the hospital. Using rows to represent the four hospitals (Albany Medical Center, Bellevue Hospital Center, Olean General Hospital, Strong Memorial Hospital), and using columns to represent the seven different days of the week, a two-way table has 28 individual cells. Using five birth weights for each of those 28 cells and using StatCrunch for two-way analysis of variance, we get the results displayed below. What do you conclude?

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