Ch. 12 - Analysis of Variance

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 12 - Analysis of Variance

Problem 12.2.5

Chapter 12, Problem 12.2.5

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?

Verified step by step guidance

Step 1: Understand the context of the problem. This is a two-way ANOVA test, which is used to analyze the effect of two independent variables (factors) on a dependent variable, as well as their interaction. The table provides the results of the analysis, including degrees of freedom (DF), sum of squares (SS), mean square (MS), F-statistic, critical F-value, and P-value for each source of variation.

Step 2: Interpret the interaction row. The interaction term tests whether the two factors interact significantly. Compare the P-value (0.01028) to the significance level (0.05). If the P-value is less than 0.05, conclude that there is a significant interaction effect between the two factors.

Step 3: Interpret the row variable. This term tests the main effect of the row variable (one of the factors). Compare the P-value (0.14832) to the significance level (0.05). If the P-value is greater than 0.05, conclude that the row variable does not have a significant main effect.

Step 4: Interpret the column variable. This term tests the main effect of the column variable (the other factor). Compare the P-value (0.01084) to the significance level (0.05). If the P-value is less than 0.05, conclude that the column variable has a significant main effect.

Step 5: Summarize the conclusions. Based on the P-values, there is a significant interaction effect between the two factors, no significant main effect for the row variable, and a significant main effect for the column variable. These conclusions should guide further analysis or decision-making.

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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 assesses not only the individual impact of each factor but also the interaction between them. This technique helps in understanding how different groups compare and whether their means are significantly different.

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 disparity between the group means relative to the variability within the groups, suggesting that at least one group mean is significantly different from the others. It is crucial for determining the significance of the results.

P-Value

The p-value is a measure that helps determine the significance of the results in hypothesis testing. It indicates the probability of observing the data, or something more extreme, assuming the null hypothesis is true. A p-value less than the significance level (e.g., 0.05) suggests that the null hypothesis can be rejected, indicating a statistically significant effect.

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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.

Clancy, Rowling, and Tolstoy Ease of Reading Pages were randomly selected from three books: The Bear and the Dragon by Tom Clancy, Harry Potter and the Sorcerer’s Stone by J.K. Rowling, and War and Peace by Leo Tolstoy. Listed below are Flesch Reading Ease Scores for those pages. Use a 0.05 significance level to test the claim that pages from books by those three authors have the same mean Flesch Reading Ease score. Given that higher scores correspond to text that is easier to read, which author appears to be different, and how is that author different?

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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.

P-VALUE If we use a 0.05 significance level in analysis of variance with the sample data given in Exercise 1, what is the P-value? What should we conclude? If the four populations have means that do not appear to be the same, does the analysis of variance test enable us to identify which populations have means that are significantly different?

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Textbook Question

Sitting Heights The sitting height of a person is the vertical distance between the sitting surface and the top of the head. The following table lists sitting heights (mm) of randomly selected U.S. Army personnel collected as part of the ANSUR II study. Using the data with a 0.05 significance level, what do you conclude? Are the results as you would expect?

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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

Cola Weights For the four samples described in Exercise 1, the sample of regular Coke has a mean weight of 0.81682 lb, the sample of Diet Coke has a mean weight of 0.78479 lb, the sample of regular Pepsi has a mean weight of 0.82410 lb, and the sample of Diet Pepsi has a mean weight of 0.78386 lb. If we use analysis of variance and reach a conclusion to reject equality of the four sample means, can we then conclude that any of the specific samples have means that are significantly different from the others?

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Textbook Question

Cola Weights Data Set 37 “Cola Weights and Volumes” in Appendix B lists the weights (lb) of the contents of cans of cola from four different samples: (1) regular Coke, (2) Diet Coke, (3) regular Pepsi, and (4) Diet Pepsi. The results from analysis of variance are shown in the Minitab display below. What is the null hypothesis for this analysis of variance test? Based on the displayed results, what should you conclude about H_knot. What do you conclude about equality of the mean weights from the four samples?

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