Chapter 7.3, Problem 87E

Solution

a.

To state: A linear regression equation for the population of garland if x is the number of years since 1980 and the below given table shows the population (in thousands) of two fast-growing suburbs.

Year	Garland (thousand)	Irving (thousands)
1980	139	110
1990	181	155
2000	216	192
2010	227	216

The resultant answer is $y\approx 2.99x+145.9$ .

Given information:

The given table is:

Year	Garland (thousand)	Irving (thousands)
1980	139	110
1990	181	155
2000	216	192
2010	227	216

Explanation:

Consider the given table:

Year	Garland (thousand)	Irving (thousands)
1980	139	110
1990	181	155
2000	216	192
2010	227	216

Use a graphing calculator; enter the time (years) starting at $x=0$ for 1980 in L1, Garland population in L2, and Irving’s population in L3.

  

Use LinReg feature (with L1 and L2) to obtain the linear regression model for the population of Garland:

  

Therefore, the resultant regression equation is: $y\approx 2.99x+145.9$

b.

To state: A linear regression equation for the population of Irving if x is the number of years since 1980 and the below given table shows the population (in thousands) of two fast-growing suburbs.

Year	Garland (thousand)	Irving (thousands)
1980	139	110
1990	181	155
2000	216	192
2010	227	216

The answer is $y\approx 3.55x+115$ .

Given information:

The given table is:

Year	Garland (thousand)	Irving (thousands)
1980	139	110
1990	181	155
2000	216	192
2010	227	216

Explanation:

Continue from part (a), use LinReg feature (with L1 and L3) to obtain the linear regression model for the population of Irving.

  

Therefore, the resultant linear regression equation is: $y\approx 3.55x+115$

c.

To graph: The models from part (a) and Part (b) and use these models to determine when the population of the two cities will be about the same.

Given information:

The models from part (a) and part (b) are $y\approx 2.99x+145.9$ and $y\approx 3.55x+115$ .

Graph:

Consider the models $y\approx 2.99x+145.9$ and $y\approx 3.55x+115$ .

Now enter $y\approx 2.99x+145.9$ and $y\approx 3.55x+115$ to graph the equations from part (a) and part (b).

And use the intersect feature to find the point of intersection:

  

The x -coordinate of the point of intersection is about 55 which corresponds to the year $1980+55=2035$ .

Therefore, the population of the two cities will be about the same in the year 2035.

Interpretation: Both the equations are linear equations and their intersection point is $\left(55.178,310.88\right)$ .