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Use linear regression to find the equation of the line that best fits the data. Predict the average retail price of a game system that is 7 years old in the spring of 1997. At what age is the game system worthless?

The regression line is:
[tex]\[ R = \square \][/tex]
(Round the coefficients to the nearest hundredth; use [tex]\( A \)[/tex] for the age and [tex]\( R \)[/tex] for the retail price.)

Average base retail price in the spring of 1997 for a game system:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Age (years)} & \text{Retail price (\$)} \\
\hline
1 & 474 \\
2 & 421 \\
3 & 325 \\
4 & 234 \\
5 & 188 \\
\hline
\end{array}
\][/tex]

Sagot :

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