Question

European population growth Many populations grow exponentially. Here are the data for the estimated population of Europe (in millions) from $1700$to $2012$. The dates are recorded as years since $1700$so that $x=312$is the year $2012$.

$\begin{array}{|ll|}\hline \text{year since}1700& \text{population (in millions)}\\ 0& 125\\ 50& 163\\ 100& 203\\ 150& 276\\ 200& 408\\ 250& 729\\ 299& 732\\ 308& 740\\ 310& 729\\ 312& 73\\ \hline\end{array}$

a. Use a logarithm to transform population size. Then calculate and state the least-squares regression line using the transformed variable.

b. Use your model from part (a) to predict the population size of Europe in $2020$.

Short answer

a). The least-squares regression line using the transformed variable is $\log y=2.0796+0.0026x$.

b). The expected size of Europe in $2020$ is $815.831$ millions.

Step-by-step solution

Part (a) Step 1: Given Information

Given data:

$\begin{array}{|ll|}\hline \text{year since}1700& \text{population (in millions)}\\ 0& 125\\ 50& 163\\ 100& 203\\ 150& 276\\ 200& 408\\ 250& 729\\ 299& 732\\ 308& 740\\ 310& 729\\ 312& 73\\ \hline\end{array}$

Part (a) Step 2: Explanation

Log of given data:

$\begin{array}{|lll|}\hline \text{year since}1700& \text{population (in millions)}& \log \text{(population)}\\ 0& 125& 2.096910013\\ 50& 163& 2.212187604\\ 100& 203& 2.307496038\\ 150& 276& 2.440909082\\ 200& 408& 2.610660163\\ 250& 547& 2.737987326\\ 299& 729& 2.862727528\\ 308& 732& 2.864511081\\ 310& 738& 2.868056362\\ 312& 740& 2.86923172\\ \hline\end{array}$

Making use of a $Ti83/84$ calculator

Step 1: select STAT;

Step 2: select 1: EDIT.

Step 3: Type the data for each year since $1700$in list L1 and the population logarithmic in list L2.

Step 4: hit STAT once more, select CALC, and then $LinReg(a+bx).$

Part (a) Step 3: Explanation

The required result:

$y=a+bx$

$a=2.0796$

$b=0.0026$

Substituting the value in $a$and $b$

$y=2.0796+0.0026x$

The year since $1700$is represented by $x$, while the population logarithm is represented by $y$.

$\log y=2.0796+0.0026x$

Part (b) Step 1: Given Information

Given data:

$\begin{array}{|ll|}\hline \text{year since}1700& \text{population (in millions)}\\ 0& 125\\ 50& 163\\ 100& 203\\ 150& 276\\ 200& 408\\ 250& 729\\ 299& 732\\ 308& 740\\ 310& 729\\ 312& 73\\ \hline\end{array}$

Part (b) Step 2: Explanation

Substituting the value $x$by $320$:

$\log y=2.0796+0.0026x$

$\log y=2.0796+0.0026\left(320\right)$

$\log y=2.9116$

Using the exponential function with a base of ten

$y={10}^{\log y}$

$={10}^{2.9116}$

$=815.831$