QCE Vault / Mathematical Methods Discrete random variables — Question 367 Original QCE Vault practice · 6 marks
Browse all questions Original practice exam Report an issue Q367 · Practice question Technology-active Complex unfamiliar 6 marks
QUESTION 367 (6 marks) A supplier compares two independent-trial inspection plans. Plan A approves a batch if at least four of five sampled items pass; each item passes with probability 0.7 0.7 0.7 . Plan B approves a batch only if all four sampled items pass; each item passes with probability 0.8 0.8 0.8 . A manager claims that the approval probability for Plan A exceeds that for Plan B by at least 0.10 0.10 0.10 . Evaluate the reasonableness of the claim. a) Support your evaluation with calculations.
[6 marks] WORKED SOLUTION
Practice marking scheme 6 marks ANSWER The claim is supported: P A = 0.52822 P_A=0.52822 P A = 0.52822 , P B = 0.4096 P_B=0.4096 P B = 0.4096 and P A − P B = 0.11862 > 0.10 P_A-P_B=0.11862>0.10 P A − P B = 0.11862 > 0.10 . Worked solution
For Plan A, X A ∼ Bin ( 5 , 0.7 ) X_A\sim\operatorname{Bin}(5,0.7) X A ∼ Bin ( 5 , 0.7 ) , so P A = P ( X A = 4 ) + P ( X A = 5 ) = 5 ( 0.7 ) 4 ( 0.3 ) + ( 0.7 ) 5 = 0.52822 P_A=P(X_A=4)+P(X_A=5)=5(0.7)^4(0.3)+(0.7)^5=0.52822 P A = P ( X A = 4 ) + P ( X A = 5 ) = 5 ( 0.7 ) 4 ( 0.3 ) + ( 0.7 ) 5 = 0.52822 . For Plan B, P B = ( 0.8 ) 4 = 0.4096 P_B=(0.8)^4=0.4096 P B = ( 0.8 ) 4 = 0.4096 . Their difference is 0.11862 0.11862 0.11862 , exceeding the claimed threshold, so the claim is reasonable under the stated models. Exact answers unless rounding is specified; equivalent forms accepted. Retain unrounded intermediate values.
Identifies the Plan A binomial model.
[1 mark] Identifies both qualifying outcomes for A.
[1 mark] Calculates the approval probability for A.
[1 mark] Calculates the approval probability for B.
[1 mark] Calculates the probability difference.
[1 mark] Evaluates the claim against 0.10.
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