Chapter 13, Problem 51E

(a)

To determine

To find: The means for each species by water combination and plot these means.

Answer to Problem 51E

Solution: For Fresh biomass, the first three means for Species-1 are 109.1, 165.1, and 168.82. The first three means for Species-2 are 116.4, 156.8, and 254.88. The first three means for Species-3 are 55.60, 78.9, and 90.3. The first three means for Species-4 are 35.13, 58.33, and 94.54.

For Dry biomass, the first three means for Species-1 are 40.56, 63.86, and 71.00. The first three means for Species-2 are 34.50, 57.36, and 79.60. The first three means for Species-3 are 26.25, 31.87, and 36.24. The first three means for Species-4 are 15.53, 23.29, and 37.05.

Explanation of Solution

Calculation: To obtain the means for the response variable Fresh biomass, Minitab is used. The steps to be followed are:

Step 1: Go to the Minitab worksheet.

Step 2: Go to Stat $>$ Basic Statistics $>$ Descriptive Statistics.

Step 3: Enter the variable ‘Fresh biomass’ in the ‘Variable’ option.

Step 4: Enter the variables ‘Species and Water’ in the ‘By Variable’ option.

Step 5: Go to statistics and select ‘Mean.

Step 6: Go to graph and select Individual value plot’.

Step 7: Click Ok.

The obtained result shows the means for each species by water level. Here, few of them are given: The first three means for Species-1 are 109.1, 165.1, and 168.82. The first three means for Species-2 are 116.4, 156.8, and 254.88. The first three means for species-3 are 55.60, 78.9, and 90.3. The first three means for Species-4 are 35.13, 58.33, and 94.54. The obtained graph is given below:

To obtain the means for the response variable Dry biomass, Minitab is used. The steps to be followed are:

Step 1: Go to the Minitab worksheet.

Step 2: Go to Stat $>$ Basic Statistics $>$ Descriptive Statistics.

Step 3: Enter the variable ‘Dry biomass’ in the ‘Variable’ option.

Step 4: Enter the variables ‘Species and Water’ in the ‘By Variable’ option.

Step 5: Go to statistics and select ‘Mean.

Step 6: Go to graph and select Individual value plot’.

Step 7: Click Ok.

The obtained result shows the means for each species by water level. Here, few of them are given: The first three means for Species-1 are 40.56, 63.86, and 71.00. The first three means for Species-2 are 34.50, 57.36, and 79.60. The first three means for species-3 are 26.25, 31.87, and 36.24. The first three means for Species-4 are 15.53, 23.29, and 37.05. The obtained graph is given below:

(b)

To determine

To find: The standard deviation for each species by water combination and plot these means.

Answer to Problem 51E

Solution: For Fresh biomass, the first three standard deviations for Species-1 are 20.9, 29.1, and 18.87. The first three standard deviations for Species-2 are 29.3, 46.9, and 13.94. The first three standard deviations for Species-3 are 13.20, 29.5, and 28.3. The first three standard deviations for Species-4 are 11.63, 6.79, and 13.93.

For Dry biomass, the first three standard deviations for Species-1 are 5.58, 7.51, and 6.03. The first three standard deviations for Species-2 are 11.61, 6.15, and 13.09. The first three standard deviations for Species-3 are 6.43, 11.32, and 11.27. The first three standard deviations for Species-4 are 4.89, 3.33, and 5.19.

Explanation of Solution

Calculation: To obtain the means for Fresh biomass, Minitab is used. The steps to be followed are:

Step 1: Go to the Minitab worksheet.

Step 2: Go to Stat $>$ Basic Statistics $>$ Descriptive Statistics.

Step 3: Enter the variable ‘Fresh biomass’ in the ‘Variable’ option.

Step 4: Enter the variables ‘Species and Water’ in the ‘By Variable’ option.

Step 5: Go to statistics and select ‘Standard deviation’.

Step 6: Click Ok.

The obtained result shows the means for each species by water level. Here, few of them are given: The first three standard deviations for Species-1 are 20.9, 29.1, and 18.87. The first three standard deviations for Species-2 are 29.3, 46.9, and 13.94. The first three standard deviations for species-3 are 13.20, 29.5, and 28.3. The first three standard deviations for Species-4 are 11.63, 6.79, and 13.93.

