Chapter 13, Problem 13.1TI

As part of an experiment to see how different types of soil cover would affect slicing tomato production,

Marist College students grew tomato plants under different soil cover conditions. Groups of three plants each had one

of the following treatments

• bare soil

• a commercial ground cover

• black plastic

• straw

• compost

All plants grew under the same conditions and were the same variety. Students recorded the weight (in grams) of

tomatoes produced by each of the n = 15 plants:

Bare: n1= 3	Ground Cover: n2= 3	Plastic: n3= 3	Straw: n4= 3	Compost: n5= 3
2,625	5,348	65,837	7,285	6,277
2,997	5,682	8,560	6,897	7,818
4,915	5,482	3,830	9,230	8,677

Table 13.4

Create the one-way ANOVA table.

To determine

To find:

The one-way anova table.

Answer to Problem 13.1TI

	Sum of square	Mean Square	Degrees of freedom	F Statistics
Between/Factor	36648560.93	9162140.233	4	4.481
Error/Residual	20446726	2044672.6	10
Total	57095286.93		14

.

Explanation of Solution

Given:

	Bare	Ground Cover	Plastic	Straw	Compost
	2625	5348	6583	7285	6277
	2997	5682	8560	6897	7818
	4915	5482	3830	9230	8677
SUM	10537	16512	18973	23412	22772

Formula used:

$\text{Sum of square factor/Between = }\sum \left[\frac{{(s_j)}^2}{n_j}\right]-\frac{{\left(\sum s_j\right)}^2}{N}$

$\text{Mean}\text{Square}\text{Factor = }\frac{\text{Sum square factor}}{\text{Degree of freedom factor}}$

$\text{Sum of square Total = }\sum \left[x^2\right]-\frac{{\left(\sum x\right)}^2}{N}$

$\text{Sum of square Error/Residual = SST - SSB}$

$\text{Mean}\text{Square}\text{Error = }\frac{\text{Sum square Error}}{\text{Degree of freedom Error}}$

$\text{F}\text{Statistic}\text{=}\frac{\text{Mean}\text{Square}\text{Between}}{\text{Mean}\text{Square within}}$

$\text{P}\text{Value}{\text{= P[F}}_{(k-1,n-k)}\text{>F statistic]}$

Calculation:

$\begin{array}{l}\text{Sum of square factor/Between = }\sum \left[\frac{{(s_j)}^2}{n_j}\right]-\frac{{\left(\sum s_j\right)}^2}{N}\\ \text{Sum of square factor/Between =}\left[\frac{{10537}^2}{3}+\frac{{16512}^2}{3}+\frac{{18973}^2}{3}+\frac{{23412}^2}{3}+\frac{{22772}^2}{3}\right]-\frac{{(10537+16512+18973+23412+22772)}^2}{15}\\ \text{Sum of square factor/Between =36648560}\text{.93}\end{array}$

$\begin{array}{l}\text{Mean}\text{Square}\text{Factor/Between = }\frac{\text{Sum square factor}}{\text{Degree of freedom factor}}\\ \text{Mean}\text{Square}\text{Factor/Between =}\frac{36648560.93}{4}\\ \text{Mean}\text{Square}\text{Factor/Between =9162140}\text{.233}\end{array}$

$\text{Sum of square Total = }\sum \left[x^2\right]-\frac{{\left(\sum x\right)}^2}{N}$

$\sum x^2=623891716$

$\begin{array}{l}\text{Sum of square Total = }\sum \left[x^2\right]-\frac{{\left(\sum x\right)}^2}{N}\\ \text{Sum of square Total =623891716}-\frac{{92206}^2}{15}\\ \text{Sum of square Total =57095286}\text{.93}\end{array}$

$\begin{array}{l}\text{Sum of square Error/Residual = SST - SSB}\\ \text{Sum of square Error/Residual =57095286}\text{.93-36648560}\text{.93}\\ \text{Sum of square Error/Residual =20446726}\end{array}$

$\begin{array}{l}\text{Mean}\text{Square}\text{Error/Residual = }\frac{\text{Sum square Error}}{\text{Degree of freedom Error}}\\ \text{Mean}\text{Square}\text{Error/Residual =}\frac{20446726}{10}\\ \text{Mean}\text{Square}\text{Error/Residual =2044672}\text{.6}\end{array}$

$\begin{array}{l}\text{F}\text{Statistic}\text{=}\frac{\text{Mean}\text{Square}\text{Factor}/\text{Between}}{\text{Mean}\text{Square}\text{Error/Residual}}\\ \text{F}\text{Statistic}\text{=}\frac{9162140.233}{2044672.6}\\ \text{F}\text{Statistic}\text{=4}\text{.481}\end{array}$

Conclusion:

	Sum of square	Mean Square	Degrees of freedom	F Statistics
Between/Factor	36648560.93	9162140.233	4	4.481
Error/Residual	20446726	2044672.6	10
Total	57095286.93		14