91Ó°ÊÓ

Consider the accompanying data on plant growth after the application of different types of growth hormone. $$ \begin{array}{llrlrl} & \mathbf{1} & 13 & 17 & 7 & 14 \\ & \mathbf{2} & 21 & 13 & 20 & 17 \\ \text { Hormone } & \mathbf{3} & 18 & 14 & 17 & 21 \\ & \mathbf{4} & 7 & 11 & 18 & 10 \\ & \mathbf{5} & 6 & 11 & 15 & 8 \end{array} $$ a. Carry out the \(F\) test at level \(\alpha=.05\). 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.)

Step by step solution

01

- Calculating Group and Total Means

Compute the mean growth for each hormone type, and the overall mean growth across all the plant observations.

02

- Calculating Sums of Squares

Compute the within-groups sum of squares (SSW) and between-groups sum of squares (SSB) using the hormone means and overall mean.

03

- Calculating Degrees of Freedom

Get the degrees of freedom for the numerator (df1) and the denominator (df2) for the F test. This can be calculated by subtracting 1 from the number of hormone types for df1, and subtracting the number of hormone types from the total number of values for df2.

04

- Performing the F test

Use the SSB, SSW, df1 and df2 to compute the F statistic.\nIf the calculated F statistic is greater than the critical F value (found by looking it up in an F-distribution table with α = .05, df1 and df2), reject the null hypothesis that all the hormones have the same mean effect on plant growth.

05

- Applying the Tukey-Kramer procedure

If the null hypothesis was rejected, perform the T-K procedure to conduct pairwise comparisons between the hormone means to see which ones differ from each other.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

F test

The F test is a statistical method used to determine if there are significant differences between group means. In the context of ANOVA (Analysis of Variance), this test evaluates whether the means of different groups (such as the effect of different hormones on plant growth) differ more than expected by random chance.

Here's how the process works:

• First, we calculate the **between-group variability** and the **within-group variability**. These provide an idea of how much the data vary due to the treatment versus random variation.
• The F statistic is then calculated by dividing the between-group variability by the within-group variability.
• If the F statistic is higher than the critical value from the F-distribution table (based on our chosen significance level \(\alpha = 0.05\), and the degrees of freedom), we reject the null hypothesis.

This rejection indicates a significant difference between the group means. In the context of this exercise, you'd use data from plant growth to compute your F statistic.

Tukey-Kramer procedure

After the F test confirms at least one significant difference among the group means, the Tukey-Kramer procedure helps identify which groups differ. This is important because the F test only tells us that a difference exists, not its source.

The Tukey-Kramer method carries out all possible pairwise comparisons between group means while controlling for Type I errors (incorrectly rejecting the null hypothesis). Key points of this procedure include:

• Calculate the difference between each pair of group means.
• Compare these differences to a critical value that considers the sample sizes involved.
• If a difference exceeds the critical value, that pair is considered significantly different.

Remember, this procedure is particularly useful because it provides a detailed picture of how group means compare to each other in a multiple comparisons scenario.

Sums of Squares

In ANOVA, the Sums of Squares help quantify the variation in the data, partitioning it into components that help us understand the source of variability. There are two main types of Sums of Squares in this context:

• **Sum of Squares Between (SSB):** This measures variability due to the interaction between different groups, like the different hormone treatments. It's based on how much each group's mean deviates from the overall mean. The larger this value, the more potential there is that the group means are different.
• **Sum of Squares Within (SSW):** This measures variability within each group, indicative of natural variation. It checks how individuals deviate from their group mean. Larger values indicate more variation due to factors other than the group intervention.

Together, SSB and SSW are used to formulate the F statistic, providing the foundation for testing the hypothesis of ANOVA.

Degrees of Freedom

Degrees of Freedom (df) are crucial in statistics for estimating variability, which are particularly important in hypothesis testing. In ANOVA, they determine the specific F distribution you should use to find your critical value.

Here's how to calculate them:

• **Degrees of Freedom Between (df1):** Calculated as the number of groups minus one. For example, if there are 5 types of hormones, df1 would be 4.
• **Degrees of Freedom Within (df2):** Calculated as the total number of observations minus the number of groups. If there are 20 observations across the 5 groups, then df2 would be 15.

Degrees of Freedom impact the shape of the F distribution and are imperative in determining whether your F statistic falls in the critical region, allowing you to make inferences about your hypothesis.