Chapter 7.3, Problem 88E

Solution

a.

To state: A linear regression equation for the population of Anaheim if x is the number of years since 1970 and the below given table shows the population (in thousands) of Anaheim, California, and Anchorage, Alaska.

Year	Anaheim (thousand)	Anchorage (thousands)
1970	166	48
1980	219	174
1990	266	226
2000	328	260
2010	336	292

The resultant answer is $y\approx 4.49x+173.2$ .

Given information:

The given table is:

Year	Anaheim (thousand)	Anchorage (thousands)
1970	166	48
1980	219	174
1990	266	226
2000	328	260
2010	336	292

Explanation:

Consider the given table:

Year	Anaheim (thousand)	Anchorage (thousands)
1970	166	48
1980	219	174
1990	266	226
2000	328	260
2010	336	292

Use a graphing calculator; enter the time (years) starting at $x=0$ for 1970 in L1, Anaheim population in L2, and Anchorage’s population in L3.

  

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

  

Therefore, the resultant regression equation is: $y\approx 4.49x+173.2$

b.

To state: A linear regression equation for the population of Anchorage if x is the number of years since 1970 and the below given table shows the population (in thousands) of two fast-growing suburbs.

Year	Anaheim (thousand)	Anchorage (thousands)
1970	166	48
1980	219	174
1990	266	226
2000	328	260
2010	336	292

The answer is $y\approx 5.74x+85.2$ .

Given information:

The given table is:

Year	Anaheim (thousand)	Anchorage (thousands)
1970	166	48
1980	219	174
1990	266	226
2000	328	260
2010	336	292

Explanation:

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

  

Therefore, the resultant linear regression equation is: $y\approx 5.74x+85.2$

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 $y\approx 4.49x+173.2$ and $y\approx 5.74x+85.2$ .

Graph:

Consider the models $y\approx 4.49x+173.2$ and $y\approx 5.74x+85.2$ .

Now enter $y\approx 4.49x+173.2$ and $y\approx 5.74x+85.2$ 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 70 which corresponds to the year $1970+70=2040$ .

Therefore, the population of the two cities will be about the same in the year 2040.

Interpretation: Both the equations are linear equations and their intersection point is $\left(70.4,489.296\right)$ .