2023 JC2 H2 Normal Distribution APQ Solutions (RVHS)

Statistics 5 Tutorial: Normal Distribution Additional Practice Questions 1. 2011 Prelim/DHS/P2/Q9 In a certain junior college, the marks (out of 100) scored by a JC 1 student in a Class Test, Common Test and Promotional Examination are denoted by C, T and S respectively. C, T and S may be modelled by normal distributions with means and standard deviations as shown in the table below. Type of assessment Mean Standard deviation Class Test, C 68  Common Test, T 65 8 Promotional Examination, S 70 10 (i) Given that P( 85) 0.05,C= determine the value of . In a particular year, a student sits for five Class Tests, a Common Test and a Promotional Examination. The average mark of the five Class Tests constitutes 20% of the overall assessment mark for the year. The Common Test and Promotional Examination consti tute 20% and 60% of the overall assessment mark for the year respectively. For the following parts, assume 10 = . (ii) Find the probability that the average mark scored by a student in the five Class Tests is more than 75. (iii) Find the probability that a student scores an overall mark of more than 80 for the year. (iv) State an assumption used in your calculations for (iii). Solution: (i) 2 2 2 ~ N(68, ) ~ N(65,8 ) ~ N(70,10 ) C T S  P( 85) 0.05 85 68P 0.05 85 68 1.6449 10.335 10.3 (3 s.f.) C Z    = − = − = == (ii) 2 2 1 2 5 ~ N(68,10 ) 10~ N 68,55 C C C CC + + +=  P( 75) 0.058762 0.0588 (3 s.f.)C = =

(iii) Let 0.2 0.2 0.6 E( ) 0.2(68) 0.2(65) 0.6(70) 68.6 X C T S X = + + = + + = 2 2 2 2 2 210Var( ) 0.2 0.2 (8 ) 0.6 (10 ) 39.365 ~ N(68.6,39.36) X X = + + =  P( 80) 0.034601 0.0346 (3 s.f.)X  = = (iv) or , and are independent.C C T S 2. 2010 Prelim/CJC/P2/Q8 The masses of snapper fish and pomfret fish sold by a fishmonger are normally distributed and independent of each other. The mean mass, standard deviation and selling price of snapper fish and pomfret fish are given in the following table: Snapper fish Pomfret fish Mean mass in kg 1 0.6 Standard deviation in kg 0.1 0.05 Selling price per kg in $ 12 7 Find the probability that the (i) total mass of 3 snapper fish and 2 pomfret fish is more than 4.5 kg; (ii) mass of 3 snapper fish exceeds twice the mass of a pomfret fish by more than 1.85 kg; (iii) total selling price of a snapper fish and 2 pomfret fish is more than $21. A customer buys 15 fish, out of which n are snapper fish and the rest are pomfret fish. The probability that the customer pays more than $150 is less than 0.7. Find the largest value of n. Solution: (i) Let X and Y be the r.v. mass of a snapper fish and a pomfret fish respectively. X ~ N(1, 0.12); Y ~ N(0.6, 0.052) X1 + X2 + X3 + Y1 +Y2 ~ N(4.2, 0.035) P[X1 + X2 + X3 + Y1 +Y2 > 4.5] = 0.0544 (ii) X1 + X2 + X3 – 2Y ~ N(1.8, 0.04) P[X1 + X2 + X3 – 2Y > 1.85] = 0.401 (iii) 12X + 7( Y1 +Y2) ~ N(20.4, 1.685) P[12X + 7( Y1 +Y