Is it possible to construct confidence intervals for a mean of weighted means using standard errors?
I have 4 different weighted means ($p_t = \sum_{i=1}^{n} w_ip_i $ where $w_i$ is the respective weight) for $t = 1,2,3,4$ time periods, with $n = 16$: \begin{equation} \bar p_1 = 378.0494 \quad \bar p_2 = 349.1045 \quad \bar p_3 = 339.0127 \quad \bar p_4 = 338.8679 \end{equation} with corresponding standard errors: \begin{equation} s_1 = 43.668 \quad s_2 = 42.133 \quad s_3 = 40.465 \quad s_4 = 40.084 \end{equation} Calculating a moving average ($\hat P_m$) of these: \begin{equation} \hat P_m = 351.2586 \end{equation} Corresponding standard error: \begin{equation} s_{\hat p} = 9.246 \end{equation}
