SPM Additional Mathematics 2017, Paper 1 (Question 23 – 25)


Question 23 (4 marks):
A set of data consists of 2, 3, 4, 5 and 6. Each number in the set is multiplied by m and added by n, where m and n are integers. It is given that the new mean is 17 and the new standard deviation is 4.242.
Find the value of m and of n.

Solution:

x =2+3+4+5+6=20 x 2 = 2 2 + 3 2 + 4 2 + 5 2 + 6 2 =90 Mean= 20 5 =4 Variance= x 2 n ( x ¯ ) 2   = 90 5 4 2 =2 New mean=17 4m+n=17 ………. ( 1 ) New standard deviation=4.242 m× 2 =4.242 m= 4.242 2 =2.99953 Substitute m=3 into ( 1 ): 4( 3 )+n=17 n=5


Question 24 (3 marks):
Diagram 9 shows the graph of binomial distribution X ~ B(3, p).

Diagram 9

(a)
Express P(X = 0) + P(X > 2) in terms of a and b.
(b) Find the value of p.


Solution:
(a)
P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 1
P(X = 0) + a + b + P(X = 3) = 1
P(X = 0) + P(X = 3) = 1 – a – b
P(X = 0) + P(X > 2) = 1 – a – b

(b)

P( X=0 )= 27 343 3 C 0 ( p 0 ) ( 1p ) 3 = 27 343 1×1× ( 1p ) 3 = ( 3 7 ) 3 1p= 3 7 p= 4 7


Question 25 (4 marks):
Diagram 10 shows a standard normal distribution graph.

Diagram 10

The probability represented by the area of the shaded region is 0.2881.
(a) Find the value of h.
(b) X is a continuous random variable which is normally distributed with a mean, μ and a variance of 16.
Find the value of μ if the z-score of X = 58.8 is h.


Solution:
(a)
P(X < h) = 0.5 – 0.2881
P(X < h) = 0.2119
P(X < –0.8) = 0.2119
h = –0.8

(b)

X=58.8 Xμ σ = 58.8μ σ    Z= 58.8μ 4    h= 58.8μ 4 0.8= 58.8μ 4 3.2=58.8μ μ=58.8+3.2 μ=62

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