Step 1 :The best estimate for \(p_{1}-p_{2}\) is the difference in the sample proportions, \(\hat{p}_{1}-\hat{p}_{2}\). So, the best estimate = \(0.75 - 0.69 = 0.06\).
Step 2 :Next, calculate the standard error (SE) for the difference in proportions. The formula for the standard error of the difference in proportions is: \(SE = \sqrt{\frac{\hat{p}_{1}(1-\hat{p}_{1})}{n_{1}} + \frac{\hat{p}_{2}(1-\hat{p}_{2})}{n_{2}}}\).
Step 3 :Substitute the given values into the formula: \(SE = \sqrt{\frac{0.75(1-0.75)}{540} + \frac{0.69(1-0.69)}{230}} = \sqrt{\frac{0.1875}{540} + \frac{0.2139}{230}} = \sqrt{0.000347 + 0.000930} = 0.035\).
Step 4 :For a 90% confidence interval, the z-score is 1.645. The margin of error (ME) is calculated as the product of the z-score and the standard error. So, \(ME = 1.645 * 0.035 = 0.057575\).
Step 5 :Finally, calculate the confidence interval as the best estimate ± the margin of error. So, the confidence interval is \(0.06 ± 0.058\).
Step 6 :\(\boxed{\text{Therefore, the best estimate for } p_{1}-p_{2} \text{ is } 0.06, \text{ the margin of error is } 0.058, \text{ and the confidence interval is from } 0.002 \text{ to } 0.118.}\)