Answer :
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
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
