Chapter 8.9, Problem 7PPS

(a)

To determine

To calculate: To make a scatter plot and draw a line of fit and to find what does the slope of each line represents

Answer to Problem 7PPS

Thescatter plot and line of best fit is drawn for State A and State B

Explanation of Solution

Given information: Table showing the population of two states for several years

Year	Population (millions)
State A	State B
1	12.4	12.3
2	12.5	12.3
3	12.6	12.3
4	12.7	12.4
5	12.7	12.4
6	12.8	12.4

Formula Used:

Line of Best fit is a straight line that best represents the data on scatter plot. Line of fit must pass through some of the points, none of the points or all the points

Calculation:

Data is given as

Year	Population (millions)
State A	State B
1	12.4	12.3
2	12.5	12.3
3	12.6	12.3
4	12.7	12.4
5	12.7	12.4
6	12.8	12.4

Scatter plot and line of best fit for State A is

  

Scatter plot and line of best fit for State B is

  

Slope of each line represents the rate of increase in population

Conclusion:

Hence, scatter plot and line of best fit is drawn for State A and State B

(b)

To determine

To calculate: To find which state’s population appear to be growing at faster rate. If two lines intersect, what does the point of intersection represent

Answer to Problem 7PPS

State A’s population appear to be growing at faster rate

Explanation of Solution

Given information: Table showing the population of two states for several years

Year	Population (millions)
State A	State B
1	12.4	12.3
2	12.5	12.3
3	12.6	12.3
4	12.7	12.4
5	12.7	12.4
6	12.8	12.4

Formula Used:

Line of Best fit is a straight line that best represents the data on scatter plot. Line of fit must pass through some of the points, none of the points or all the points

Calculation:

Data is given as

Year	Population (millions)
State A	State B
1	12.4	12.3
2	12.5	12.3
3	12.6	12.3
4	12.7	12.4
5	12.7	12.4
6	12.8	12.4

From the above data, population of state A is growing at faster rate.

The point of intersection of the two lines represent the year when the population of State A and State B was same.

Slope of each line represents the rate of increase in population

Conclusion:

Hence, State A’s population appear to be growing at faster rate

(c)

To determine

To calculate: To write an equation for each line of fit

Answer to Problem 7PPS

Equation of line of fit for State A is $y=0.08x+12.32$

Equation of line of fit for State B is $y=0.02x+12.28$

Explanation of Solution

Given information: Table showing the population of two states for several years

Year	Population (millions)
State A	State B
1	12.4	12.3
2	12.5	12.3
3	12.6	12.3
4	12.7	12.4
5	12.7	12.4
6	12.8	12.4

Formula Used:

Equation of line passing through two points is given as

  $y-y_2=\frac{y_2-y_1}{x_2-x_1}\left(x-x_2\right)$

Slope intercept of line is given as

  $y=mx+c$

Where,

  $m=$ Slope

  $c=$ y-intercept

Calculation:

Data is given as

Year	Population (millions)
State A	State B
1	12.4	12.3
2	12.5	12.3
3	12.6	12.3
4	12.7	12.4
5	12.7	12.4
6	12.8	12.4

Consider State A

Point on line are as follows:

  $\left(1,12.4\right)$ and $\left(6,12.8\right)$

Equation of line passing through above points is

  $\begin{array}{l}y-12.4=\frac{12.8-12.4}{6-1}\left(x-1\right)\\ y-12.4=\frac{0.4}{5}\left(x-1\right)\\ y-12.4=0.08\left(x-1\right)\\ y-12.4=0.08x-0.08\\ y=0.08x+12.32\end{array}$

Consider State B

Point on line are as follows:

  $\left(1,12.3\right)$ and $\left(6,12.4\right)$

Equation of line passing through above points is

  $\begin{array}{l}y-12.3=\frac{12.4-12.3}{6-1}\left(x-1\right)\\ y-12.3=\frac{0.1}{5}\left(x-1\right)\\ y-12.3=0.02\left(x-1\right)\\ y-12.3=0.02x-0.02\\ y=0.02x+12.28\end{array}$

Conclusion:

Hence,

Equation of line of fit for State A is $y=0.08x+12.32$

Equation of line of fit for State B is $y=0.02x+12.28$

(d)

To determine

To calculate: To estimate the population of both states in year 16

Answer to Problem 7PPS

Population of State A in year 16 is $13.44$ millions

Population of State B in year 16 is $12.6$ millions

Explanation of Solution

Given information: Table showing the population of two states for several years

Year	Population (millions)
State A	State B
1	12.4	12.3
2	12.5	12.3
3	12.6	12.3
4	12.7	12.4
5	12.7	12.4
6	12.8	12.4

Formula Used:

Equation of line passing through two points is given as

  $y-y_2=\frac{y_2-y_1}{x_2-x_1}\left(x-x_2\right)$

Slope intercept of line is given as

  $y=mx+c$

Where,

  $m=$ Slope

  $c=$ y-intercept

Calculation:

Equation of line of fit for State A is given as

  $y=0.08x+12.32$

In order to find population of State A in year 16, substitute $x=16$ in above equation:

  $\begin{array}{l}y=0.08\left(16\right)+12.32\\ y=1.12+12.32\\ y=13.44\end{array}$

Equation of line of fit for State B is given as follows:

  $y=0.02x+12.28$

In order to find population of State B in year 16, substitute $x=16$ in above equation:

  $\begin{array}{l}y=0.02\left(16\right)+12.28\\ y=0.32+12.28\\ y=12.6\end{array}$

Conclusion:

Hence,

Population of State A in year 16 is $13.44$ millions

Population of State B in year 16 is $12.6$ millions