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 lower limit.
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:
- CI is the confidence interval
- p1 and p2 are the proportions of the two populations (52/175 for Cereal A and 35/150 for Cereal B)
- Z is the Z-value for the desired confidence level (90% confidence level corresponds to a Z-value of 1.645)
- n1 and n2 are the sample sizes for the two populations (175 for Cereal A and 150 for Cereal B)
Calculating the confidence interval:
p1 = 52/175 = 0.297
p2 = 35/150 = 0.233
Z = 1.645
n1 = 175
n2 = 150
CI = (0.297 - 0.233) ± 1.645 * sqrt((0.297 * (1 - 0.297) / 175) + (0.233 * (1 - 0.233) / 150))
Calculating the standard error:
SE = sqrt((0.297 * (1 - 0.297) / 175) + (0.233 * (1 - 0.233) / 150))
Substituting the values:
CI = (0.064) ± 1.645 * (0.028)
Calculating the confidence interval:
CI = 0.064 ± 0.046
To find the lower limit, subtract the margin of error from the difference:
Lower limit = 0.064 - 0.046 = 0.018
Therefore, the lower limit of the 90% confidence interval for the difference between the two population proportions is 0.018.
To construct a confidence interval for the difference between two population proportions, we can use the formula:
(p1 - p2) ± Z * √((p̂1 * (1 - p̂1) / n1) + (p̂2 * (1 - p̂2) / n2))
where:
p1 and p2 are the sample proportions
n1 and n2 are the sample sizes
Z is the z-score corresponding to the desired confidence level
In this case, for a 90% confidence level, the z-score is 1.645.
p̂1 = 52/175 = 0.2971
p̂2 = 35/150 = 0.2333
n1 = 175
n2 = 150
Substituting the values into the formula, we get:
(0.2971 - 0.2333) ± 1.645 * √((0.2971 * (1 - 0.2971) / 175) + (0.2333 * (1 - 0.2333) / 150))
(0.0638) ± 1.645 * √((0.0883 / 175) + (0.0704 / 150))
(0.0638) ± 1.645 * √(0.000504 + 0.000469)
(0.0638) ± 1.645 * √(0.000973)
(0.0638) ± 1.645 * 0.0312
(0.0638) ± 0.0513
The lower limit of the confidence interval is:
0.0638 - 0.0513 = 0.0125
Therefore, the lower limit of the 90% confidence interval for the difference between the two population proportions is 0.0125