OK since someone requested:
1.a.i. ![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?E(D)=E(4L-5S))
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?=E(4L)-E(5S))
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?=4E(L)-5E(S))
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?=(4)(65)-(5)(50))
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?=260-250)
ii. ![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?Var(D)=Var(4L+(-5)S))
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?=Var(4L)+Var((-5)S))
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?=4^{2}Var(L)+(-5)^{2}Var(S))
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?=(16)(16)+(25)(9))
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?=256+225)
b. First define your variables. The letters in the first part are a clue.
Let L be the weight of a large egg
Let S be the weight of a standard egg
Next, turn what they're asking into a mathematical formula. Just disect the question word by word and this shouldn't be hard. Probability indicates that obviously, you're finding a probability, so you'll need Pr(something). Next, since it's asking for the probability that a standard egg is greater than four-fifths of the weight of the large egg, using your variables, this reads as
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?S>\frac{4}{5}L)
, so combine the two to get
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?Pr(S>\frac{4}{5}L))
So working from there, you're actually allowed to move terms inside the Pr, so multiply both sides by 5 and transpose to get:
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?Pr(5S-4L>0))
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?=Pr(4L-5S<0))
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?=Pr(D<0))
<---recognise the link between this question and the first one
Then using the values you worked out before, use the normal distribution to calculate it.
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?=Pr(Z<\frac{0-10}{\sqrt{481}}))
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\approx 0.324)