7.11: Two Population Means with Known Standard Deviations
- Page ID
- 4608
Even though this situation is not likely (knowing the population standard deviations is very unlikely), the following example illustrates hypothesis testing for independent means with known population standard deviations. The sampling distribution for the difference between the means is normal in accordance with the central limit theorem. The random variable is \(\overline{X_{1}}-\overline{X_{2}}\). The normal distribution has the following format:
\[\textbf{The standard deviation is:}\nonumber\]
\[\sqrt{\frac{\left(\sigma_{1}\right)^{2}}{n_{1}}+\frac{\left(\sigma_{2}\right)^{2}}{n_{2}}}\nonumber\]
\[\textbf{The test statistic (z-score) is:}\nonumber\]
\[Z_{c}=\frac{\left(\overline{x}_{1}-\overline{x}_{2}\right)-\delta_{0}}{\sqrt{\frac{\left(\sigma_{1}\right)^{2}}{n_{1}}+\frac{\left(\sigma_{2}\right)^{2}}{n_{2}}}}\nonumber\]