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.

Verified step by step guidance

1. Step 1: Identify the null and alternative hypotheses for the test. The null hypothesis (H0) is that all treatment means are equal, i.e., \( \mu_1 = \mu_2 = \mu_3 = \mu_4 \). The alternative hypothesis (Ha) is that at least one treatment mean is different.
2. Step 2: Perform an ANOVA test to determine if there is a statistically significant difference among the treatment means. This involves calculating the between-group variability and within-group variability and then computing the F-statistic.
3. Step 3: Use the significance level \( \alpha = 0.05 \) to compare the p-value from the ANOVA test. If the p-value is less than 0.05, reject the null hypothesis, indicating that not all treatment means are equal.
4. Step 4: Since the ANOVA test indicates a difference, use the Bonferroni test to perform pairwise comparisons between treatment means. The Bonferroni test adjusts the significance level to control for Type I error when making multiple comparisons.
5. Step 5: For each pairwise comparison, calculate the confidence intervals or p-values using the Bonferroni correction. If any confidence interval does not include zero or any p-value is less than the adjusted significance level, conclude that those treatment means differ significantly.

Key Concepts

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare the means of three or more groups to determine if at least one group mean is significantly different. It tests the null hypothesis that all group means are equal by analyzing variance within and between groups. A significant ANOVA result suggests differences among treatments.

Bonferroni Test

The Bonferroni test is a post-hoc multiple comparison procedure used after ANOVA to identify which specific group means differ. It adjusts the significance level to control the overall Type I error rate when making multiple pairwise comparisons, ensuring more reliable conclusions about treatment effects.

Significance Level and Hypothesis Testing

The significance level (commonly 0.05) is the threshold for deciding whether to reject the null hypothesis. It represents the probability of making a Type I error, or falsely detecting an effect. In this context, it helps determine if the treatments have statistically different mean weights.