Normal Distribution and CLT

My teacher hasn't taught us how to do this problem but still wants us to do it--

Suppose that the distribution of the weight of a prepackaged 1-pound bag of carrots is N(1.18,.07^2) and the distribution of the weight of a prepackaged 3-pound bag of carrots is N(3.22,.09^2). Selecting bags at random, find the probability that the sum of three 1-pound bags exceeds the weight of one 3-pound bag. HINT: First determine the distribution of Y, the sum of the three, and then compute P(Y>W), where W is the weight of the 3-pound bag.

Now, I know Y=X1+X2+X3. Is it also equal to Chi-Square with 3 degrees freedom? I'm completely lost on this problem.