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, 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.

Explanation / Answer

As we can get the information from the problem:

Null Hypothesis H0: P1 = P2

Alternate Hypothesis Ha: P1 P2

Significance Level: 0.01

Sample Data
n1: sample size for Group 1 = 400
n2: sample size for Group 2 = 380

Proportion p1: = 24 / 400 = 0.06
Proportion p2: = 40 / 380 = 0.105

where p1 = sample proportion in sample 1 , p2 = sample proportion in sample 2

Pooled Sample Proportion
p = (p1 * n1 + p2 * n2) / (n1 + n2)

p = [(0.06*400) + (0.105*380)] / (400 + 380)

p = 63.9 / 780

p = 0.08192

Standard Error:
SE = sqrt{p * (1 - p ) * [(1/n1) + (1/n2)]}

SE = sqrt{0.81923 * (1 - 0.081923) * [ (1/400) + (1/380)]}

SE = sqrt (0.081923 * 0.918077 * 0.00513157)

SE = sqrt (0.0003859537)

SE = 0.0196457

Test Statistic z-score: z = (p1 - p2) / SE

z = (0.06 - 0.105) / 0.0196457

z = - 0.045 / 0.0196457

z = - 2.2905

For two-tailed test, the p-value is the probability that the z-score is less than -2.2905 and more than 2.2905


Use the Normal Distribution Table to find P(z < -2.2905) = 0.01101, and P(z > 2.2905) = 0.01101

For a two-tailed test:
The P-value = 0.1101 + 0.1101 = 0.2202

Since the P-value (0.2202) is more than the significance level (0.01) the Null Hypothesis H0: P1 = P2 must be accepted. Infested rates are same with Pernod 5 and Action.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.