Question

The Longley family owns a large grape vineyard in the Niagara Peninsula. 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: Datif and Censor. To test their effectiveness, 8 long rows were selected. 4 rows were sprayed with Datif and 4 were sprayed with Censor. When the grapes ripened, 350 of the vines treated with Datif were checked for infestation. Likewise, a sample of 250 vines sprayed with Censor were checked. The results are:

		Number of
		Vines Checked		Number of
Insecticide		(sample size)		Infested Vines
Datif		350		59
Censor		250		67

Can we conclude that there is a difference in the proportion of vines infested using Datif as opposed to Censor? Test at the 0.05 significance level.

a)

State the hypotheses.

	Reject H0 in favour of H1 if the computed value of the statistic is less than -zα/2 or is greater than zα/2.
	Reject H0 in favour of H1 if the computed value of the statistic is less than -zα/2.
	Reject H0 in favour of H1 if the computed value of the statistic is greater than zα/2.
	Reject H0 in favour of H1 if the computed value of the statistic is between -zα/2 and zα/2.

b) Compute the pooled proportion.
For full marks your answer should be accurate to at least two decimal places.

Pooled proportion = 0

c) Compute the value of the test statistic.
For full marks your answer should be accurate to at least three decimal places.

Test statistic = 0

d) Determine the p-value.
For full marks your answer should be accurate to at least four decimal places.

P-value = 0

e)

What is your decision regarding the null hypothesis?

	There is sufficient evidence, at the given significance level, to reject H0.
	There is insufficient evidence, at the given significance level, to reject H0.
	There is insufficient evidence to reject or not reject the null hypothesis.

Answer 1

From the given

Datif censor

n1=350. n2=250

x1=59. x2=67

Alpha=0.05

P1=x1/n1 = 59/350 = 0.1686

P2= x2/n2 = 67/250 =0.268

H0:P1=P2

V/s H1: P1 P2

A) reject H0 in fever of H1 if the computed value of the test statistic is less than -Z_alpha/2 or greater than Z_alpha/2

B) pooled proportion

P=(x1+x2) /(n1+n2)

=(59+67)/(350+250)

=0.21

C)

under null hypothesis test statistic is

Z

Z=

Z= - 2.948

D) p-value = p(z < - Z) + p(z > Z)

=2*p(z > Z)

= 2*p(z > 2.948)

= 0.0032

E)

P-value < alpha

There is sufficient evidence at the given significance level to reject H0.