Confidence Interval - 2

Confidence Intervals for Difference and Sum

Let \(S_1\) and \(S_2\) are two statistic with a normal or near to normal sampling distribution , the confidence limits for the difference/sum of the population paramters are given by \(S_1-S_2\) +/- \(z_c\sigma{s_1-s_2}\), we can replace \(\sigma{s_1-s_2}\) by \(\sqrt{{\sigma_{S_1}}^2 + \sigma{_{S_2}}^2}\), so the new formula \[S_1-S_2 - z_c\sqrt{{\sigma_{S_1}}^2 + \sigma{_{S_2}}^2}\] and \[S_1-S_2 + z_c\sqrt{{\sigma_{S_1}}^2 + \sigma{_{S_2}}^2}\]

For example , the confidence limit of two population means, in case where the population is infinite are given by,

\[ \bar{X_1}-\bar{X_2} - z_c\sqrt{{\sigma_{X_1}}^2 + \sigma{_{X_2}}^2} =\bar{X_1}-\bar{X_2} - z_c\sqrt{\frac{{\sigma_1}^2}{N_1} + \frac{{\sigma_2}^2}{N_2}} \]

and \[ \bar{X_1}-\bar{X_2} + z_c\sqrt{{\sigma_{X_1}}^2 + \sigma{_{X_2}}^2} =\bar{X_1}-\bar{X_2} + z_c\sqrt{\frac{{\sigma_1}^2}{N_1} + \frac{{\sigma_2}^2}{N_2}}\]

Similarly , confidence limit for the difference of two population proportion, where population is infinite.

\[ P_1-P_2 - z_c\sqrt{{\sigma_{X_1}}^2 + \sigma{_{X_2}}^2} = P_1-P_2 - z_c\sqrt{\frac{p_1q_1}{N_1} + \frac{p_2q_2}{N_2}}\]

and \[ P_1-P_2 + z_c\sqrt{{\sigma_{X_1}}^2 + \sigma{_{X_2}}^2} =P_1-P_2 + z_c\sqrt{\frac{p_1q_1}{N_1} + \frac{p_2q_2}{N_2}}\]