In: Math
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
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
----------------------------
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)