You have not given your radio buttons values:
<input type=\"radio\" name=\"weight[$question_num][]\" data-role=\"none\" style=\"text-align:justify\">
should be something like:
<input type=\"radio\" name=\"weight[$question_num][]\" value='Value of radio button 1' data-role=\"none\" style=\"text-align:justify\">
Now you can get the value of the selected radio button in php.