Question

1. over the years 1950 - 2020, the total world population can be modeled by a linear function. selected values for the total world population p, in billions, are given in the table above, where t represents the number of years since 1950.

a) use the regression capabilities on your calculator to find a linear model of the form y = a + bx for the world population (in billions) x years since 1950.

b) according to the model found in part a, what was the world population in 1979, the year mr. passwater was born?

c) what is the residual of the total world population for the year 1990? did our model underestimate or overestimate the total world population for the year 1990?

Explanation:

Step1: Find the value of \(x\) for 1990

Since \(x\) represents the number of years since 1950, for 1990, \(x = 1990 - 1950=40\).

Step2: Substitute \(x = 40\) into the model \(y=a + bx\)

Given \(y = 2.5+0.16x\), substitute \(x = 40\):
\(y=2.5+0.16\times40\)
\(y=2.5 + 6.4\)
\(y = 8.9\) (This is the predicted value from the model)

Step3: Find the actual value for 1990

Looking at the table, when \(x = 40\), the actual population \(y_{actual}=5.32\)

Step4: Calculate the residual

The formula for the residual \(e=y_{actual}-y_{predicted}\)
\(e = 5.32-8.9=- 3.58\)

Answer:

The residual is \(-3.58\)