To find variance and covariance for a double sampling problem - Cross Validated most recent 30 from stats.stackexchange.com 2019-10-22T01:55:10Z https://stats.stackexchange.com/feeds/question/96709 https://creativecommons.org/licenses/by-sa/4.0/rdf https://stats.stackexchange.com/q/96709 1 To find variance and covariance for a double sampling problem A.Chakraborty https://stats.stackexchange.com/users/44431 2014-05-07T07:41:19Z 2014-05-07T07:41:19Z <p>A simple random sample of size $n=n_1 + n_2$ is drawn without replacement from a finite population of size $N$. Further a simple random sample of size $n_1$ is drawn without replacement from the first sample. Let $\bar y$ and $\bar {y_1}$ be the respective sample means.</p> <p>Find V($\bar {y_1}$) and V ($\bar {y_2}$), where $\bar {y_2}$ is the mean of the remaining $n_2$ units in the first sample. Also find Cov($\bar {y_1}$,$\bar {y_2}$). Assume that $S^2$ is the population variance.</p> <hr> <p>This is what I've been able to do thus far:</p> <p>Note that $\bar {y_2}$ = $\frac{(n\bar y) - n_1\bar {y_1}}{n_2}$.</p> <p>Also, V($\bar {y_1}$) = $E_1[V_2(\bar {y_1})] + V_1[E_2(\bar {y_1})]$. After simplification, this has turned out to be $(\frac1{n_1} - \frac1N)S^2$. Now am I on the right track? Also, some help in finding $V(\bar y_2)$ and the covariance would be appreciated.</p>