SOLUTION: 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 i

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Question 978005: 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 400 vines sprayed with Action were checked. The results are:
----------------------------------Number of
---------------------------------Vines Checked
Insecticide-----------------------(sample size)-----------Number of
-------------------------------------------------------Infested Vines
Pernod 5 -----------------------------400--------------------35
Action--------------------------------400--------------------37
At the .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 as the first sample.
1.State the decision rule. (Round your answers to 2 decimal places.)
H0 is rejected if z <__ or z >__ .
Compute the pooled proportion. (Round your answer to 3 decimal places.)
Pooled proportion=__?
2.Compute the value of the test statistic. (Round your answer to 2 decimal places.)
Value of the test statistic=__?
3.What is your decision regarding the null hypothesis?
Decision =(reject/do not reject)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Hypothesis:
H0: p1 = p2 which is the same as saying p1 - p2 = 0
H0: p1 =/= p2 which is the same as saying p1 - p2 =/= 0

This is a two tailed test
=============================================================================
1)
Decision Rule: H0 is rejected if z < -2.58 or z > 2.58

Stated another way, if z is NOT between -2.58 and 2.58, then reject H0. Otherwise, we fail to reject.

How did I find these critical values? There are multiple ways. Most involve either a table or a calculator. I used this table. Go to the Z row at the bottom. Then find the value in that row that is over the "99%" confidence level. That value is 2.576. That rounds to 2.58 (since it says to round to 2 decimal places). So the critical values are the plus/minus of that number.

note: if alpha = 0.01, then C = 1-alpha = 1-0.01 = 0.99
C = confidence level (0.99 = 99% confidence level)
All of this is only possible since this is a two tailed test.

---------------------------------------------------------------
Pooled Proportion







The Pooled Proportion is 0.09
=============================================================================
2)







---------------------------------------------

Sample Proportions




---------------------------------------------

Standard Error:






---------------------------------------------

Test Statistic:









=============================================================================
3)

Summary:
Critical values: -2.58 and 2.58
Test Statistic: -0.25

Question: is the test statistic between the critical values?
Answer: Yes it is. So we fail to reject the null hypothesis.

Alternatively, if we go back to the decision rule in part a), none of the inequalities are made true if z = -0.25. So a different route to get to the fail to reject decision.

-------------------------------------------------------

What does the fail to reject decision mean? Well if you go back to the very top where the hypotheses were set up, you would see

Hypothesis:
H0: p1 = p2 which is the same as saying p1 - p2 = 0
H0: p1 =/= p2 which is the same as saying p1 - p2 =/= 0

Since we failed to reject H0, we effectively accept it. There wasn't enough statistically significant evidence to prove it was false. So that means the two proportions p1 and p2 are the same. So the two sprays are equally effective.

Question: can we conclude that there is a difference in the proportion of vines infested using Pernod 5 as opposed to Action?

Answer: No we cannot conclude there is a difference in the proportions.