Question

A farmer is interested in comparing three different brands of fertilizer, Grow'em Big, Jim's Fertilizer and Sally's Best. He planted wheat in 4 different fields, one with each brand of fertilizer and the fourth one with no fertilizer as a control. Each field is 69 acres in size. The farmer recorded the yield in bushels for each acre of every field. They are interested in knowing whether certain brands of fertilizer have an impact on yield. Assume that the mean of the control is μ₁, Grow'em Big is μ₂, Sally's Best is μ₃ and Jim's Fertilizer is μ₄. The summary statistics for the farmer's experiment are as follows:

	Control	Grow'em Big	Jim's Fertilizer	Sally's Best
Mean	22.01	25.35	45.41	23.50
Sum of Squares:	33951.24	78820.37	221676.00	260258.25

How many pairwise comparisons is the farmer making?

Assuming a confidence level of 95 %, what would be the value of α^*

^{(I tried 0.05 its incorrect)}

For μ₁−μ₄ (control vs. Jim's Fertlizer) do we reject the null hypothesis? For μ₁−μ₃ (control vs. Sally's Best), do we reject the null hypothesis? And for μ₁−μ₂ (control vs. Grow'em Big), do we reject the null hypothesis?

Answer 1

pairwise comparisons = 3

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α* = 0.05/3 = 0.0167

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no. of treatments,k=   4
DF error =N-k=   272
MSE=   2186.4186
t-critical value,t(α/2,df)=   2.4089

					confidence interval
	mean difference			critical value	lower limit	upper limit
µ1-µ2	-3.34			19.18	-22.5166	15.8366
µ1-µ3	-1.49			19.18	-20.6666	17.6866
µ1-µ4	-23.40			19.18	-42.5766	-4.2234

if confidence interval do not contains zero, then means are significantly different.

For μ₁−μ₄ (control vs. Jim's Fertlizer) do we reject the null hypothesis?

answer: YES

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For μ₁−μ₃ (control vs. Sally's Best), do we reject the null hypothesis?

answer: NO

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And for μ₁−μ₂ (control vs. Grow'em Big), do we reject the null hypothesis?

answer: NO