急! More about Dispersion

(a)
You can use 2 methods to calculate the new s.d.

Method 1. Find all the new marks S, and then calculate s.d. by calculator.

Method 2. Calculator for s.d. of the original marks M and follow this step:
S = 0.9*M + 15
s.d. of S = s.d. of (0.9*M + 15)
　　　　= s.d. of (0.9*M)
　　　　= 0.9* [s.d. of M]

(i) Therefore, you can get the answer as about 19.7
(ii) Since (s.d. of S) = 0.9* [s.d. of M] < (s.d. of M), you can tell that the s.d. of the adjusted marks will become smaller.

(b)
(i) Problem of (1): If the original mark is over 91, then the new mark will be over 100 which is very odd.
　Problem of (2): If the original mark is over 75, then the new mark will even be lower than the original mark, which is not what the teacher wants.
[You can think of this by setting S = 0.8M+15 < M and solve for M.]

(ii) S = 0.8M + 20 (as the textbook answer) solves the problem of (2) because only those higher than 100 (which is impossible) would become less.