To construct the confidence interval for the difference between two population proportions, we can use the formula:
CI = (p1 - p2) ± z * sqrt((p1 * (1 - p1)/ n1) + (p2 * (1 - p2)/ n2))
where:
p1 = proportion of consumers who would buy cereal A
p2 = proportion of consumers who would buy cereal B
n1 = size of test panel for cereal A
n2 = size of test panel for cereal B
z = z-score corresponding to the desired level of confidence
Given:
p1 = 52/175 = 0.2971
p2 = 35/150 = 0.2333
n1 = 175
n2 = 150
Confidence level = 90% (Corresponding z-score = 1.645)
Substituting the values into the formula, we have:
CI = (0.2971 - 0.2333) ± 1.645 * sqrt((0.2971 * (1 - 0.2971)/175) + (0.2333 * (1 - 0.2333)/150))
CI = 0.0638 ± 1.645 * sqrt((0.08796725/175) + (0.17890689/150))
CI = 0.0638 ± 1.645 * sqrt(0.0005022 + 0.0011927)
CI = 0.0638 ± 1.645 * sqrt(0.0016949)
CI = 0.0638 ± 1.645 * 0.04117
CI ≈ 0.0638 ± 0.0676
The upper limit of the 90% confidence interval for the difference between the two population proportions is approximately 0.0638 + 0.0676 = 0.1314.
Fifty two of a test panel of 175 consumers say that they would buy cereal A if it is presented on the market and 35 of another test panel of 150 consumers say that they would buy cereal B. Construct a 90% confidence interval for the difference between the two population proportions. Give the upper limit.
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