To obtain the means for Dry biomass, Minitab is used. The steps to be followed are:

Step 1: Go to the Minitab worksheet.

Step 2: Go to Stat $>$ Basic Statistics $>$ Descriptive Statistics.

Step 3: Enter the variable ‘Dry biomass’ in the ‘Variable’ option.

Step 4: Enter the variables ‘Species and Water’ in the ‘By Variable’ option.

Step 5: Go to statistics and select ‘Standard deviation’.

Step 6: Click Ok.

The obtained result shows the means for each species by water level. Here, few of them are given: The first three standard deviations for Species-1 are 5.58, 7.51, and 6.03. The first three standard deviations for Species-2 are 11.61, 6.15, and 13.09. The first three standard deviations for species-3 are 6.43, 11.32, and 11.27. The first three standard deviations for Species-4 are 4.89, 3.33, and 5.19

The highest SD from the Fresh biomass data is 108.01 and the lowest SD is 6.79. Thus,

$\begin{array}{c}{\sigma }_{\max }<2\times {\sigma }_{\min }\\ \frac{108.01}{6.79}<2\\ 15.90<2\end{array}$

The highest SD from the Dry biomass data is 35.76 and the lowest SD is 3.12. Thus,

$\begin{array}{c}{\sigma }_{\max }<2\times {\sigma }_{\min }\\ \frac{35.76}{3.12}<2\\ 11.46<2\end{array}$

Hence, it is reasonable to pool the standard deviation.

(c)

To determine

To test: A two-way ANOVA for Fresh biomass and Dry biomass

Answer to Problem 51E

Solution: A two-way ANOVA for Fresh biomass is provided below:

Source of Variation	Degree of freedom	Sum of squares	Mean sum of squares	F- value	P- value
Species	3	458295	152765	81.45	0.000
Water	6	491948	81991	43.71	0.000
Interaction	18	60334	3352	1.79	0.040
Error	84	157551	1876
Total	111	1168129

A two-way ANOVA for Dry biomass is provided below:

Source of Variation	Degree of freedom	Sum of squares	Mean sum of squares	F- value	P- value
Species	3	50524	16841.3	79.93	0.000
Water	6	56624	9437.3	44.79	0.000
Interaction	18	8419	467.7	2.22	0.008
Error	84	17698	219.7
Total	111	133265

Explanation of Solution

Calculation: To perform a Two-way ANOVA for Fresh biomass, Minitab is used. The steps to be followed are:

Step 1: Go to the Minitab worksheet.

Step 2: Go to Stat $>$ ANOVA $>$ Two-Way ANOVA.

Step 3: Enter the variable ‘Fresh biomass’ in the ‘Response’ option.

Step 4: Enter the variables ‘Species’ in the ‘Row Factor and ‘Water’ in the Column Factor’.

Step 5: Click Ok.

The obtained results are provided below:

Source of Variation	Degree of freedom	Sum of squares	Mean sum of squares	F- value	P- value
Species	3	458295	152765	81.45	0.000
Water	6	491948	81991	43.71	0.000
Interaction	18	60334	3352	1.79	0.040
Error	84	157551	1876
Total	111	1168129

To perform a Two-way ANOVA for Dry biomass, Minitab is used. The steps to be followed are:

Step 1: Go to the Minitab worksheet.

Step 2: Go to Stat $>$ ANOVA $>$ Two-Way ANOVA.

Step 3: Enter the variable ‘Dry biomass’ in the ‘Response’ option.

Step 4: Enter the variables ‘Species’ in the ‘Row Factor and ‘Water’ in the Column Factor’.

Step 5: Click Ok.

The obtained results are provided below:

Source of Variation	Degree of freedom	Sum of squares	Mean sum of squares	F- value	P- value
Species	3	50524	16841.3	79.93	0.000
Water	6	56624	9437.3	44.79	0.000
Interaction	18	8419	467.7	2.22	0.008
Error	84	17698	219.7
Total	111	133265

Conclusion: From ANOVA table above, for both Fresh biomass and Dry biomass, main effects and interaction are significant. The interaction for Fresh biomass and Dry biomass has the P-value 0.04 and 0.008, respectively.