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 distr