Question

Discuss the problem of finding a confidence interval for the difference \(\mu_{1}-\mu_{2}\) between the two means of two normal distributions if the variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\) are known but not necessarily equal.

Short answer

To find the confidence interval for the difference between two means of two normal distributions when the variances are known but not necessarily equal, you can use the formula \( CI = (\mu_{1}-\mu_{2}) ± Z* \sqrt{(\sigma_{1}^{2}/n_{1}) + (\sigma_{2}^{2}/n_{2})} \) where \( CI \) represents the confidence interval, \( \mu_{1}-\mu_{2} \) is the difference of the population means, \( Z* \) is the z-score that corresponds to the desired confidence level, \( \sigma_{1}^{2} \) and \( \sigma_{2}^{2} \) are the variances, and \( n_{1} \) and \( n_{2} \) are the sample sizes. Simply plug in the appropriate values into this formula to compute the confidence interval.

Step-by-step solution

Understand Confidence Interval Formula

The general formula for a confidence interval for the difference between two means is \( CI = (\mu_{1}-\mu_{2}) ± Z* \sqrt{(\sigma_{1}^{2}/n_{1}) + (\sigma_{2}^{2}/n_{2})} \). Here, \( n_{1} \) and \( n_{2} \) represent the sample sizes, \( Z* \) is the z-score that corresponds to your chosen level of confidence, \( \mu_{1}-\mu_{2} \) is the difference of the means, and \( \sigma_{1}^{2} \) and \( \sigma_{2}^{2} \) are the variances of the two populations.

Apply Known Variances

Next, you apply the known variances \( \sigma_{1}^{2} \) and \( \sigma_{2}^{2} \) into the formula. This becomes \( CI = (\mu_{1}-\mu_{2}) ± Z* \sqrt{(\sigma_{1}^{2}/n_{1}) + (\sigma_{2}^{2}/n_{2})} \).

Calculate Confidence Interval

Lastly, to calculate the confidence interval, you will need to know or estimate other factors in the equation such as the sample sizes \( n_{1} \) and \( n_{2} \), the z-score for your desired confidence level, and the difference between the two means \( \mu_{1}-\mu_{2} \). Once these are known or estimated, put them in the equation to compute the confidence interval.