3.) You consult for a small artificial intelligence company. They claim that the
ID: 3012493 • Letter: 3
Question
3.) You consult for a small artificial intelligence company. They claim that their product won't be marketable for another two years but it is currently a hot item for investors. Investment capital is being acquired at a rate of $40,000/month, yet meanwhile cost for company employees, office space, etc. is $90,000/month. If after two years investment stops, and revenue from their product is $478,000/month, when will they break-even?
4.) Find the line of best fit to the data set {(5,6), (-9,1), (1,2)}, then find the correlation coefficient. Is is justifiable to fit a line to these points? If the points were exactly on a line, what value(s) of the correlation coefficient would you expect to find?
Show work please.
Explanation / Answer
Solution for Q3
For the first two years, net cost per month = cost per month - investment per month = 90000 - 40000 = 50000. Hence accumulated cost for the first two years = 50000 x 24 = 1200000
From the 25th month, revenue is more than the cost and hence the net revenue per month = 478000 - 90000 = 388000. This surplus must offset the accumulated cost of first 24 months. i.e., 1200000 must be offset at the rate of 388000 per month. Hence, number of months required to offset the accumulated cost = 1200000/388000 = 3.09.
ANSWER: The break-even is 3 monhts after revenue generation.
Solution for Q4
First of all, it is not justifiable to use 3 sets of points to extimate best line of fit.
However, it is being worked out here just demonstrate the method.
Denoting the first value in each set as x and second value as y,
Step 1: Mean x =(5 + - 9 + 1)/3 = - 1; Mean y = (6 + 1 + 2)/3 = 3
Step 2: Deviation from mean: for x: {(5 - - 1) + (- 9 - -1) + (1 - - 1)} = 6, - 8, 2
Step 3: Deviation from mean: for y: {(6 - 3) + (1 - 3) + (2 - 3)} = 3, - 2, - 1
Step 4 Squared Deviation from mean: for x: 62 + 82 + 22 = 36 + 64 + 4 = 104 = Sx
Step 5: Squared Deviation from mean: for y: 32 + (- 2)2 + (- 1)2 = 9 + 4 + 1 = 14 = Sy
Step 6: Cross-product of Deviation from mean: for x and Deviation from mean: for y = (6 x 3) + (- 8 x - 2) + (2 x - 1) = 18 + 16 + - 2 = 32 = Sxy
Step 7: Regression coefficient, b = Sxy/Sx = 32/104 = 0.3076 ~ 0.31, constant, a = Mean y - (b x Mean x) = 3 - (0.31 x - 1) = 3 + 0.31 = 3.31
Step 8: Line of best fit is: y = a + bx i.e., y = 3.31 + 0.31x ANSWER
Cprrelation coefficient, r = Sxy/sq.rt of(Sx x Sy) = 32/sq.rt(104 x 14) = 32/sq.rt(1456) = 32/38.16 = 0.8385 ~ 0.84
ANSWER Correlation coefficient = 0.84
ANSWER for the last part: if all the points lie on the straight line, the correlation coefficient is +1 or -1. In this, as x decreases y also decreases, r must be + 1.
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