7 Manual Work a) 0 for all values of x Therefore, it is a proper probability distribution f (x) S f (x) = 1 b) Probability x = 30 is f (30) = .25 c) Probability x 25 is f (20) + f (25) = .20 + .15 = .35 d) Probability x > 30 is f (35) = .40 S Excel Function x f(x) 20 25 30 35 b) c) d) PROB(B17:B20,C17:C20,30) PROB(B17:B20,C17:C20,20,25) PROB(B17:B20,C17:C20,35) 0.2 0.15 0.25 0.4 0.25 0.35 0.4 Q13 Manual Work a) b) c) Yes, the function is valid because f(x) are all non-negative and f (2) = 2/6 = .333 f (2) + f (3) = 2/6 + 3/6 = .833 S f (x) = f (1) + f (2) + f (3) = 1/6 + 2/6 + 3/6 = 1 Excel Function a) b) c) 1/6+2/6+3/6 2/6 2/6+3/6 1 0.333 0.833 Q14 Manual Work a) f (200) = 1 - f (-100) - f (0) - f (50) - f (100) - f (150) = 1 - 0.95 = 0.05 This is the probability MRA will have a $200,000 profit which is 0.05. This is least probable b) P(Profit) = f (50) + f (100) + f (150) + f (200) = 0.3 + 0.25 + 0.1 + 0.05 = 0.7 c) P(at least 100) = f (100) + f (150) + f (200) = .25 + .10 +.05 = .40 x f(x) -100 0 50 100 150 200 Excel Function a) PROB(B17:B22,C17:C22,200) b) PROB(B17:B22,C17:C22,50,200) c) PROB(B17:B22,C17:C22,100,200) 0.1 0.2 0.3 0.25 0.1 0.05 0.05 0.7 0.4 0.02 Q15 Manual x 3 6 9 f (x) 0.25 0.5 0.25 1 x f (x) 3*0.25=0.75 6*0.5=3 9*0.25=2.25 0.75+3+2.25=6 x x-m (x - m)2 f (x) (x - m)2 f (x) 3 6 9 -3 0 3 9 0 9 0.25 0.5 0.25 2.25 0 2.25 4.5 f (x) 0.25 0.5 0.25 1 x f (x) 0.75 3 2.25 6 E(x) = m = 6 Var(x) = s2 = 4.5 s = 4.50 = 2.12 Excel Function x 3 6 9 a) Expected Value = 6 SUM(D27:D29) x c) (x - m)2 f (x) (x - m)2 f (x) 3 6 9 b) x-m -3 0 3 9 0 9 0.25 0.5 0.25 2.25 0 2.25 4.5 Variance= 4.5 2.12 SQRT(D43) SUMPRODUCT(D37:D39,E37:E39) Q15 Manual a) # of times # of students 1 721,769 2 601,325 3 166,736 4 22,299 5 6,730 Probability Distribution p(x) 721769/1518859= 601325/1518860= 166736/1518861= 22299/1518862= 6730/1518863= Total # of students= 0.4752 0.3959 0.1098 0.0147 0.0044 0.475205 0.395906 0.109777 0.014681 0.004431 1,518,859 b) P(x > 1) = 1 - f(1) = 1 - .4752 = .5248 Over 50% of the students take the SAT more than 1 time. c) P(x > 3) = f(3) + f(4) + f(5) = .1098 + .0147 + .0044 = .1289 x-m x f (x) x f (x) 1 2 3 4 5 0.4752 0.3959 0.1098 0.0147 0.0044 0.4752 0.7918 0.3294 0.0588 0.022 -0.6772 0.3228 1.3228 2.3228 3.3228 (x - m)2 0.4586 0.1042 1.7498 5.3954 11.041 0.4752+0.7918+0.3294+0.05 88+0.022=1.6772 c) (x - m)2 f (x) 0.2179 0.0412 0.1921 0.0792 0.0489 0.5794 Expected Value = 1.6772 The mean number of times a student takes the SAT is 1.6772, or approximately 1.7 times d) Variance 0.5792 SUMPRODUCT(F21:F25,C21:C25) e) Standard Deviation 0.7611 SQRT(C31) Excel Function a) # of times # of students Probability Distribution p(x) 1 721,769 0.4752 2 601,325 0.3959 3 166,736 0.1098 4 22,299 0.0147 5 6,730 0.0044 Total # of students= b) 0.5248 0.4752 0.3959 0.1098 0.0147 0.0044 1,518,859 PROB(B38:B42,D38:D42,2,5) Over 50% of the students take the SAT more than 1 time. c) 0.1289 PROB(B38:B42,D38:D42,3,5) x f (x) x f (x) 1 2 3 4 5 0.4752 0.3959 0.1098 0.0147 0.0044 0.4752 0.7918 0.3294 0.0588 0.022 1.67720 x-m -0.6772 0.3228 1.3228 2.3228 3.3228 c) Expected Value = 1.6772 d) Variance 0.5792 SUMPRODUCT(F21:F25,C21:C25) e) Standard Deviation 0.7611 SQRT(C31) (x - m)2 0.4586 0.1042 1.7498 5.3954 11.041 SUM(D21:D25) The mean number of times a student takes the SAT is 1.6772, or approximately 1.7 times (x - m)2 f (x) 0.2179 0.0412 0.1921 0.0792 0.0489 0.5794 Q22 Manual Unit Demand Probability 300 400 500 600 Expected Value 0.2 0.3 0.35 0.15 300 (.20) + 400 (.30) + 500 (.35) + 600 (.15) = 445 Cost Revenue 445 @ $50 300 @ $70 Excel Function Unit Demand =22250 =21000 $22,250 Loss of $1,250 Probability 300 400 500 600 0.2 0.3 0.35 0.15 60 120 175 90 445 Expected Value(Monthly Order)= 445 Actual Demand Per unit revenue Revenue Generated $ $ $ 300.00 70.00 21,000.00 =C28*C29 Actual purchased Actual cost Total Cost $ $ $ 445.00 50.00 22,250.00 =C32*C33 $ (1,250.00) Profit/Loss Generated Each Month SUM(D18:D21) Company has a loss of $1,250 each month