Some researchers have conjectured that stem-pitting disease in peach-tree seedli

Question

Some researchers have conjectured that stem-pitting disease in peach-tree seedlings might be controlled with weed and soil treatment. An experiment is conducted to compare peach-tree seedling growth when the soil and weeds are treated with one of two herbicides. In a field containing 25 seedlings, 15 are randomly selected throughout the field and assigned to receive Herbicide A. The remainder of the seedlings is assigned to receive Herbicide B. Soil and weeds for each seedling are treated with the appropriate herbicide, and at the end of the study period the height in centimeters is recorded for each seedling. The following results are obtained:

a) Suppose we wish to determine if there tends to be a difference in height for the seedlings treated with the different herbicides. Please state your hypotheses.

b) Based on equal variance assumption, please give the test statistic, calculate the p-value and give your conclusion on hypotheses.

c) Based on unequal variance assumption, please give the test statistic, calculate the p-value and give your conclusion on hypotheses.

Herbicide A Mean1 = 94.5 cm s1 = 10 cm Herbicide B Mean2 = 109.1 cm s2 = 9 cm

Explanation / Answer

(a) Null Hypothesis : H0 : There is no significanct difference between heights for different seedlings. 1 = 2

Alternative Hypothesis : Ha : There is significant difference between heights for these different seedlings.  1 2

(b) Here we have equal variance assumption

so Pooled standared deviation sp = sqrt [(n1-1)s1 2 + (n2 -1)s22 / (n1 + n2 -2)] =

= sqrt [( 14 * 10 * 10 + 9 * 9 * 9)/23] = 9.621

Test statistic

t = (x2 - x1)/[sp * sqrt(1/n1 + 1/n2)] = (109.1 - 94.5)/ [9.621 * sqrt (1/15 + 1/10)] = 14.6/3.9277 = 3.717

so, Here p - value = 2 * Pr(t > 3.717 ; dF = 23) = 0.00113 < 0.05

so here we can reject the null hypothesis and can say that there is signficant difference between seedlings treated with the different herbicides.

(c) Here with unequal vairance assumption

test statistic

t = (x2 - x1)/[sqrt(s12/n1 + s12/n2)] = (109.1 - 94.5)/ [sqrt (102/15 + 92/10)] = 3.80

Here dF = [(s12/n1 + s22/n2)2 ]/ [(s12/n1 )/(n1 -1) +  (s22/n1 )/(n2 -1)] = 20.8373 or 21

so p - value = 2 * Pr(t > 3.80 ; dF = 21) = 0.0010 < 0.05

so here we can reject the null hypothesis and can say that there is signficant difference between seedlings treated with the different herbicides.

Herbicide A Mean1 = 94.5 cm s1 = 10 cm Herbicide B Mean2 = 109.1 cm s2 = 9 cm

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