WordPress; update and store session variable using jquery and ajax

I am fairly new to WordPress and am having some issues that I can’t solve on my own.

My goal is to update the session variable with the input the user entered in a textbox.

I then want to store the updated session variable to a text file.

I am using jquery and ajax. My code is not working. Any help would be great. Thanks.

I will do my best to explain my code.
I will be happy to add additional information if I am not clear.

The form contains the following code:

<input type="text" id="question_in" />
<input type="button" id="btn" value="Add" />

My wordpress header.php contains the following:

<script src="//ajax.googleapis.com/ajax/libs/jquery/1.6.4/jquery.min.js" type="text/javascript"></script>

<script>
$(document).ready(function() {
 jQuery('#btn').click(function() {    
  var ajaxurl = '<?php echo admin_url('admin-ajax.php'); ?>';  
  updateSession();
});

function updateSession(){
 alert ("in updateSession!"); //for testing this displays
 var new_question = $("#question_in").val();    alert ("input is "+new_question); // for testing this works 

// This does the ajax request
$.ajax({
   type:"POST",
   url: "write-test.php",  
   data: "question=" + new_question,   
   success:function(){
    alert ("in success!");  //this displays
    },
    error: function(errorThrown){
        alert ("There is an error!");
    }
});    
}
</script>

Here is the code for write-test.php

<?php
  session_start(); 
  if(isset($_POST['question'])) {

  $theQuestion = $_POST['question'];

  $myfile = fopen("newfile.txt", "w") or die("Unable to open file!");
  fwrite($myfile, $theQuestion);
  fclose($myfile);
?>