Question

T12.10We record data on the population of a particular country from 1960 to 2010. A
scatterplot reveals a clear curved relationship between population and year. However, a different scatterplot reveals a strong linear relationship between the logarithm (base 10) of the population and the year. The least-squares regression line for the transformed data is
$log\widehat{(population)=}-13.5+0.01\left(year\right)$
Based on this equation, which of the following is the best estimate for the population of the country in the year 2020?
a. 6.7
b. 812
c. 5,000,000
d. 6,700,000
e. 8,120,000

Short answer

The correct answer is option (c) $5,000,000$.

Step-by-step solution

Given information

To determine the best estimate for the population of the country in the year $2020$.

Explanation

A scatterplot shows that the population and year have a clear curving relationship. A separate scatterplot, on the other hand, displays a significant linear link between the population logarithm and the year.
The following is the linear regression line:
$\ln (\tilde{\mathrm{P}}\text{opulation)}=-13.5+0.01\text{(Year)}$
By $2020$, the year will be replaced, and it will be:
$\ln \left(\tilde{\mathrm{P}}\text{opulation}\right)=-13.5+0.01\text{(Year)}$
$\Rightarrow \ln \left(\text{Population}\right)=-13.5+0.01\left(2020\right)$
$=6.7$
Take each side's exponential:
$\text{Population}={10}^{6.7}$
$=5011872$
$\approx 5000000$
As a result, option (c)$5000000$ is the correct option.