Chapter 12.2, Problem 45E

(a)

To determine

To Calculate: and state the least square regression line.

Answer to Problem 45E

  $\log y=2.0796+0.0026x$

Explanation of Solution

Given:

year since 1700	population (in millions)
0	125
50	163
100	203
150	276
200	408
250	547
299	729
308	732
310	738
312	740

Calculation:

Taking the log of the given data

year since 1700	population (in millions)	log (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

Using Ti83/84 calculator

Step 1: press on STAT,

Step 2: choose 1: EDIT

Step 3: type the data of year since 1700 in list L1 and in L2 logarithmic of population in the list L2.

Step 4: again press on STAT, chose CALC and chose LinReg(a+bx)

Final result

  $\begin{array}{l}y=a+bx\\ a=2.0796\\ b=0.0026\end{array}$

Putting the value in a and b

  $y=2.0796+0.0026x$

  $x$ is representing the year since 1700 and y is representing the logarithm of population

  $\log y=2.0796+0.0026x$

  $x$ Represents the years since 1700 and y represent the population

(b)

To determine

To Explain: the prediction the population size of Europe in 2020 using the part (a)

Answer to Problem 45E

815.831 millions

Explanation of Solution

Given:

year since 1700	population (in millions)
0	125
50	163
100	203
150	276
200	408
250	547
299	729
308	732
310	738
312	740

Calculation:

From the part (b)

  $\log y=2.0796+0.0026x$

Putting the value x by 320

  $\begin{array}{l}\log y=2.0796+0.0026x\\ \log y=2.0796+0.0026\left(320\right)\\ \log y=2.9116\end{array}$

Taking the exponential with base 10

  $\begin{array}{c}y={10}^{\log y}={10}^{2.9116}\\ =815.831\end{array}$

Therefore the expected size of Europe in 2020 is 815.831 millions