Ch. 12 - Analysis of Variance

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 12 - Analysis of Variance

Problem 12.2.7

Chapter 12, Problem 12.2.7

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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Step 1: Understand the problem. The table shows distances (in mm) between pupils for U.S. Army personnel categorized by gender (Female/Male) and handedness (Right-Handed/Left-Handed). The ANOVA results are provided to test for interaction effects, row effects (gender), and column effects (handedness) at a significance level of 0.05.

Step 2: Analyze the interaction effect. Look at the 'Interaction' row in the ANOVA table. The p-value is 0.07489, which is greater than the significance level of 0.05. This indicates that there is no statistically significant interaction between gender and handedness.

Step 3: Analyze the row variable (gender). The p-value for the 'Row Variable' is 0.03433, which is less than the significance level of 0.05. This suggests that gender has a statistically significant effect on the distance between pupils.

Step 4: Analyze the column variable (handedness). The p-value for the 'Column Variable' is 0.15388, which is greater than the significance level of 0.05. This indicates that handedness does not have a statistically significant effect on the distance between pupils.

Step 5: Conclude the findings. Based on the analysis, gender significantly affects pupil distance, but handedness does not. Additionally, there is no significant interaction between gender and handedness. These results align with expectations if gender differences in physical characteristics are known to influence pupil distance.

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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 assesses not only the individual impact of each factor but also the interaction between them. In this case, the factors are handedness (right-handed vs. left-handed) and gender (male vs. female), and the dependent variable is the distance between pupils.

F-Statistic

The F-statistic is a ratio used in ANOVA to compare the variance between group means to the variance within the groups. A higher F-statistic indicates a greater degree of variance between the groups relative to the variance within the groups, suggesting that at least one group mean is significantly different. In the provided results, the F-statistics for the interaction, row, and column variables help determine the significance of the effects.

P-Value

The p-value is a measure that helps determine the statistical significance of the results obtained from an ANOVA test. It indicates the probability of observing the data, or something more extreme, if the null hypothesis is true. A p-value less than the significance level (0.05 in this case) suggests that the null hypothesis can be rejected, indicating a significant effect of the factors being studied. The provided p-values for the interaction, row, and column variables guide the conclusions drawn from the analysis.

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Step 3: Get P-Value

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?

Textbook Question

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?

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?

Textbook Question

Pancake Experiment Listed below are ratings of pancakes made by experts (based on data from Minitab). Different pancakes were made with and without a supplement and with different amounts of whey. The results from two-way analysis of variance are shown. Use the displayed results and a 0.05 significance level. What do you conclude?

Textbook Question

Tukey Test A display of the Bonferroni test results from Table 12-1 (which is part of the Chapter Problem) is provided here. Shown on the top of the next page is the SPSS-generated display of results from the Tukey test using the same data. Compare the Tukey test results to those from the Bonferroni test.

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Textbook Question

Car Crash Test Measurements If we use the data given in Exercise 1 with two-way analysis of variance and a 0.05 significance level, we get the accompanying display. What do you conclude?