A random sample of 100 males shows that 31 males have a certain minor blood disorder, while a random sample of 100 females shows that only 24 have the disorder. Construct a 99% confidence interval for the diffrences between the two population proportions. Give the the upper limit.
To construct the confidence interval for the difference between the two population proportions, we can use the formula:
Margin of Error (ME) = Z * sqrt(p1(1-p1)/n1 + p2(1-p2)/n2)
Where:
- ME is the margin of error
- Z is the z-score associated with the desired level of confidence (in this case, for a 99% confidence interval)
- p1 is the proportion in sample 1 (males)
- p2 is the proportion in sample 2 (females)
- n1 is the sample size of sample 1 (males)
- n2 is the sample size of sample 2 (females)
First, let's calculate the proportions and sample sizes:
p1 = 31/100 = 0.31
p2 = 24/100 = 0.24
n1 = 100
n2 = 100
Next, we need to find the z-score associated with a 99% confidence level. For a 99% confidence level, the z-score is approximately 2.576.
Plugging in the values:
ME = 2.576 * sqrt(0.31 * (1-0.31)/100 + 0.24 * (1-0.24)/100)
ME ≈ 0.0779
Finally, we can construct the confidence interval:
CI = (p1 - p2) ± ME
CI = (0.31 - 0.24) ± 0.0779
Since we want the upper limit of the confidence interval, let's calculate the upper limit:
Upper Limit = (0.31 - 0.24) + 0.0779
Upper Limit ≈ 0.1479
Therefore, the upper limit of the 99% confidence interval for the difference between the two population proportions is approximately 0.1479.
To construct a confidence interval for the difference between two population proportions, you can use the following formula:
CI = (p1 - p2) ± z * √[(p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2)]
Where:
- p1 and p2 are the sample proportions from each group
- n1 and n2 are the sample sizes from each group
- z is the z-value for the desired confidence level
Given that the sample size for males is 100, with 31 males having the disorder (p1 = 31/100 = 0.31), and the sample size for females is also 100, with 24 females having the disorder (p2 = 24/100 = 0.24), we can calculate the confidence interval.
A 99% confidence interval corresponds to a z-value of 2.576. Plugging these values into the formula, we get:
CI = (0.31 - 0.24) ± 2.576 * √[(0.31 * (1 - 0.31) / 100) + (0.24 * (1 - 0.24) / 100)]
Calculating this expression will give us the confidence interval.