The Damon family owns a large grape vineyard in western New York along Lake Erie

Question

The Damon family owns a large grape vineyard in western New York along Lake Erie. The grapevines must be sprayed at the beginning of the growing season to protect against various insects and diseases. Two new insecticides have just been marketed: Pernod 5 and Action. To test their effectiveness, three long rows were selected and sprayed with Pernod 5, and three others were sprayed with Action. When the grapes ripened, 425 of the vines treated with Pernod 5 were checked for infestation. Likewise, a sample of 410 vines sprayed with Action were checked. The results are:

At the 0.10 significance level, can we conclude that there is a difference in the proportion of vines infested using Pernod 5 as opposed to Action? Hint: For the calculations, assume the Pernod 5 as the first sample.

a. State the decision rule. (Negative amounts should be indicated by a minus sign. Do not round the intermediate values. Round your answers to 2 decimal places.)

b. Compute the pooled proportion. (Do not round the intermediate values. Round your answer to 2 decimal places.)

c. Compute the value of the test statistic. (Negative amount should be indicated by a minus sign. Do not round the intermediate values. Round your answer to 2 decimal places.)

d. What is your decision regarding the null hypothesis?

O Fail to reject

O Reject

Insecticide Number of Vines Checked (sample size) Number of Infested Vines Pernod 5 425 50 Action 410 44

Explanation / Answer

here we use z-test and z=(p1-p2)/SE(p1-p2)

p1=x1/n1=50/425=0.1176, p2=x2/n2=44/410=0.1073

var(p1-p2)=var(p1)+var(p2)=p1(1-p2)/n1+p2(1-p2)/n2=(0.1176*(1-0.1176)/425 + 0.1073*(1-0.1073)/410)=0.000478

SE(p1-p2)=0.0219

here null hypothesis H0:p1=p2 and alternate hypothesis Ha:p1 is not equal to p2 ( this is two tailed test)

(a) here we use two tailed critical z(0.1)=1.6449

(b) pooled proportion=p=(50+44)/(425+410)=0.1125

(c) z=(0.1176-0.1073)/0.0219=0.47

(d) fail to reject null hypothesis

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