RI_8865_2022_Prelim_Stats Solutions_vetted

H1 Mathematics 2022 Year 6 Preliminary Examination (Solution) Page 1 of 7 Question 6 No. Solution (i) Let  and  denote the mean and standard deviation of X. P (X  0) = 0.06  P  Z  0 −   = 0.06 where ( )~ 0,1ZN  0 −   = −1.55477   − 1.55477 = 0 --- (1) P (X  15) = 0.029  P  Z  15 −   = 0.029 where ( )~ 0,1ZN  15 −   = 1.89570   + 1.89570 = 15 --- (2) Solving (1) and (2),  = 6.76 and  = 4.35 (3 sf ) (ii) 6.76 8 −8

H1 Mathematics 2022 Year 6 Preliminary Examination (Solution) Page 2 of 7 Question 7 No. Solution (i) P (AB) = 1 6  P (A  B) P (B) = 1 6  P (B) = 6 k P ( A  B ) = P ( A ) + P ( B ) − P ( A  B )  9 20 = 1 5 + 6 k − k  k = 1 20 (ii) P (A'  B) = P (A  B) − P ( A ) = 9 20 − 1 5 = 1 4 (iii) P (A' )  P ( B ) = 4 5  6 k = 6 25 Since P (A'  B)  P (A' )  P ( B ), the events A' and B are not independent. Question 8 No. Solution (i) Let X denote the number of defective lamps in a sample of 30 lamps X ~ B (30, 0.05) The expected number of defective lamps is E(X ) = 30  0.05 = 1.5 (ii) P (X  1) = 1 − P (X = 0) = 0.785 (3 sf ) (iii) Required probability = P (X  2X  1) = ( ) ( ) P 1 2 P1 X X   = P (X = 1) + P (X = 2) P (X  1) = 0.59754 0.78536 = 0.761 (3 sf )

H1 Mathematics 2022 Year 6 Preliminary Examination (Solution) Page 3 of 7 Question 9 No. Solution (i) (ii) From GC, r = 0.945 (3 sf ) r is close to 1. There is a strong positive linear correlation between the number of years, x, of experience of the prosecutors and the percentage, y, of their cases that ended in guilty pleas. As the number of years of experience of the prosecutors increases, the percentage of their cases that ended in guilty pleas increases. (iii) From GC, y = 1.77 x + 70.4 (3 sf ) The percentage of a prosecutor’s cases that ended in guilty pleas is expected to increase by 1.77 % when the number of years of experience of the prosecutor increases by 1. (iv) From GC, when x = 20, y = 106 (3 sf ) An estimate for the percentage of cases that ended in guilty pleas for a prosecutor with 20 years of experience is 106 %. The estimate is not reliable because (1) x = 20 is outside the data range of 1  x  15 and (2) the estimated percentage of 106 does not make sense since it exceeds 100 %. x y 15 1 70 95

H1 Mathematics 2022 Year 6 Preliminary Examination (Solution) Page 4 of 7 Question 10 No. Solution (i) Let S (in grams) denote the mass of fries in a small packet. S ~ N (80, 2 2) P (S  77) = 0.066807 Required probability = [ P (S  77) ] 2 = 0.00446 (3 sf )