If you look at the source of the web page you can see it's in a form tag:
<form action="#" method="get">
So Adding this between the form tags would get the desired results: Live Example Link
<form action="#" method="get">
<div data-role="fieldcontain">
<label for="select-choice-1" class="select">Choose shipping method:</label>
<select name="select-choice-1" id="select-choice-1">
<option value="standard">Standard: 7 day</option>
<option value="rush">Rush: 3 days</option>
<option value="express">Express: next day</option>
<option value="overnight">Overnight</option>
</select>
</div>
</form>
Another problem it could be:
Try adding this tag to the dropdown: Live Example Link
data-native-menu="false"
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