Question

please answer question #2

2. John Smith constructed a market analysis that showed that the probability that the demand will be "good" is 50%, the probability that it will be "fair" is 30% and the probability that it will be "poor" is 20%. On that basis of this new information, by using a decision tree, answer the following questions a. What is the expected value of each alternative? Which alternative should Smith choose from this point of view? b. What is the risk factor (o) of each alternative? Which alternative has minimum risk? c. Smith needs to make at least $20K to make a living. What is the probability of this event occurring for each alternative? (Use the Z-table) d. How about making at least $40K? (Use the Z-Table) e. Which alternative is the optimal one from the point of view of the coefficient of variation (cov)?

Answer 1

Solution: (a) Obtain the expected value. From the given information, it is clear that the probability value for good is 0.5, for fair is 0.3, and poor is 0.2. Therefore, the expected value is derived as follows: Good: # = E(x) = xP(x) =(0.5x50)+(0.5 X80)+(0.5x100)+(0.5x300) = 25+40+50+150 = 265 Thus, the expected value for the good demand is $265. Fair: =E(x) = xP(x) =(0.3x20)+(0.3x30)+(0.3x30)+(0.3x25) = 6+9+9+7.5 = 31.5 Thus, the expected value for the fair demand is $31.5.

Poor: i=E(x) = I xP(x) =(0.2x-10)+(0.2x-20)+(0.2x-40)+(0.2x-160) =(-2)+(-4)+(-8)+(-32) =-46 Thus, the expected value for the poor demand is $ - 46. From the obtained results, the alternative, good demand is higher. Thus, Smith should choose good alternative from this point of view. (b) Obtain the standard deviation. Therefore, the standard deviation value is derived as follows: Good: o=vE(x-E(x)) = {(x-*P(x)) = V167675 = 330 Thus, the standard deviation value for the good demand is $330).

Fair: 0= E(x-E(x)) = V(2(x-2xP(x))) = V179 = 13.38 Thus, the standard deviation value for the fair demand is $13.38). Poor: o= E(x-E(x)) = {(x- _ xP(x))) = V15004 = 122.49 Thus, the standard deviation value for the poor demand is $122.49. From the obtained results, the alternative, fair demand is lower. Thus, the alternative, Fair has minimal risk.

Good: From the given information, it is clear that the value for mean is 265 and the standard deviation value is 330. Therefore, the probability value is calculated as follows: DX-u 20- 20–265 = P(22785) =P(Z 2-1.49) =1-P(Z <-1.49) =1-0.0681 = 0.9319 Thus, the probability of this event occurring for good, alternative is 0.9319. Fair: From the given information, it is clear that the value for mean is 31.5 and the standard deviation value is 13.38. Therefore, the probability value is calculated as follows: DX-u 20- = P(Z 2-1.72) =1-P(Z <-1.72) =1-0.0427 = 0.9573 Thus, the probability of this event occurring for fair, alternative is 0.9573

Poor: From the given information, it is clear that the value for mean is -46 and the standard deviation value is 122.49. Therefore, the probability value is calculated as follows: X -2 =P(Z21.08) =1-P(Z<1.08) =1-0.8599 = 0.1401 Thus, the probability of this event occurring for poor, alternative is 0.1401. (d) Good: From the given information, it is clear that the value for mean is 265 and the standard deviation value is 330. Therefore, the probability value is calculated as follows: u 40-265 pl - Plz2-223 14-165) 14) =P(Z2-1.36) =1-P(Z <-1.36) =1-0.0869 = 0.9131 Thus, the probability of this event occurring for good, alternative is 0.9131

Fair: From the given information, it is clear that the value for mean is 31.5 and the standard deviation value is 13.38. Therefore, the probability value is calculated as follows: = P(Z 1.27) =1-P(Z <1.27) =1-0.8980 = 0.1020 Thus, the probability of this event occurring for fair, alternative is 0.1020. Poor: From the given information, it is clear that the value for mean is -46 and the standard deviation value is 122.49. Therefore, the probability value is calculated as follows: X-u 40-(-46) =plz 86) 4 = P(Z21.40) =1-P(Z <1.40) =1-0.9192 =0.0808 Thus, the probability of this event occurring for poor, alternative is 0.0808

(e) Obtain the Coefficient of variation. Good: From the given information, it is clear that the value for mean is 265 and the standard deviation value is 330. Therefore, the C.V. value is calculated as follows: CV = x100 330 = 265x100 = 124.53% Thus, the coefficient of variation for good, alternative is 124.53% Fair: From the given information, it is clear that the value for mean is 31.5 and the standard deviation value is 13.38. Therefore, the C.V. value is calculated as follows: CV = 3x100 13.38 x100 31.5 = 42.48% Thus, the coefficient of variation for Fair, alternative is 42.48%.

Poor: From the given information, it is clear that the value for mean is-46 and the standard deviation value is 122.49. Therefore, the C.V. value is calculated as follows: CV = x100 122.49 -X100 46 = 266.28% Thus, the coefficient of variation for Poor, alternative is 266.28%. From the obtained results, the alternative, Fair demand is vey lesser. Thus, the alterative, Fair is the optimal one form the point of view of the coefficient of variation.