RI T2W6+Normal+Distribution+%28Soln%29

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 _____________________________________ Y6 H2 Math T2W3 Math Focus: Vectors 2 and 3 Page 1 of 6 Term 2 Week 6 Math Focus Topic: Normal Distribution 1 EJC Prelim 9758/2020/02/Q11(i), (ii), (iii) [modified] The masses, in pounds, of apples and pears have independent normal distributions with means and standard deviations as shown in the following table. Mean Standard deviation Apple 0.33 σ Pear 0.45 0.08 (i) The probability of a randomly chosen apple weighing less than t pounds and that weighing between t and 0.35 pounds is 0.34 each. Find the values of σ and t. [4] (ii) Three pears are chosen at random. Find the probability that one of them weighs less than the population mean mass and each of the other two pears has a mass within one standard deviation of the population mean mass. [2] The apples cost $2.20 per pound and the pears cost $3.10 per pound. (iii) Find the probability that the total cost of 3 randomly chosen apples and 2 randomly chosen pears is more than $5. [3] (i) ( ) 2 Let be the mass of an apple (in pounds). N 0.33, X X σ ( ) ( ) P 0.35 1 0.34 0.34 0.32 0.35 0.33P 0.32 From GC, P 0.46770 0.32 0.02 0.46770 0.042762 0.0428 (3 s.f) X Z Z σ σ σ > = −−= − >= >= ∴= ≈= ( )P 0.34 From GC, 0.312 (3 s.f) Xt t <= =

Raffles Institution H2 Mathematics 2025 Year 6 _________________________________________________________________________________________ _____________________________________ Y6 H2 Math T2W6 Math Focus: Normal Distribution Page 2 of 6 Alternatively, ( ) ( ) ( ) ( ) P 0.35 0.34 P 0.35 P 0.34 P 0.35 0.34 0.34 0.68 0.35 0.33P 0.68 0.35 0.33From GC, 0.46770 0.042762 0.0428 (3s.f.) tX X Xt X Z σ σ σ << = < − <= < =+= − <= − = = = (ii) ( ) 2 Let be the mass of a pear (in pounds). N 0.45,0.08 Y Y  ( )( ) ( ) ( ) ( ) 23 1 23 1 Required probability P 0.45 P 0.45 0.08 0.45 0.08 0.5 P 0.37 0.53 0.699 (3 s.f) CY Y CY = <  − << +   =  <<  = (iii) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 12 2 2 2 22 2 Let 2.2 3.1 E 2.2 3E 3.1 2E 2.2 3 0.33 3.1 2 0.45 4.968 Var 2.2 3E 3.1 2E 2.2 3 0.042762 3.1 2 0.08 0.14956 N 4.968,0.14956 P 5 0.467 (3 s.f) T X X X YY TX Y TX Y T T = ++ + + =  +  = × × + × × =   =  +  = × × + × × = >= 

Raffles Institution H2 Mathematics 2025 Year 6 _________________________________________________________________________________________ _____________________________________ Y6 H2 Math T2W6 Math Focus: Normal Distribution Page 3 of 6 2 EJC Prelim 9758/2021/02/Q10 (modified) In this question, you should state the parameters of any distributions that you use. A stationery factory manufactures pens for sale. The diameters (in mm) of the pens have distribution ( )N 10,0.003 . The pens are packed in sets of 24 into a box. Within the box, the pens are laid out side by side. The widths of the boxes (in mm) are normally distributed with mean 240.5 mm and variance 0.02 mm2. (i) Find the probability that a random sample of 24 pens would fit into a randomly selected box. [3] A pen is considered “defective” if its diameter is not within 0.1mm of the population mean. (ii) Find the probability that a randomly selected pen is defective. [2] Pens are produced and inspected in batches. For each batch, a sample of 12 pens is randomly selected and checked. • If there are fewer than 2 defective pens in this sample of 12, the batch passes the inspection. • If there are exactly 2 defective pens in this sample, a second sample of 12 pens is randomly selected and checked. If there are no defective pens in the second sample, the batch passes the inspection. • Otherwise, the batch does not pass the inspection. (iii) Show that the probability of a batch of pens passing the inspection is 0.871, correct to 3 significant figures. [3] (iv) Find the probability that not more than 3 defective pens were found in a batch of pens during the inspection process, given that the batch of pens did not pass the inspection. [3] (i) Let Y be the width (in mm) of a box. Let W be the width (in mm) of a pen. ( )~ 240.5,0.02YN ( )~ 10,0.003WN ( ) ( ) ( ) ( ) ( ) ( ) 1 2 24Let ... E 24 E E 24(10) 240.5 0.5 Var 24 Var Var 24(0.003) 0.02 0.092 CW W W Y C WY C WY =++ − =  − = − =− =  + = + = ( )~ 0.5,0.092CN − ( ) ( ) ( ) 1 2 24pens fit into box ... 0 0.95037 0.950 (to 3 sf) P PW W W Y PC = ++ < = < = =

