The answer is shown in the next step. Please refer the below image for the detailed solution.

The answer is shown in the next step. Please refer the below image for the detailed solution.

To calculate the weighted average of the two samples, we first need to find the sample variances for each sample. The sample variance is given by:

$[$S$\_\{$x$,$y$\}=\backslash$frac$\{1\}\{$n$\_\{$x$\}\}\backslash$sum$\_\{$i$=1\}^\{$n$\_\{$x$\}\}($x$\_\{$i$\}-$ x$\_\{$avg$,$x$\})^\{2\}]$

$[$S$\_\{$y$,$y$\}=\backslash$frac$\{1\}\{$n$\_\{$y$\}\}\backslash$sum$\_\{$i$=1\}^\{$n$\_\{$y$\}\}($y$\_\{$i$\}-$ y$\_\{$avg$,$y$\})^\{2\}]$

We then calculate the weighted average of the samples by taking the average of the samples, weighted by their respective sample variances:

$[$avg $=($x$\_\{$avg$,$x$\}+$ y$\_\{$avg$,$y$\})/($n$\_\{$x$\}+$ n$\_\{$y$\})]$

$[$avg $=(55+65)/(10+15)]$

$[$avg $=59/25]$

Therefore, the weighted average of the two samples is $59.$

Final answer: The answer is shown in the next step. Please refer the below image for the detailed solution.

Hope you understand the above explanation.

Hence the problem is solved.

Let me know if you have any doubts or if you need anything to change.

Thank you!