Final answer:
To construct a 90% confidence interval for the difference between two population proportions, calculate the sample proportions, then the standard error, and use the z* value corresponding to the 90% confidence to find the margin of error. Finally, add and subtract this margin of error from the difference of sample proportions to get the interval.
Step-by-step explanation:
To construct a confidence interval for the difference between two population proportions (p1 - p2), we first need to calculate the sample proportions: p1' = x1/n1 and p2' = x2/n2. Then we find the standard error of the difference in sample proportions (SE) using the formula: SE = √[(p1'(1-p1')/n1) + (p2'(1-p2')/n2)]. Once SE is calculated, we can find the critical value (z*) corresponding to the desired level of confidence from the standard normal distribution. For a 90% level of confidence, z* is approximately 1.645. The margin of error (E) is then calculated as E = z* × SE. Finally, the confidence interval is given by (p1' - p2') ± E.
Given the question values:
- x1 = 355
- n1 = 513
- x2 = 408
- n2 = 575
We calculate the sample proportions as follows:
- p1' = 355/513
- p2' = 408/575
The standard error (SE) is then calculated using the sample proportions and sample sizes. We compute E by multiplying SE with the z* value for 90% confidence, which is 1.645. Finally, the confidence interval for p1 - p2 is found by adding and subtracting E from the difference in sample proportions (p1' - p2').