RI S3 Normal Distribution_Tutorial Qns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 ___________________________ Tutorial S3: Normal Distribution Page 1 of 3 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  . [(a) 0.306 (b) 0.0821 (c) 0.293 (d) 0.161 (e) 0.839 (f) 0.683] 2 Given that ~ N(22,5)Y and ~N ( 0 , 1 )Z , find the value(s) of k such that (a) P( ) 0.35Yk , (b) P( ) 0.8Yk , (c) P( 25) 0.46kY  , (d) P(| | ) 0.27Zk , (e) P(22 22 ) 0.67kY k   , (f) P( ) 0.3Yk . [(a) 21.1 (b) 20.1 (c) 21.7 (d) 1.10 (e) 2.18 (f) k > 20.8] 3 The random variable X follows a normal distribution with mean  and standard deviation  . Given that P( 20) 0.14X  and that P( 50) 0.65X  , calculate the values of  and  . [ = 42.1,  = 20.5] 4 The independent random variables X and Y are each normally distributed with means 6 and 8 respectively, and variances 9 and 16 respectively. Find (a) (i) P( )XY (ii) P( 1.5)XY . (b) 12, XX and 3X are three independent observations of X and 12 and YY are two independent observations of Y. Find (i) 123P( 15)XXX , (ii) 12 1 2P( )XXY Y  , (iii) 12P(2 )XYY . [(a)(i) 0.655 (ii) 0.218 (b)(i) 0.282 (ii) 0.286 (iii) 0.314] 5 The random variables X and Y are normally distributed with known means and variances. A student was aske d to find the probability that sum of 3 independent observations of X differ from 3 times an observation of Y by at least 1. State an assumption that the student has to make to solve the question. Part of the student’s solution to this question is given as follows:    3 3 ~ N 3E 3E , 9Var 9VarXY X Y X Y  P 3 3 1 ....XY  Point out 3 mistakes that the student has made. [3]

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ________________________ Tutorial S3: Normal Distribution Page 2 of 3 6 The pineapples in a large warehouse have mass es with a normal distribution. The mean of this distribution is 1.52 kg and the standard deviation is 0.070 kg. Six randomly chosen pineapples are packed in a box of mass 2.15 kg. Find the probability that the total mass of the filled box is less than 11.2 kg. Find the probability that, out of 3 randomly ch osen filled boxes, at least 1 has total mass less than 11.2 kg. [4] [0.342 , 0.715] 7 A company sells bags of cement in two sizes. The mass of a large bag of cement can be modelled by a normal distribution with mean 50 kg and standard deviation 2 kg. The mass of a small bag of cement can be modelled by a normal distribution with mean 30 kg and standard deviation 1.5 kg. (i) Find the mass that is exceeded by 99% of the large bags. [2] (ii) Fi