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Question

1) You are estimating the cost ($K) of optical sensors based on the power output of the sensor. Using the preliminary calculations from a data set of 8 sensors, determine the equation of the line. (Round your intermediate calculations to 3 decimal places) ΣY = 2575 ΣX=680 ΣXY=241400 ΣX2=62600 Cost = -77.005 + 4.693 (Power) Cost = 3.786 + 6.763 (Power) Cost = 6.763 + 3.786 (Power) Cost = 4.693 + (-77.005) (Power)

Answer 1

It seems we're trying to calculate the linear regression equation y=mx+b.

The given summations enable us to calculate what we need to find m and b. And we're given that the dataset has 8 pairs of (X,Y) elements. We use this number to find mean values. ∑X/8 = Xm the mean or average of the X's. ∑Y/8 = Ym the mean of the Y's. ∑X^2/8 = Xms the mean of the squares of X. ∑XY/8 = XYm the mean of the products of X and Y.

m=(XmYm-XYm)/((Xm)^2-Xms). Xm=680/8=85; Ym=2575/8=321.875; XYm=241400/8=30175; Xms=62600/8=7825. So m=(85*321.875-30175)/(7225-7825)=4.693 approx.

b=Ym-mXm=321.875-4.693*85=-77.03; y=4.693x-77.03 is the linear regression equation, in which y is the cost and x is the power.

We can write this Cost=4.693*Power-77.03 so that it's clear what the variables are. From this it seems the first answer in the list of possible answers is the right one: Cost=-77.005+4.693(Power).  (The small discrepancy in the constant is due to rounding. A more accurate estimate of m is 4.6927. This value gives b=-77.0045.)