Consider the accompanying data on plant growth after the...

Consider the accompanying data on plant growth after the application of five different types of growth hormone. $$ \begin{array}{lllll} \mathbf{1} & 13 & 17 & 7 & 14 \\ \mathbf{2} & 21 & 13 & 20 & 17 \\ \mathbf{3} & 18 & 14 & 17 & 21 \\ \mathbf{4} & 7 & 11 & 18 & 10 \\ \mathbf{5} & 6 & 11 & 15 & 8 \end{array} $$ a. Carry out the ANOVA $F$ test using a significance level of $\alpha=0.05$ b. What happens when the T-K procedure is applied? (Note: This "contradiction" can occur when $H_{0}$ is "barely" rejected. It happens because the test and the multiple comparison method are based on different distributions. Consult your friendly neighborhood statistician for more information.)

Consider the accompanying data on plant growth after the application of five different types of growth hormone.
$$
\begin{array}{lllll}
\mathbf{1} & 13 & 17 & 7 & 14 \\
\mathbf{2} & 21 & 13 & 20 & 17 \\
\mathbf{3} & 18 & 14 & 17 & 21 \\
\mathbf{4} & 7 & 11 & 18 & 10 \\
\mathbf{5} & 6 & 11 & 15 & 8
\end{array}
$$
a. Carry out the ANOVA $F$ test using a significance level of $\alpha=0.05$
b. What happens when the T-K procedure is applied? (Note:
This "contradiction" can occur when $H_{0}$ is "barely" rejected. It happens because the test and the multiple comparison method are based on different distributions. Consult your friendly neighborhood statistician for more information.)

Key Concepts

Tukey-Kramer Procedure

The Tukey-Kramer procedure is a specific post-hoc multiple comparison test used after an ANOVA when sample sizes are unequal. It compares all possible pairs of group means while controlling the family-wise error rate. Because the test is more conservative than the overall F test, it sometimes fails to find significant pairwise differences even when the ANOVA indicates overall significance, particularly when the rejection of the null hypothesis is borderline.

Multiple Comparison Procedures

Multiple comparison procedures are techniques applied after an overall test (like ANOVA) to determine exactly which pairs of group means exhibit significant differences. These procedures are crucial because they account for the increased probability of Type I errors when making several pairwise comparisons, ensuring that the overall confidence level is maintained.

Significance Level

The significance level, often denoted as alpha (?), is the threshold probability used to decide whether to reject the null hypothesis. In many studies, including this case, ? is set at 0.05, which means there is a 5% risk of mistakenly rejecting a true null hypothesis (Type I error). This level guides the interpretation of p-values in hypothesis testing.

F Test

The F test is the foundational statistical test used in ANOVA to compare the variance between groups to the variance within groups. Specifically, it calculates an F statistic, a ratio that indicates if the group means are more spread out than would be expected by random variation alone. A sufficiently high F statistic relative to a critical value signals a rejection of the null hypothesis.

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to determine if there are any statistically significant differences between the means of three or more independent groups. It works by partitioning the overall variability in the data into variability due to differences between group means (between-group variation) and variability within each group (within-group variation). The test concludes whether the observed differences in sample means could have arisen by random chance under the null hypothesis of equal group means.

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