5.(14) How much corn should be planted per acre for a farmer to get the highest

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5.(14) How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will compete with each other for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study, and three acres were assigned at random to each of the planting densities. A summary of the data follows Plants (per acre 12,000 16,000 20,000 24,000 5 2.0 72.8 12 3.0 4 1.6 Assume the data are four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. Perform a Global ANOVA test. a) (11) Complete the following ANOVA table, then state conclusion base on the p-value Sum of squares SS MS Source of variation Factor Error Total DF Mean square P-value (1pt) (lpt) (1pt) (1pt) (1pt) (1pt) (1pt) (1pt) (1pt)

Explanation / Answer

Use the given information to calculate the SS(Factor)=n1(y1bar-ybar)^2+n2(y2bar-ybar)^2+n3(y3bar-ybar)^2+n4(y4bar-ybar)^2, where, ybar=(y1bar+y2bar+y3bar+ybar4)/4=(5+7+12+4)/4=7

=3(5-7)^2+3(7-7)^2+3(12-7)^2+3(4-7)^2=114

MS(Factor)=SS(Factor)/(k-1), where, k is number of groups.

=114/(4-1)=38

SSE=(n1-1)s1^2+(n2-1)s2^2+(n3-1)s3^2+(n4-1)s4^2

=(3-1)2^2+(3-1)2.8^2+(3-1)3^2+(3-1)1.6^2=46.8

MSE=SSE/(n-k)=46.8/(12-4)=5.85

F=MS(Factor)/MSE=38/5.85=6.50

P value at (3,8) df and F=6.50 is 0.0154

The p value is less than 0.05. Per rejection rule based on p value reject null hypothesis, if p value is less than 0.05. Thus reject null hypothesis and conclude that there is significant difference between two mean planting densities per acre.

b. 3 The endpoints of the confidenc einterval for 12000 and 24000 planting densities is as follows:

(y1bar-y4bar)+-qalpha/sqrt 2.s sqrt[1/n1+1/n4], where, qalpha for q curve for k=4, and v=n-k=8 at alpha=0.05, s=sqrt MSE=sqrt 5.85=2.4187

=(5-4)+-(4.53/sqrt2)(2.4187)sqrt(1/3+1/3)

=(-5.326,7.326)

Source of variation DF SS MS F p Factor 3 114 38 6.50 0.0154 Error 8 46.8 5.85 Total 11

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