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.
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.
To construct a 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 and p2 are the sample proportions (in this case, the proportion of consumers who would buy cereal A and cereal B, respectively)
- n1 and n2 are the sample sizes (the number of consumers in each test panel)
- Z is the z-value corresponding to the desired confidence level (90% confidence interval corresponds to z = 1.645)
First, we need to calculate the sample proportions:
p1 = 52 / 175
p2 = 35 / 150
p1 = 0.2971
p2 = 0.2333
Next, we substitute these values into the formula:
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((0.2095 / 175) + (0.1663 / 150))
CI = 0.0638 ± 1.645 * sqrt(0.0012 + 0.0011)
CI = 0.0638 ± 1.645 * sqrt(0.0023)
CI = 0.0638 ± 1.645 * 0.0481
CI = 0.0638 ± 0.0791
Finally, we can calculate the upper limit of the confidence interval:
Upper limit = 0.0638 + 0.0791 = 0.1429
Therefore, the upper limit of the 90% confidence interval for the difference between the two population proportions is 0.1429.