Question

The following table gives projections of the U.S. population from 2000 to 2100.
a) Find a linear function that models the data (the equation of best fit) with x equal to the number of years after 2000 and f(x) equal to the population in millions.

Find f(65) and state what it means.
Find the slope of the line. What does it mean in the context of the problem?

Answer 1

The equation of best fit is $f(X) = 2.91964X + 265.86364$.

$f(65) = 455.64024$. It means that the population, in millions, after $65$ years from $2000$ is $455.64024$.

The slope of the line is $2.91964$ which means population is increasing at the rate of $2.91964$ per year.

Sum of $X = 550$

Sum of $Y = 4530.3$

Mean $X = 50$

Mean $Y = 411.8455$

Sum of squares (SSX) $= 11000$

Sum of products (SP) $= 32116$

Regression Equation $= y = bX + a$

$b = \frac{SP}{SSX} = \frac{32116}{11000} = 2.91964$

$a = MY - bMX = 411.85 - (2.92*50) = 265.86364$

$y = 2.91964X + 265.86364$

$f(X) = 2.91964X + 265.86364$

$f(65) = 2.91964\times 65 + 265.86364$

$f(65) = 189.7766 + 265.86364$

$f(65) = 189.7766 + 265.86364$

$f(65) = 455.64024$

It means that the population, in millions, after $65$ years from $2000$ is $455.64024$.

From the equation, $f(X) = 2.91964X + 265.86364$

the slope of the line is $2.91964$ which means population is increasing at the rate of $2.91964$ per year.

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