php dynamic select/option from mysql table

First, add an id to age group

<select name="age_group" id="age_group">
Then

$( document ).ready(function() {
    $('#age_group').change(function(){
        $.ajax({
            type: "POST",
            url: "ajax.php",
            data: { group_id: $(this).val() },
            dataType: "html"
        })
        .done(function( msg ) {
            $('#hobbies').html(msg);
        });
    });
});
In ajax.php you get all hobbies where group_id = $_POST['group_id']

$sql = "SELECT name FROM tbl_hobbies WHERE group_id = ".$_POST['group_id'];
// Execute your query and put the result in a variable (example $result_query)
// Then loop it
foreach($result_query as $row) {
    echo '<option>'.$row['name'].'</option>';
}
NB : The loop may change according to your method to retrieve data