RI S3 Normal Distribution Add Prac Solns

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 ______________________________________________________ Additional Practice Questions for Chapter S3: Normal Distribution Page 1 of 15 Additional Practice Questions for Chapter S3: Normal Distribution (Solutions) 1 8864/2007/01/Q6 A manufacturer produces packets of margarine. The mass of margarine in a packet has a normal distribution with mean 502g and standard deviation 0.8g. Find the proportion of packets which contain less than 500g of margarine. [2] The manufacturer increases the mean amount of margarine in a packet to  g. The standard deviation remains unchanged. Only 1 packet in 1000, on average, now contains less than 500g. Find  , correct to 1 decimal place. [3] Solution Let X be the mass of a packet of margarine in grams. Then  2N 502, 0.8X Required proportion= P X 500 0.00621 (3 s.f)  Let Y be the mass of a packet of margarine with new mean mass  in grams. Then 2N( ,0.8 )Y  Given that P( 500) 0.001Y    500 P 0.0010.8 From GC, P 3.09023 0.001 500 3.090230.8 500 2.4722 502.5 (1 d.p) Z Z                

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________ Additional Practice Questions for Chapter S3: Normal Distribution Page 2 of 15 2 9205/1995/02/Q9 (modified) The length of time which an ordinary light-bulb will last may be taken to have a normal distribution with mean 600 hours and standard deviation 100 hours. The length of time for which a new ‘long life’ bulb will last may be taken to have a normal distribution with mean 2000 hours and standard deviation 200 hours. (i) One ordinary bulb is chosen at random. Find the probability that it will last for more than 450 hours. [2] (ii) Two ordinary bulbs are chosen at random. Find the probability that the sum of the times for which they last will be less than 1100 hours. [3] (iii) One ordinary bulb and one long-life bulb are chosen at random. Find the probability that the long-life bulb lasts for more than three times as long as the ordinary bulb. [3] The total time in which 10 ‘long life’ bulbs will last is denoted by W hours. Find the value of t (to the nearest hour) such that P( ) 0.75W t  . [3] Solution (i) Let X be the length of time for which an ordinary light-bulb will last, in hours. Then 2N 600,100X  Let Y be the length of time for which a new ‘long-life’ light-bulb will last, in hours. Then 2N 2000, 200Y  P( 450) 0.933X  (3 s.f) (ii) 2 1 2 N 2(600), 2(100) X X  1 2i.e. N 1200, 20000X X  P( 1 2 1100X X  ) = 0.240 (3 s.f) (iii)   2 2 23 N 2000 3(600), 200 3 (100 ) i.e. 3 N 200, 130000 Y X Y X       P( 3 0) 0.710Y X   (3 s.f)   1 2 10 2 Let ... . Then N 10(2000),10(200) W Y Y Y W      i.e. N 20000, 400 000W P( ) 0.75W t  From the G.C, P( 19573) 0.75W   19573t (to the nearest hour)

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________ Additional Practice Questions for Chapter S3: Normal Distribution Page 3 of 15 3 Math B/1982/01/Q12 (Modified) [The notation 2N( , )  denotes a normal distribution with mean  and variance 2 .] The school bus leaves the stop near Ben’s home at X minutes past 8.00 a.m., where X is an 2N(20,3 ) random variable. Ben reaches the bus stop at Y minutes after 8.00 a.m., where Y is an 2N(15,2 ) random variable, X and Y being independent. Find the probability (to three decimal places) that Ben misses the bus. [2] The ride from this stop to the school lasts T minutes, where T is an 2(30,( 7) )N random variable, independent of X. The random variable W is the number of minutes before 9.00 a.m. at which the bus arrives at the school. Express W in terms of X and T. [1] Calculate the mean and variance of W and find, to three significant figures, the probability that the bus arrives at the school after 9.00 a.m. [2] Solution ~ N(5,13) P(Ben misses the bus) P( 0) 0.083 (3 d.p) X Y X Y      22 60 E( ) 60 20 30 10 Var( ) 3 7 16 W X T W W           i.e ~ N(10,16)W P( the bus arrives after 9) P( 0) 0.00621 (3 s.f.) W  

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________ Additional Practice Questions for Chapter S3: Normal Distribution Page 4 of 15 4 9740/2007/02/Q8 Chickens and turkeys are sold by weight. The masses, in kg, of chickens and turkeys are modeled as having independent normal distributions with means and standard deviations as shown in the table. Chickens are sold at $3 per kg and turkeys at $5 per kg. (i) Find the probability that a randomly chosen chicken has a selling price exceeding $7. [2] (ii) Find the probability of the event that both a randomly chosen chicken has a selling price exceeding $7 and a randomly chosen turkey has a selling price exceeding $55. [3] (iii) Find the probability that the total selling price of a randomly chosen chicken and a randomly chosen turkey is more than $62. [4] (iv) Explain why the answer to part (iii) is greater than the answer to part (ii). [1] Solution (i) Let X and Y be the mass of a chicken and turkey in kg respectively. Then 2N(2.2,0.5 )X  and 2N(10.5,2.1 )Y . Let C be the selling price, in $, of a chicken. Then 2 23 N 3 2.2, 3 0.5C X   , i.e. N(6.6, 2.25)C . Let T be the selling price, in $, of a turkey. Then 2 25 N(5 10.5, 5 2.1 )T Y   , i.e. N(52.5, 110.25)T  . P( 7) 0.395C  (3 s.f.) or 7P 3X    (ii) P( 7 and 55) P( 7)P( 55)C T C T     , since C and T are independent = 0.160 (3 s.f.) or 7P 3X    55P 5Y    (iii) E( ) E( ) E( ) 6.6 52.5 59.1C T C T      2 2Var( ) Var( ) Var( ) 1.5 10.5 112.5C T C T      N(59.1, 112.5)C T  P( 62) 0.392C T   (3 s.f.) (iv) Part (iii) includes the cases in part (ii) as well as other cases. For example, 6 7C  and 56T so that 62C T  is included in part (iii) but not in part (ii). Hence, the answer for (iii) would be greater than the answer in (ii). Mean mass Standard deviation Chickens 2.2 0.5 Turkeys 10.5 2.1

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________ Additional Practice Questions for Chapter S3: Normal Distribution Page 5 of 15 5 HCI Prelim 9740/2014/02/Q7 The speed of a randomly chosen passenger car travelling from point A to point B on an expressway has a normal distribution with mean 85 km/h and standard deviation 20 km/h. It is assumed that a passenger car travels at a constant speed from point A to point B. (i) For passenger cars travelling from point A to point B, find the probability that the total speed of 3 randomly chosen cars differs from twice the speed of a randomly chosen car by at most 50 km/h. [3] (ii) The speeds of 80 randomly chosen passenger cars which travelled from point A to point B are recorded. Find the probability that the total distance travelled in 5 minutes is at most 550 km. State an assumption you had made in your calculation. [4] Solution (i) Let S denote the speed of a randomly chosen passenger car in km/h. S ~ N(85, 202) Let 1 2 3 2X S S S S    . E(X) = 3E(S)  2E(S) = E(S) = 85 Var(X) = 3Var(S) + 22Var(S) = 7Var(S) = 2800  N 85, 2800X    P 50 P 50 50 0.249