Caro, there are many factors that can affect this sort of issue with a laptop's adapter.
Have you definitely set the Voltage on the new adapter to 20V? All I did was plug it in.
So, did you see if the sales assistant set the device to the correct Voltage? Did he/she even mention anything about this to you?!
If so, then try setting it to 19V and try again. How do I set it?
Load the product link you posted in #3 and click on 'Download Manuals' on the left of the product information page. Download the 'Quick install manual' at the top of the table shown, which will open a PDF file. There is a round part to the device, which has a dial for setting the correct Voltage, as shown in Diagram 1 - Find your laptop's Voltage.
What about the tip, are you sure you're using the correct tip size for your laptop's power socket? It fits very well so I thought it was ok.
Is the old adapter the original laptop one or an old replacement? An old replacement.
Is your laptop's power socket 'loose' in any way when you connect the adapter's tip into it - the old or new one? The new one seems better than the old one which had a tendency to fall out quite easily. The new one seems more secure.
Does the light on the new adapter come on when plugged into the mains? Yes
You state that the adapter worked in the store, but how did you test this? Was there an exact same model open in the store or just another adapter? The sales assistant asked if I'd definitely buy it if it worked and when I said yes he plugged it into my lappy. The power light flashed amber as it usually does when charging so I thought it was ok.
You can also double-check the correct ratings for your laptop (i.e. Voltage & AMP) on the battery and the bottom of your laptop (if still visible) just to make sure what Voltage you should set the new adapter to.
Rgds.