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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 12.1.18b
Chapter 12, Problem 12.1.18b

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?
Table showing Bonferroni test results from SPSS comparing treatments with mean differences, standard errors, significance, and confidence intervals.

Verified step by step guidance
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Step 1: Understand the context of the Bonferroni test. It is a post-hoc multiple comparison test used after an ANOVA to determine which specific group means are significantly different from each other while controlling the overall Type I error rate.
Step 2: Identify the treatments being compared. The treatments are No Treatment (1), Fertilizer (2), Irrigation (3), and Fertilizer and Irrigation (4). The Bonferroni results table compares the mean differences between No Treatment (1) and each of the other treatments.
Step 3: Interpret the 'Mean Difference (I-J)' column. This shows the difference in average weights between the No Treatment group and each other treatment group. For example, the difference between No Treatment and Fertilizer is -0.02200, indicating the Fertilizer group has a slightly lower mean weight than No Treatment.
Step 4: Look at the 'Sig.' (significance) column to determine if the differences are statistically significant. A value less than the chosen alpha level (commonly 0.05) indicates a significant difference. Here, the significance values for comparisons with Fertilizer and Irrigation are 1.000 and 0.393 respectively, which are not significant, but the comparison with Fertilizer and Irrigation combined is 0.039, which is significant.
Step 5: Examine the 95% confidence intervals for the mean differences. If the interval includes zero, it suggests no significant difference. For the Fertilizer and Irrigation combined treatment, the confidence interval does not include zero (-1.6549 to -0.0331), confirming the significant difference found in the significance column.

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Key Concepts

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

Bonferroni Test

The Bonferroni test is a multiple comparison procedure used after an ANOVA to control the overall Type I error rate when making pairwise comparisons. It adjusts the significance level by dividing it by the number of comparisons, making it more stringent to declare differences significant. This helps prevent false positives when testing multiple hypotheses simultaneously.
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Interpretation of SPSS Bonferroni Output

The SPSS Bonferroni output shows pairwise mean differences between treatment groups, their standard errors, significance values (p-values), and confidence intervals. A significant p-value (typically < 0.05) indicates a statistically significant difference between the two treatment means. Confidence intervals that do not include zero also support significant differences.
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Multiple Comparisons in Experimental Design

In experiments with multiple treatment groups, multiple comparisons are necessary to identify which groups differ. However, conducting many tests increases the chance of Type I errors. Methods like the Bonferroni correction adjust for this by controlling the family-wise error rate, ensuring that the overall probability of making one or more false discoveries remains low.
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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?


Textbook Question

Interaction


b. In general, when using two-way analysis of variance, if we find that there is an interaction effect, how does that affect the procedure?


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?


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

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.