Question

1. One further question of interest to WMRT is whether the number of times that some saw the advertisement (if at all) is related to the long-run impact of the advertisement on awareness. To assess this, WMRT collected data on how many times CSG’s ad was seen and whether or not the household was aware of the advertisement six months later after the campaign had ended.    The data from this study can be summarized in the following table from a random sample of 850 potential customers.

	Was Aware of CSG	Not Aware of CSG
Did not see the advertisement	80	120
Saw the advertisement between 1-3 times	140	150
Saw the advertisement more than 3 times	220	140

(2c) Construct a 99% confidence interval for the proportion of households that are aware of CSG. (1 pt)

(2d) Test whether the number of times that a household saw the advertisement is independent of whether or not they are aware of CSG. (Provide the relevant test statistic and p-value.) What might this result indicate about the long-term effect of advertising, if anything? (1 pt)

Answer 1

	Was Aware of CSG	Not Aware of CSG	Total
Did not see the advertisement	80	120	200
Saw the advertisement between 1-3 times	140	150	290
Saw the advertisement more than 3 times	220	140	360
Total	440	410	850

2c)

99% confidence interval for the proportion of households that are aware of CSG

Proportion of households that are aware of CSG = 440/850 = 0.5176

α = 0.01, Zc​ = Z1−α/2​ =2.576

CI = P +/- Zc * SQRT(P*(1-P)/n)

CI = 0.5176 +/- 2.576 * SQRT(0.5176*(1-0.5176)/850)

Lower	0.473449
Upper	0.561751

CI = (0.4734, 0.5618)

2d)

The number of times that some saw the advertisement (if at all) is related to the long-run impact of the advertisement on awareness

The number of times that a household saw the advertisement is independent of whether or not they are aware of CSG.

Hypothesis :

H0​: The two variables are independent

Ha​: The two variables are dependent

Critical value :

df = (3−1)×(2−1) = 2

α = 0.01

X^2 critical = 9.21

Test :

	Was Aware of CSG	Not Aware of CSG	Total
Did not see the advertisement	80	120	200
Saw the advertisement between 1-3 times	140	150	290
Saw the advertisement more than 3 times	220	140	360
Total	440	410	850
Expected	Was Aware of CSG	Not Aware of CSG	Total	Rt*Ct/T
Did not see the advertisement	103.5294118	96.47058824	200
Saw the advertisement between 1-3 times	150.1176471	139.8823529	290
Saw the advertisement more than 3 times	186.3529412	173.6470588	360
Total	440	410	850
	Was Aware of CSG	Not Aware of CSG	Total	(O-E)^2/E
Did not see the advertisement	5.347593583	5.738880918	11.08647
Saw the advertisement between 1-3 times	0.681910382	0.731806263	1.413717
Saw the advertisement more than 3 times	6.075163399	6.51968755	12.59485
Total		X^2	25.09504

X^2 stat = 25.09504

Decision:

X^2 = 25.095 > X^2 critical ​= 9.21, reject H0

The P-Value is < .00001. The result is significant at p < .01 and reject H0

Conclusion:

There is enough evidence to conclude that the number of times that a household saw the advertisement is dependent of whether or not they are aware of CSG, at the 0.01 significance level.

Above solution for whether the number of times that some saw the advertisement (if at all) is related to the long-run impact of the advertisement on awareness

If particularly asked about only saw the advertisement whether or not they are aware of CSG, you have to remove Did not see the advertisement row and do the same procedure