You can't.
Serial numbers were not required before 1968. Check the barrel and receiver.
http://www.scottcreative.net/Default.asp?ID=135
Hallet Davis & Co. started with serial 163000 in 1938. Your serial 163344, then, would have been produced fairly early on in the year. As a pre-1960 build, we can also assume this is a grand piano; Hallet Davis did not start producing uprights until after 1960.
$50 - $80
About $150
on hallet davis co. upright piano seriel #69465
Try www.bluebookofpianos.com Pre-1900 according to the above website.
Jan Davis has written: 'Formulating Wealth in the Food and Drink Industries'
Just sold a derringer 25 Davis for $75
50-100
50-75 USD