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 the means of two normal distributions, the formula \(\mu_{1} - \mu_{2} ± z\sqrt{\sigma_{1}^2/n_{1} + \sigma_{2}^2/n_{2}}\) is used when variances are known but not necessarily equal. The resulting interval will give the range within which the difference between the means is likely to fall with a certain level of confidence.

Step-by-step solution

Understanding the Formula

When the question asks for the confidence interval for \( \mu_{1} - \mu_{2} \), it indicates we need to find the range in which the difference of means of two normal distributions will lie, with a certain level of confidence. Given that the variances are known but not necessarily equal, the formula for the confidence interval would be: \(\mu_{1} - \mu_{2} ± z\sqrt{\sigma_{1}^2/n_{1} + \sigma_{2}^2/n_{2}}\) where \( \sigma_{1}^2\) and \( \sigma_{2}^2\) are population variances, \( n_{1}\) and \( n_{2} \) are the sample sizes, and \( z\) represent the z-value from the standard normal distribution corresponding to the desired level of confidence.

Finding Z-score

The z-score or z-value is based on the level of confidence we get from the standard normal distribution table. For example, if the confidence level is 95%, the z-value is 1.96.

Applying the formula

Given that we have the means, variances and sample sizes of the two samples, we can plug the values in to the formula we defined in Step 1. The result will be the confidence interval for the difference between two means.