I'm doing some homework over the break (!) so I don't have access to my usual lines of help. I've hit a wall:

I assume the "N(12.2, .04)" notation refers to that the distribution of the boxes has mean 12.2 and variance .04. I think 'a' has something to do with the normal distribution, how standard deviations mark percentages of the curve. 'b' I just have no clue on how to proceed.

Also:

What do they mean by triangular?

The last portion of the class I would hazard a guess that I could integrate 'f(w)' to get the CDF 'W', but I'm not sure if that applies to this. The answer is supposed to be, 'mean = 0' and 'variance = 1' but damned if I can get it to come out that way just by manipulating the numbers.

I'd wager a guess that it just means the probability increases as w increases according to the line 2w. f(w) = 2w will give you a triangle, after all. Naturally, it still needs to be normalized, though.

N(12.2,.04) means a normal distribution with mean 12.2 and standard deviation (not variance) .04.

#1a) Integrate your distribution between 0 and 12.
#1b) Try using one of the tests you learned.
#2a) Okay.
#2b) You have formulas for this.

As for the "triangluar" you can get the answers by ignoring this term
"triangular" (If you shade in the area under the pdf it is a triangle but who cares).

the expected value of W is
[tex] E(W)=\int_0^1 w f(w) = \int_0^1 2w^2= 2/3 [/tex]
the variance of W is
[tex] (\int_0^1 w^2f(w) )- (E(W)^2) = (\int_0^1 2w^3) - (\frac{2^2}{3^2})= \frac{1}{18} [/tex]

you should now realise that you are close to the answer just use the formulas E(aW+b) = aE(W)+ b
and Var(aW+b) = a^2 Var(W)

Hint E(a(W-b)) = a E(W-b) = a ( E(W)-b))
subbing crap in gives
E( sqrt(18) (W-2/3) ) = sqrt18( E(W)-2/3) =0

I'm sorry, but those explanations do nothing for me. We haven't learned T-tests as far as I know.

Damned charming :) -

I don't have power point, thanks though.

I can't get those pictures (?) to paste, but you get the idea. How did you do that anyway? That's kool.

Anyway, I got the first two answers correctly, those ideas go back several chapters, I have a firm grasp on those. But I'm not sure, exactly how they relate to part 'b'. I've tried doing the integrations the same way, but to no avail. The mean comes out correctly, to zero, but the variance is some strange number (sqrt(2)/6).

We've done this stuff, but shouldn't the integral come out to the same solution as the shortcut here?

[tex] \sqrt{2}/6 = \sqrt{ \frac{1}{18}} [/tex] which is the
standard deviation when the variance 1/18, as my integration confirms.
I am not sure what you mean when you say when the integration should be same as the short cut. An alternative method using integration is
[tex] E(W-2/3) = \int_0^1 ( W- 2/3)2W = 0 [/tex]

PS if you want to know how I have done the integrals I have attached a txt file containing exactly what I typed.