Chapter 12.2, Problem 46E

(a)

To determine

To Calculate: and state the least square regression line.

Answer to Problem 46E

  $\log y=0.1565+0.0079x$

Explanation of Solution

Given:

year since 1700	population (in millions)
0	2
50	2
100	7
150	26
200	82
250	172
299	307
308	337
310	345
312	351

Calculation:

Taking the log of the given data

year since 1700	population (in millions)	log (population)
0	2	0.301029996
50	2	0.301029996
100	7	0.84509804
150	26	1.414973348
200	82	1.913813852
250	172	2.235528447
299	307	2.487138375
308	337	2.527629901
310	345	2.537819095
312	351	2.545307116

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

  $\begin{array}{l}y=a+bx\\ a=0.1565\\ b=0.0079\end{array}$

Putting the value in a and b

  $y=0.1565+0.0079x$

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

  $\log y=0.1565+0.0079x$

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

(b)

To determine

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

Answer to Problem 46E

483.615 millions

Explanation of Solution

Given:

year since 1700	population (in millions)
0	2
50	2
100	7
150	26
200	82
250	172
299	307
308	337
310	345
312	351

Calculation:

From the part (b)

  $\log y=0.1565+0.0079x$

Putting the value x by 320

  $\begin{array}{c}\log y=0.1565+0.0079x\\ \log y=0.1565+0.0079\left(320\right)\\ \log y=2.6845\end{array}$

Taking the exponential with base 10

  $\begin{array}{c}y={10}^{\mathrm{l}2.6845}={10}^{2.6845}\\ =483.615\end{array}$

Therefore the expected size of North America in 2020 is 483.615 millions.