Step 1 :First, we calculate the sample proportions for New Yorkers and Californians. The sample proportion for New Yorkers, denoted as \(p_{1}\), is calculated as \(\frac{220}{702} = 0.313\). The sample proportion for Californians, denoted as \(p_{2}\), is calculated as \(\frac{379}{924} = 0.410\).
Step 2 :Next, we calculate the difference in sample proportions, \(p_{1} - p_{2}\), which is \(0.313 - 0.410 = -0.097\).
Step 3 :We then calculate the standard error for the difference in proportions. The standard error, denoted as SE, is calculated as \(\sqrt{\frac{p_{1}(1-p_{1})}{702} + \frac{p_{2}(1-p_{2})}{924}} = 0.024\).
Step 4 :We use the Z-score for a 99% confidence interval to calculate the margin of error. The Z-score for a 99% confidence interval is approximately 2.576. The margin of error, denoted as ME, is calculated as \(Z \times SE = 2.576 \times 0.024 = 0.061\).
Step 5 :Finally, we subtract and add this margin of error from the difference in sample proportions to get the confidence interval. The lower limit of the confidence interval is \(-0.097 - 0.061 = -0.158\), and the upper limit of the confidence interval is \(-0.097 + 0.061 = -0.035\).
Step 6 :So, the 99% confidence interval for the difference between the two population proportions, \(p_{1}-p_{2}\), is \(\boxed{[-0.158, -0.035]}\).