Raffles Institution H2 Mathematics 2025 Year 6 _________________________________________________________________________________________ _____________________________________ Y6 H2 Math T2W6 Math Focus: Normal Distribution Page 4 of 6 (ii) Let W be the width (in mm) of a pen. ( )~ 10,0.003WN ( ) ( )pen is defective 1 9.9 10.1 1 0.93211 0.067889 0.0679 (to 3sf)P PW= − << = − = = (iii) Let X be the number of defective pens in a box of 12. ( )~ 12,0.067889XB ( ) ( ) ( ) ( ) ( )( ) batch passes inspection 1 20 0.806087 0.150598 0.430142 0.870865 0.871(to 3 sf) P PX PX PX= ≤+ = = = + = = (iv) ( ) ( ) ( ) ( ) ( ) ( )( ) 3 defective pens found batch did not pass inspection 21 3 batch did not pass inspection 0.150598 0.375945 0.036562 1 0.870865 0.0931785 0.722 (to 3 sf)0.129135 P PX PX PX P ≤ = = +== += − = =

Raffles Institution H2 Mathematics 2025 Year 6 _________________________________________________________________________________________ _____________________________________ Y6 H2 Math T2W6 Math Focus: Normal Distribution Page 5 of 6 3 HCI JC2 Prelim 9758/2019/02/Q7 A cafe sells sandwiches in 2 sizes, “footlong” and “6 -inch”. The lengths in inches of “footlong” loaves have the distribution ( )N 12.2,0.04 and the lengths in inches of “6-inch” loaves have the distribution ( )N 6.1,0.02 . (i) Is a randomly chosen “footlong” loaf more likely to be less than 12 inches in length or a randomly chosen “6-inch” loaf more likely to be less than 6 inches in length? [2] (ii) Find the probability that two randomly chosen “6- inch” loaves have total length more than one randomly chosen “footlong” loaf. [2] Sue buys a “6-inch” sandwich 3 times a week. (iii) Find the probability that Sue gets at most one sandwich that is less than 6 inches in length in a randomly chosen week. [2] (iv) Given that Sue gets more than four sandwiches that are less than 6 inches in length in a randomly chosen 4-week period, find the probability that she gets exactly one such sandwich in the first week. [3] (i) Let X and Y denote the length (in inches) of a “footlong” and a “6 -inch” loaf respectively. ( )~ N 12.2,0.04X , ( )~ N 6.1,0.02Y ( ) ( )P 6 0.23975 0.240 3sfY <≈ = ( ) ( ) ( )P 12 0.15866 0.159 3sf P 6XY<≈ = < < ∴ A “6-inch” loaf is more likely to be less than 6 inches. (ii) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 12 12 E 2E E 2 6.1 12.2 0 Var 2Var Var 2 0.02 0.04 0.08+ YY X Y X YY X Y X +− = − = − = +− = = + = ( )12 ~ N 0,0.08YY X+− ( ) ( )12 12P P 0 0.500 (3 s.f)YY X YY X+> = +−>= (iii) Let A be the number of “6 -inch” sandwiches less than 6 -inches in length , out of 3 bought in a week. ( )~ B 3,0.23975A ( ) ( )P 1 0.855 3 s.fA≤= (iv) Let C and D denote the number of “6-inch” sandwiches less than 6-inches in length in a 3-week and 4-week period respectively. ( )~ B 9,0.23975C and ( )~ B 12,0.23975D

Raffles Institution H2 Mathematics 2025 Year 6 _________________________________________________________________________________________ _____________________________________ Y6 H2 Math T2W6 Math Focus: Normal Distribution Page 6 of 6 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P 1P 3P14 P4 P 11P 3 1P 4 0.41571 0.14710 0.13721 0.446 3 s.f ACAD D AC D = >= >= > =−≤= −≤ ×≈ =