Q20. - Algebra 1

Question

The table shows the number of children from Russia adopted by U.S. citizens.

Years Since 2000	0	1	2	3	4
Number of children	4269	4279	4939	5209	6936

a. Write the slope-intercept form of the equation for the regression line.

b. Predict the number of children from Russia who will be adopted in 2025.

Step-by-Step Solution

Answer

a. The equation that fits the data in the table is $\mathit{Y}\mathbf{=}\mathbf{626}\mathbf{.4}\mathit{X}\mathbf{+}\mathbf{3873}\mathbf{.6}$.

b. The number of children from Russia who will be adopted in 2025 is 19533.

Part a. Step 1. Define the equation for the regression line .

The regression equation is given by $Y=bX+a$, 

where $b=\frac{SP}{SSX}$ and $a=\left(\text{Mean\hspace{0.17em}\hspace{0.17em}}Y\right)-b\left(\text{Mean\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}X\right)$.

Part a. Step 2. Calculate the equation of the regression line.

Let years since 2000 be represented by $X$ and the number of children is represented by $Y$, then the equation for the regression line can be calculated by performing the following calculations.

Sum of $X = 10$

Sum of $Y = 25632$

Mean $X = 2$

Mean $Y = 5126.4$

Sum of squares $\left(SSX\right) = 10$

Sum of products $\left(SP\right) = 6264$

Regression Equation =$Y=bX+a$

Here,

 $$\begin{gathered} b=\frac{SP}{SSX} \\ =\frac{6264}{10} \\ =\text{ 626.4} \end{gathered}$$

And, 

$$\begin{gathered} a=\left(Meany\right)-b\left(MeanX\right) \\ =5126.4-\left(626.4\times 2\right) \\ =3873.6 \end{gathered}$$

Substitute the values of $a$ and $b$.

The equation of the regression line is $Y=626.4X+3873.6$.

Part a. Step 3. Plot the regression line.

The plot for the regression line is:

 

Part b. Step 1. Define the equation for the regression line .

The regression equation is given by $Y=bX+a$, where $b=\frac{SP}{SSX}$ and $a=\left(\text{Mean\hspace{0.17em}\hspace{0.17em}}Y\right)-b\left(\text{Mean\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}X\right)$.

Part b. Step 2. Calculate the equation of the regression line.

Let years since 2000 be represented by $X$ and the number of children is represented by $Y$, then the equation for the regression line can be calculated by performing the following calculations.

Sum of $X = 10$

Sum of $Y = 25632$

Mean $X = 2$

Mean $Y = 5126.4$

Sum of squares $\left(SSX\right) = 10$

Sum of products $\left(SP\right) = 6264$

Regression Equation = $Y=bX+a$

Here,

$$\begin{gathered} b=\frac{SP}{SSX} \\ =\frac{6264}{10} \\ =\text{ 626.4} \end{gathered}$$

And, 

$$\begin{gathered} a=\left(Meany\right)-b\left(MeanX\right) \\ =5126.4-\left(626.4\times 2\right) \\ =3873.6 \end{gathered}$$

Substitute the values of $a$ and $b$.

The equation of the regression line is $Y=626.4X+3873.6$.

Part b. Step 3. Calculate the number of children from Russia who will be adopted in 2025.

In the equation above, substitute 25 for $X$.

 $$\begin{gathered} Y=626.4\times 25+3873.6 \\ =19533.6 \end{gathered}$$

Therefore, a total of 19533.6 children will be adopted in the year 2025.