RI S3 Normal Distribution_Tutorial Solns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 ___________________________ Tutorial S3: Normal Distribution Page 1 of 11 Tutorial S3: Normal Distribution 1 Given that ~ N(1.5,2.5)X , evaluate (a) P( 0.7)X , (b) P( 3.7)X , (c) P(1.1 2.3)X  , (d) P(| | 0.5)X , (e) P(| | 0.5)X (f) P( 1.5 2.5)X  . Solution (a) to (d) can be evaluated from GC directly. (e) P(| | 0.5)X 1 P( 0.5 0.5) 0.839 (3 s.f) X      Or: P(| | 0.5) P( 0.5) P( 0.5)X X X     (f) P( 1.5 2.5)X  P( 2.5 1.5 2.5) P(1.5 2.5 1.5 2.5) 0.683 (3 s.f) X X            Or: 1.5P( 1.5 2.5) P( 1) P( 1) 0.6832.5 XX Z        2 Given that ~ N(22,5)Y and ~ N(0,1)Z , find the value(s) of k such that (a) P( ) 0.35Y k  , (b) P( ) 0.8Y k  , (c) P( 25) 0.46k Y   , (d) P(| | ) 0.27Z k  , (e) P(22 22 ) 0.67k Y k     , (f) P( ) 0.3Y k  . 1.5 1.5 +2.5 1.5 2.5

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ________________________ Tutorial S3: Normal Distribution Page 2 of 11 Solution ~ N(22,5)Y and ~ N(0,1)Z (a) P( ) 0.35Y k  From G.C, 21.1 (3 s.f)k k = invNorm(0.35,22,5, LEFT) = 21.1 (b) P( ) 0.8Y k  From G.C, 20.1 (3 s.f)k k = invNorm(0.8,22,5,RIGHT) = 20.1 (c) P( 25) 0.46k Y   From the diagram, P( 25) P( ) 0.46 0.91014 P( ) 0.46 P( ) 0.45014 Y Y k Y k Y k          From G.C, 21.7 (3 s.f)k k = invNorm(0.4501,22,5,LEFT) = 21.7 (d) P(| | ) 0.27Z k  From the diagram, P( ) 0.73Z k  From G.C, 1.10 (3 s.f)k k = invNorm(0.73,0,1,CENTER) =1.10 25 22 0.46 k k 22 0.8 0.35 k 22 k 0 0.135 0.135 k

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ________________________ Tutorial S3: Normal Distribution Page 3 of 11 (e) P(22 22 ) 0.67k Y k     From G.C, 22 19.822 and 22 24.178 2.18 (3 s.f) k k k       using invNorm(0.67,22,5,CENTER), 22 – k =19.822 and 22 + k = 24.178 (f) P( ) 0.3Y k  From G.C, P( 20.8) 0.3Y  From the diagram, 20.8 (3 s.f)k invNorm(0.3,22,5,LEFT) = 20.8 22 0.165 0.67 22+k 22 k 0.165 0.3 20.8 22

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ________________________ Tutorial S3: Normal Distribution Page 4 of 11 3 The random variable X follows a normal distribution with mean 