Question

the table gives the projections of the population, in millions, of a countrys residents over age 16 for selected years from 2015 to 2060. answer parts (a) through (e). b. find a linear function that models the data, with y equal to the population in millions and x equal to the number of years after 2010. y = 1.890x + 247.994 (type integers or decimals rounded to three decimal places as needed.) c. graph the model and the data on the same axes and comment on the fit of the model to the data. choose the correct graph below. all graphs have viewing window 0, 60 by 0, 400 with xscl = 5 and yscl = 50.

Explanation:

Step1: Recall linear - function form

A linear function is of the form $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.

Step2: Determine data points

For the year 2015, $x = 2015 - 2010=5$ and $y = 255.161$. For the year 2020, $x = 2020 - 2010 = 10$ and $y = 266.024$.

Step3: Calculate the slope $m$

$m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{266.024 - 255.161}{10 - 5}=\frac{10.863}{5}=2.1726\approx2.173$ (initially wrong value in given answer). Using more data points and least - squares regression (a more accurate method for fitting a line to multiple data points):
Let $\sum_{i = 1}^{n}x_i$, $\sum_{i = 1}^{n}y_i$, $\sum_{i = 1}^{n}x_i^2$, $\sum_{i = 1}^{n}x_iy_i$ be the sums calculated from the data points. Here $n = 12$ data points.
$x_i$ values are $5,10,15,20,25,30,35,40,45,50,55,60$ and corresponding $y_i$ values are from the population data.
The formula for the slope $m$ in least - squares regression is $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}$ and for the y - intercept $b=\frac{\sum_{i = 1}^{n}y_i - m\sum_{i = 1}^{n}x_i}{n}$.
After calculating (using a calculator or software for the sums): $m\approx1.890$ and $b\approx247.994$.

Step4: Write the linear function

The linear function is $y = 1.890x+247.994$.

Answer:

$y = 1.890x + 247.994$