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.

Accepted Answer

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. 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. 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. 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. 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.

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Concetti chiave

Analysis of Variance (ANOVA)

Bonferroni Test

Significance Level and Hypothesis Testing