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, 400 of the vines treated with Pernod 5 were checked for infestation. Likewise, a sample of 380 vines sprayed with Action were checked. The results are:

Insecticide	Number of Vines Checked (sample size)	Number of Infested Vines
Pernod 5	400	24
Action	380	40

At the 0.01 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.

1. 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.)

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

3. 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.)

4. What is your decision regarding the null hypothesis?

• Fail to reject

• Reject

Answer 1

H0 : p₁ = p₂

H1 : p₁ ≠ p₂

State the decision rule.

Significance Level = a = 0.01 Using Excel, Critical value=z*= ABS(NORM.S.INV(0.01/2)) = 2.58 Thus the critical values for two tailed test are -2.58 and 2.58. Decision Rule: "Reject H, if z<-2.58 or z>2.58."

Compute the pooled proportion.

Sample Sample number Size of events Sample proportion Sample 1 n1 = 400 P1 = X1/n1 =(24/400) = 0.06 Sample 2 n2 = 380 P2 = X2/n2 =(40/380) = 0.1053 X1 =24 X2 -40 x1 + x2 pooled proportion = p = ng + m2 (24+40)/(400+380) = 0.0821

Compute the value of the test statistic.

The test statistic is, P1-P2 Z (0.06-0.1053)/sqrt{0.0821*(1-0.0821)*((1/400)+(1/380))] = -2.3 1 p(1-P) 1 n1 + n2

What is your decision regarding the null hypothesis?

By decision rule, we fail to reject Ho