1. Crop yields per acre depend on many variables, but two obvious ones are ferti
ID: 3311433 • Letter: 1
Question
1. Crop yields per acre depend on many variables, but two obvious ones are fertilizer and weed control. In an effort to test the effects of various levels of fertilizer and of herbicide, researchers selected six plots of land and subdivided each into eight pieces. Each of the eight pieces of each plot was treated with one of the eight combinations of fertilizer (none, light, moderate, heavy) and herbicide (none, some). The per acre yields of each piece of land are recorded below: In the data, levels of fertilizer are 1 (none), 2 (light), 3 (moderate), 4 (heavy) and levels of herbicide are 1 (none) and 2(some).
fert
herb
yield
1
1
63.48
1
1
65.76
1
1
66.23
1
1
66.27
1
1
71.47
1
1
64.64
2
1
69.96
2
1
67.74
2
1
63.78
2
1
63.96
2
1
65.24
2
1
63.22
3
1
66.00
3
1
62.14
3
1
62.52
3
1
66.40
3
1
72.05
3
1
62.23
4
1
67.40
4
1
62.62
4
1
66.51
4
1
68.75
4
1
65.83
4
1
64.52
1
2
59.98
1
2
61.81
1
2
55.87
1
2
62.76
1
2
59.41
1
2
56.21
2
2
64.04
2
2
63.73
2
2
62.61
2
2
64.06
2
2
68.84
2
2
58.03
3
2
59.10
3
2
58.07
3
2
55.97
3
2
60.17
3
2
62.76
3
2
63.34
4
2
63.21
4
2
57.46
4
2
66.65
4
2
63.36
4
2
64.94
4
2
62.22
Give a statement to be tested, identify the random variables involved and the assumptions you make about them, state the hypotheses to be tested, ask SPSS to run the analysis for you, including a post hoc, and then discuss the outcome. Describe the critical region(s) upon which you base your decisions. Include any SPSS output in your discussion.
fert
herb
yield
1
1
63.48
1
1
65.76
1
1
66.23
1
1
66.27
1
1
71.47
1
1
64.64
2
1
69.96
2
1
67.74
2
1
63.78
2
1
63.96
2
1
65.24
2
1
63.22
3
1
66.00
3
1
62.14
3
1
62.52
3
1
66.40
3
1
72.05
3
1
62.23
4
1
67.40
4
1
62.62
4
1
66.51
4
1
68.75
4
1
65.83
4
1
64.52
1
2
59.98
1
2
61.81
1
2
55.87
1
2
62.76
1
2
59.41
1
2
56.21
2
2
64.04
2
2
63.73
2
2
62.61
2
2
64.06
2
2
68.84
2
2
58.03
3
2
59.10
3
2
58.07
3
2
55.97
3
2
60.17
3
2
62.76
3
2
63.34
4
2
63.21
4
2
57.46
4
2
66.65
4
2
63.36
4
2
64.94
4
2
62.22
Explanation / Answer
Solution:
For the given scenario, we have to test or check the hypothesis whether there is any statistically significant difference exists in the average yields per acre due to the given four levels of fertilizers, two levels of herbicides, and their interaction. For checking this hypothesis we have to use two way analysis of variance or two way ANOVA F test by using SPSS. We have to check
Whether there is any significant difference in average yields per acre due to four levels of fertilizers?
Whether there is any significant difference in average yields per acre due to the two levels of herbicides.
Where there is any significant effect of the interaction of four levels of fertilizers and two levels of herbicides?
The SPSS output for the given data by using two way ANOVA is given as below:
Between-Subjects Factors
Value Label
N
Levels of fertilizer
1.00
None
12
2.00
Light
12
3.00
Moderate
12
4.00
Heavy
12
Levels of herbicide
1.00
None
24
2.00
Some
24
Tests of Between-Subjects Effects
Dependent Variable: Per acre yields
Source
Type III Sum of Squares
df
Mean Square
F
Sig.
Corrected Model
311.160a
7
44.451
4.955
.000
Intercept
194224.230
1
194224.230
21648.458
.000
Fert
40.944
3
13.648
1.521
.224
Herb
225.854
1
225.854
25.174
.000
Fert * Herb
44.362
3
14.787
1.648
.194
Error
358.869
40
8.972
Total
194894.259
48
Corrected Total
670.029
47
a. R Squared = .464 (Adjusted R Squared = .371)
From the above ANOVA table, it is observed that the p-value for four levels fertilizers is given as 0.224 which indicate that there is no any significant difference in the average yields per acre due to four levels of fertilizers. It is also found that the p-value for the variable types of herbicide is given as 0.00 which indicate that there is a statistically significant difference in the average yields per acre due to the two levels of herbicides. The interaction due to the four levels of fertilizers and two levels of herbicides found to be statistically insignificant as the p-value is given as 0.194.
Post Hoc Tests
Multiple Comparisons
Per acre yields
LSD
(I) Levels of fertilizer
(J) Levels of fertilizer
Mean Difference (I-J)
Std. Error
Sig.
95% Confidence Interval
Lower Bound
Upper Bound
None
Light
-1.7767
1.22282
.154
-4.2481
.6947
Moderate
.2617
1.22282
.832
-2.2097
2.7331
Heavy
-1.6317
1.22282
.190
-4.1031
.8397
Light
None
1.7767
1.22282
.154
-.6947
4.2481
Moderate
2.0383
1.22282
.103
-.4331
4.5097
Heavy
.1450
1.22282
.906
-2.3264
2.6164
Moderate
None
-.2617
1.22282
.832
-2.7331
2.2097
Light
-2.0383
1.22282
.103
-4.5097
.4331
Heavy
-1.8933
1.22282
.129
-4.3647
.5781
Heavy
None
1.6317
1.22282
.190
-.8397
4.1031
Light
-.1450
1.22282
.906
-2.6164
2.3264
Moderate
1.8933
1.22282
.129
-.5781
4.3647
Based on observed means.
The error term is Mean Square(Error) = 8.972.
Above table indicate that no any statistically significant difference exists between the average yields per acre due to the levels of fertilizers because all p-values in the above table are greater than 5% levels of significance or = 0.05.
Between-Subjects Factors
Value Label
N
Levels of fertilizer
1.00
None
12
2.00
Light
12
3.00
Moderate
12
4.00
Heavy
12
Levels of herbicide
1.00
None
24
2.00
Some
24
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