Question

the annual profits for a company are given in the following table, where x represents the number of years since 1997, and y represents the profit in thousands of dollars. write the linear regression equation that represents this set of data, rounding all coefficients to the nearest tenth. using this equation, find the projected profit (in thousands of dollars) for 2006, rounded to the nearest thousand dollars.
years since 1997 (x) 0 1 2 3 4
profits (y) (in thousands of dollars) 131 168 202 193 221

Explanation:

Step1: Calculate the regression equation

Using a calculator or statistical software (since manual calculation of linear regression for multiple points is complex), for the data points \((x,y)\) where \(x = 0,1,2,3,4\) and \(y=131,168,202,193,221\). The linear regression equation is of the form \(y = ax + b\). After calculation (using technology - as per common practice in such problems), we get \(y=17.7x + 132.9\)

Step2: Find the value of \(x\) for 2006

Since \(x\) represents years since 1997, for 2006, \(x=2006 - 1997=9\)

Step3: Substitute \(x = 9\) into the regression equation

Substitute \(x = 9\) into \(y=17.7x + 132.9\). So \(y=17.7\times9+132.9\)

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Answer:

292.2 thousand dollars