Question

1-Television viewers often express doubts about the validity of certain commercials. In an attempt to answer their critics, a large advertiser wants to estimate the true proportion of consumers who believe what is shown in commercials. Preliminary studies indicate that about 40% of those surveyed believe what is shown in commercials. What is the minimum number of consumers that should be sampled by the advertiser to be 95% confident that their estimate will fall within 2% of the true population proportion?

2- An auditor for a hardware store chain wished to compare the efficiency of two different auditing techniques. To do this he selected a sample of nine store accounts and applied auditing techniques A and B to each of the nine accounts selected. The number of errors found in each of techniques A and B is listed in the table below:

Errors in A	Errors in B
25	11
28	17
26	19
28	17
32	34
30	25
29	29
20	21
25	30

Select a 90% confidence interval for the true mean difference in the two techniques.

a) [0.261, 8.627]

b) [-4.183, 4.183]

c) [2.195, 6.693]

d) [3.050, 5.838]

e) [2.584, 6.304]

f) None of the above

Answer 1

1. Proportion, p = 0.4

Margin of error, E = 0.02

Confidence Level, CL = 0.95

Significance level, α = 1 - CL = 0.05

Critical value, z = NORM.S.INV(0.05/2) = 1.96

Sample size, n = (z² * p * (1-p)) / E² = (1.96² * 0.4 * 0.6)/ 0.02²

= 2304.8 = 2305

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2.

Errors in A	Errors in B	Difference
25	11	14
28	17	11
26	19	7
28	17	11
32	34	-2
30	25	5
29	29	0
20	21	-1
25	30	-5

Sample mean of the difference using excel function AVERAGE(), x̅d = 4.4444

Sample standard deviation of the difference using excel function STDEV.S(), sd = 6.7474

Sample size, n = 9

90% Confidence interval for the true mean difference in the two techniques:

At α = 0.10 and df = n-1 = 8, two tailed critical value, t-crit = T.INV.2T(0.10, 8) = 1.860

Lower Bound = x̅d - t-crit*sd/√n = 4.4444 - 1.86 * 6.7474/√9 = 0.261

Upper Bound = x̅d + t-crit*sd/√n = 4.4444 + 1.86 * 6.7474/√9 = 8.627

(0.261, 8.6268)

Answer a)