Question 978005
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

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1)

Decision Rule: H0 is rejected if z < <font color="red">-2.58</font> or z > <font color="red">2.58</font>


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 <a href="http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf">this table</a>. 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. 


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Pooled Proportion *[Tex \LARGE \overline{p}]


*[Tex \LARGE \overline{p} = \frac{x_1 + x_2}{n_1+n_2}]


*[Tex \LARGE \overline{p} = \frac{35+37}{400+400}]


*[Tex \LARGE \overline{p} = 0.09]


The Pooled Proportion is <font color="red">0.09</font>

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2)


*[Tex \LARGE \overline{q} = 1-\overline{p}]


*[Tex \LARGE \overline{q} = 1-0.09]


*[Tex \LARGE \overline{q} = 0.91]


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Sample Proportions
*[Tex \LARGE \hat{p}_1 = \frac{x_1}{n_1} = \frac{35}{400} = 0.0875]


*[Tex \LARGE \hat{p}_2 = \frac{x_2}{n_2} = \frac{37}{400} = 0.0925]


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Standard Error:
*[Tex \LARGE SE = \sqrt{ \frac{\overline{p}*\overline{q}}{n_1}+\frac{\overline{p}*\overline{q}}{n_2} }]


*[Tex \LARGE SE = \sqrt{ \frac{0.09*0.91}{400}+\frac{0.09*0.91}{400} }]


*[Tex \LARGE SE = 0.02023610634484]


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Test Statistic:


*[Tex \LARGE z = \frac{(\hat{p}_1-\hat{p}_2) - (p_1 - p_2)}{SE}]


*[Tex \LARGE z = \frac{(0.0875-0.0925) - (0)}{0.02023610634484}]


*[Tex \LARGE z = -0.2470831055538]


*[Tex \LARGE z = -0.25]


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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 <font color="red">fail to reject</font> 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.


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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.