Freight car loadings over a 12-year period at a busy port are as follows: Week Number Week Number Week Number 1 220 7 350 13 460 2 245 8 360 14 475 3 280 9 400 15 500 4 275 10 380 16 510 5 300 11 420 17 525 6 310 12 450 18 541 a. Determine a linear trend line for expected freight car loadings. b. Use the trend equation to predict expected loadings for weeks 20 and 21. c. The manager intends to install new equipment when the volume exceeds 800 loadings per week. Assuming the current trend continues, the loading volume will reach that level in approximately what week?

a)
Linear trend y = a+bx
a = intercept
b = slope
x = time period
y = forecast for demand for period x
b = (sum xy – nbar{x}bar{y})/(sum x^{2}-nbar{x}^{2})
a=bar{y}-bbar{x}
n = no of peiods
bar{x}=sum x/n = the mean of the x values
bar{y}=sum y/n = the mean of the y values
https://d2vlcm61l7u1fs.cloudfront.net/media%2Feaf%2Feafc365c-e796-4469-8a46-b7f682c3b780%2FphpOwiJOj.png
ope b = (75713-(18*388.94*9.5))/(2109-(18*9.5*9.5)) = 19
Intercept a = 388.94 – (9.5*19) = 208.44
Y = 208.44 + 19x
b)
Forecast for week 20 and 21
Y (x= 20) = 208.44 + 19* 20 = 588.44
Y (x= 21) = 208.44 + 19* 21 = 607.44
c)
Y = 800
800 = 208.44 + 19*x
x = 31.13 weeks
 
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