How to display a specific range of numbers in an SQL query

So there's just a little bit more to this than the title suggests. I've got a table which contains a name column and a qty column, I want to display the results who have a TOTAL sum qty between 500 and 1500.
Example:
brad 100
cody 300
kylie 1100
brad 800
nelson 1200
bob 600
nelson 100
kylie 600

I want the result to display
brad 900
nelson 1300
bob 600

Hopefully that makes sense. This is the query I've got
SELECT name, SUM(qty) AS Total FROM t1 NATURAL JOIN t2 WHERE qty BETWEEN 500 AND 1500 GROUP BY name ORDER BY name

The issue is that it seems to only be summing the fields in qty that are between 500 and 1500, rather than display only the Total fields that are in that range. I tried doing "...WHERE SUM(qty) BETWEEN...." but this results in a compilation error (i'm using SQLite studio)

This is a homework problem for a database class I'm in, I'm looking to learn, not just get the answer. Thanks!