Question

1An agronomist was interested in finding out the amount of potassium in corn leaves, not including the stems, after four different fertilizers had been used in a field of corn

A. If 5 plots of corn were used per fertilizer in a completely randomized manner, show the layout of an experiment.

B. If in the 5 plots per fertilizer, 2 stalks of corn per plot were to be sampled from the experimental area, show the Anova table and state the assumptions necessary to test the effects of fertilizer.

C. Comment on this design (in part(B)) and analysis.

* part A, B & C please with as much detail as possible

Answer 1

i) The four fertilizers were assigned completely at random to the plots.Each fertilizer were replicated 5 times.

A	B	A	C
D	C	B	D
C	A	D	B
B	C	D	A
C	B	A	D

ii)

Source of Variation	Degree of Freedom(df)	Sum of Squares(SS)	Mean Sum of Squares(MSS)	F-ratio
Treatments	k-1	SST	SST/k-1
Error	n-k	SSE	SSE/n-k
Total	n-1	TSS

For this problem,

Number of treatment k = 4

Number of replication = 5

Total number of experimental units n = 20

Source of Variation	Degree of Freedom(df)	Sum of Squares(SS)	Mean Sum of Squares(MSS)	F-ratio
Treatments	k-1= 4-1=3	SST	SST/3
Error	n-k = 20-4 =16	SSE	SSE/16
Total	n-1 = 20-1 = 19	TSS

The assumptions necessary to test the effect of fertilizers are

1.the observations are normally distributed within the treatment groups

2.the observations should have homoscedasticity (have equal variance)

iii)The effects of four different fertilizers can be studied through the above completely randomised design(CRD). In CRD, treatments are assigned to experimental units at random so that each unit has the same chance of receiving any one treatment.

After assigning treatments to the units, the results are recorded and the analysis of variance is performed to find the effect of fertilizers.

ANOVA is performed as

1. NULL HYPOTHESIS:

All the means are equal (i.e.,)All the fertilizers have the same effect on the crops.

2.ALTERNATIVE HYPOTHESIS:

All the means are not equal (i.e.,)All the fertilizers do have the same effect on the crops.

3.TEST STATISTIC

where SST is Treatment Sum of Square

SSE is Error Sum of Square

TSS is Total Sum of Square

Total Sum of Square = Treatment Sum of Square + Error Sum of Square

MST is Mean Square Sum of Treatments

MSE is Mean Square Sum of Error

4.INFERENCE

The computed test statistic is compared with the critical value obtained from F-table for (k-1,n-k)d.f at specified level of significance

If Test Statistic is less than the critical value (), null hypothesis is accepted.

Otherwise Null Hyothesis is rejected and alternative is accepted

In case if Alternative hypotheis is accepted , it means that the fertilizers have some effect on the crops. By post-hoc comparison, we can find out which fertilizers has more effect on the crops.