{Continued from the previous question} A botanist used a completely randomized design to allocate 45 individually potted eggplant plants to five different soil treatments. The observed variable was the total plant dry weight without roots (g) after 31 days of growth. The treatment means were as shown in the table. The [tex]$SS _{ E }\left(= SS _{\text {within }}\right)$[/tex] of this dataset was 8.9840.
| | A | B | C | D | E |
| :---- | :---- | :---- | :---- | :---- | :---- |
| Mean | 4.37 | 4.76 | 3.70 | 5.41 | 5.38 |
| [tex]$n$[/tex] | 9 | 9 | 9 | 9 | 9 |
(c) What's the Tukey's critical value at alpha [tex]$=0.05$[/tex] for this problem? Use the Tukey table shown in class.
(d) Next, use the Tukey's table value to calculate the "critical" value for pairwise mean comparison.
A. 4.04; 0.6382
B. [tex]$3.79 ; 0.2246$[/tex]
C. [tex]$4.23 ; 0.3478$[/tex]
D. [tex]$4.04 ; 0.1008$[/tex]
Answer (1)
Calculate degrees of freedom for error: d f E = 45 − 5 = 40 .
Calculate the Mean Square Error: MSE = 40 8.9840 = 0.2246 .
Use the formula for the critical value for pairwise mean comparison: Critical Value = q × n MSE .
With q = 4.04 , calculate the critical value: 4.04 × 9 0.2246 ≈ 0.6382 .
Explanation
Understand the problem and provided data We are given a completely randomized design with 45 eggplant plants allocated to 5 soil treatments. The error sum of squares ( S S E ) is 8.9840. We need to find Tukey's critical value at alpha = 0.05 and then calculate the critical value for pairwise mean comparison.
Calculate degrees of freedom for error First, we need to determine the degrees of freedom for error ( d f E ). Since there are 45 total plants and 5 treatments, we calculate it as: d f E = \t Total number of plants − Number of treatments = 45 − 5 = 40 So, d f E = 40 .
Determine the number of treatments and Tukey's critical value Next, we determine the number of treatments ( k ). In this case, k = 5 . We need to look up Tukey's critical value ( q ) in the Tukey's table for alpha = 0.05, k = 5 , and d f E = 40 . From the given options, we will test each one to see which one leads to a correct critical value for pairwise mean comparison.
Calculate the Mean Square Error Now, we calculate the Mean Square Error ( MSE ) using the formula: MSE = d f E S S E = 40 8.9840 = 0.2246 So, MSE = 0.2246 .
Calculate the critical value for pairwise mean comparison We calculate the critical value for pairwise mean comparison using the formula: Critical Value = q × n MSE where n is the sample size for each treatment, which is 9.
Let's test the first option where Tukey's critical value q = 4.04 :
Critical Value = 4.04 × 9 0.2246 = 4.04 × 0.0249555... ≈ 4.04 × 0.15797 ≈ 0.6382
This matches the first option: 4.04; 0.6382
Examples
Understanding the differences between soil treatments can help farmers optimize their crop yields. By using statistical methods like Tukey's test, botanists can determine if the observed differences in plant growth are statistically significant or simply due to random variation. This information is crucial for making informed decisions about which soil treatments to use in order to maximize plant growth and overall agricultural productivity. For example, if treatment D and E are significantly better than A, B, and C, the farmer might choose to use treatments D or E.
