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, 390 of the vines treated with Pernod 5 were checked for infestation. Likewise, a sample of 420 vines sprayed with Action were checked. The results are:

At the 0.02 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 desicion rule. (H0 is rejected if Z< ___ or Z>____)

B. Compute the pooled proportion.

C. Compute value of test statistic.

D. What is your desicion regarding the null hypothesis? (Reject/ Do not reject)

Insecticide Number of Vines Checked (sample size) Number of Infested Vines Pernod 5 390 23 Action 420 46

Explanation / Answer

H0 : There is no diference in proprtion of vines infested using Pernod 5 as opposed to action.

Ha : There is significant diference in proprtion of vines infested using Pernod 5 as opposed to action.

A. State the desicion rule. (H0 is rejected if Z< ___ or Z>____)

Here as the test is two sided test we shall reject the null hypothesis for alpha = 0.02 is Z < -2.33 or Z > 2.33

(b) Pooled proportion p0= (x1 + x2)/ (n1 + n2) = (23 + 46)/ (390 + 420) = 0.0852

standard error of pooled proportion = sqrt [p0(1-p0) * (1/n1 + 1/n2)] = sqrt [0.0852 * 0.9148 * (1/390 + 1/420)] = 0.0196

Here p1^ = 23/390 = 0.059

p2^ = 46/420 = 0.1095

so Test statistic

Z = (p1^ - p2^)/ se0 = (0.1095 - 0.059)/ 0.0196 = 2.5765

so Here Z > Zcr so we shall reject the null hypothesis.

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