To find the equation of the line that best fits the given data using linear regression, we need to determine the linear relationship between the age of the game system ([tex]\(A\)[/tex]) and its retail price ([tex]\(R\)[/tex]).
Given data:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Age} (\text{years}) & \text{Retail Price} (\$) \\
\hline
1 & 474 \\
2 & 421 \\
3 & 325 \\
4 & 234 \\
5 & 188 \\
\hline
\end{array}
\][/tex]
First, perform linear regression on the given data to find the slope ([tex]\(m\)[/tex]) and the intercept ([tex]\(b\)[/tex]) of the best fitting line. The result from the linear regression analysis gives us:
[tex]\[
\text{Intercept } (b) = 556.1
\][/tex]
[tex]\[
\text{Slope } (m) = -75.9
\][/tex]
Therefore, the equation of the regression line can be written as:
[tex]\[
R = -75.9A + 556.1
\][/tex]
Next, we need to predict the average retail price of a game system that is 7 years old.
For [tex]\( A = 7 \)[/tex]:
[tex]\[
R = -75.9 \cdot 7 + 556.1
\][/tex]
[tex]\[
R = -531.3 + 556.1
\][/tex]
[tex]\[
R = 24.8
\][/tex]
Thus, the predicted average retail price of a game system that is 7 years old is \[tex]$24.8.
Finally, we need to determine at what age the game system becomes worthless (i.e., when \( R = 0 \)).
Set \( R = 0 \) in the regression equation:
\[
0 = -75.9A + 556.1
\]
Solve for \( A \):
\[
75.9A = 556.1
\]
\[
A = \frac{556.1}{75.9}
\]
\[
A = 7.33
\]
Therefore, the game system becomes worthless at approximately 7.33 years old.
To summarize:
- The regression line equation is: \( R = -75.9A + 556.1 \)
- The predicted average retail price of a game system that is 7 years old is \$[/tex]24.8
- The game system becomes worthless at an age of 7.33 years
