As per Link: http://bavotasan.com/2011/style-select-box-using-only-css/ there are lot of extra rework that needs to be done(Put extra div and position the image there. Also the design will break as the option drilldown will be mis alligned to the the select.
Here is an easy and simple way which will allow you to put your own dropdown image and remove the browser default dropdown.(Without using any extra div). Its cross browser as well.
HTML
<select class="dropdown" name="drop-down">
<option value="select-option">Please select...</option>
<option value="Local-Community-Enquiry">Option 1</option>
<option value="Bag-Packing-in-Store">Option 2</option>
</select>
CSS
select.dropdown {
margin: 0px;
margin-top: 12px;
height: 48px;
width: 100%;
border-width: 1px;
border-style: solid;
border-color: #666666;
padding: 9px;
font-family: tescoregular;
font-size: 16px;
color: #666666;
-webkit-appearance: none;
-webkit-border-radius: 0px;
-moz-appearance: none;
appearance: none;
background: url('yoururl/dropdown.png') no-repeat 97% 50% #ffffff;
background-size: 11px 7px;
}