[Astrokel]

hi guys, this is rather urgent because i have 20 more days to A level. i wanna clarify a doubt on stats as i don't have to attend school anymore until the day of exam so i cannot ask my teacher. I will give an example so it is easier to ask.

An examination consists of two parts: a written paper and a coursework. The marks obtained on each part by the large number of candidates taking the examination may be assumed to follow indepdent normal distribution, with means as shown below.

      mean  standard deviation
WR  48       16
CW  63       10

the marks may be regarded as continous variables so that no continuity are necessary. For a randomly chosen candidate find the probability that

i) the sum of the marks obtained on the two parts of examination will be greater than 150
ii) the marks score on the written paper will be higher than the mark scored on coursework

It is decided to caclulate a candidate's total mark for the examination by adding twice the written pp mark to 3 times the coursework mark. find the probability that a randomly chosen candidate's total calculated in this way will be greater than 200.

State where in your calculations you have used the assumption that a candidate's marks on the two parts of exam are indepdent. explain briefly why this assumption may not be valid in practice.

My problem lies in 3rd part, i let X~N(48, 16²)   Y~N(63,10²)

my working:
E(2X+3Y)  = 2E(X) + 3E(Y) = 96 + 189 = 285
Var(2X+3Y) = 2²Var(x) + 3²Var(Y) = 1924

P((2X+3Y)>200) = 0.974 which is the correct answer

my problem is the bold part and i was taught 2 different situation leading to different variance.

If it is indepdent variable... X1+X2+X3....+Xn ~N(nmean, nvariance)
If it is random variable nX~N(nmean,n²variance)

apparently this question is case 2 but why? When do i use case 1?

i have make this as clear as possible, thanks for any help

[Astrokel]

anyone could help me with the doubt? thanks