To solve this question, we need to find the linear regression equation that best fits the given data points and then use it to predict the retail price of the game system for an age of 6 years. Additionally, we need to determine the age at which the retail price of the game system will be zero, implying it is worthless.
### Step 1: List the Given Data
We have the following data points for the age and retail price of the game system:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Age (years)} & \text{Retail Price (\$)} \\
\hline
1 & 496 \\
2 & 392 \\
3 & 342 \\
4 & 257 \\
5 & 166 \\
\hline
\end{array}
\][/tex]
### Step 2: Perform Linear Regression
To find the equation of the line that best fits the data, we need to perform a linear regression analysis. The general form of the linear regression equation is:
[tex]\[ R = mA + b \][/tex]
where:
- [tex]\( R \)[/tex] is the retail price,
- [tex]\( A \)[/tex] is the age,
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the y-intercept.
After performing the linear regression analysis, we have:
[tex]\[ m = -79.5 \][/tex]
[tex]\[ b = 569.1 \][/tex]
So, the equation of the regression line is:
[tex]\[ R = -79.5A + 569.1 \][/tex]
### Step 3: Predict the Retail Price for a 6-Year-Old Game System
We need to substitute [tex]\( A = 6 \)[/tex] into the regression equation to find the predicted retail price [tex]\( R \)[/tex].
[tex]\[
R = (-79.5 \times 6) + 569.1
\][/tex]
[tex]\[
R = -477 + 569.1
\][/tex]
[tex]\[
R = 92.1
\][/tex]
Thus, the predicted retail price for a game system that is 6 years old is [tex]\( \$92.10 \)[/tex].
### Step 4: Determine the Age at Which the Game System is Worthless
To find the age at which the game system becomes worthless, we set the retail price [tex]\( R \)[/tex] to 0 and solve for [tex]\( A \)[/tex].
[tex]\[
0 = -79.5A + 569.1
\][/tex]
Solving for [tex]\( A \)[/tex]:
[tex]\[
79.5A = 569.1
\][/tex]
[tex]\[
A = \frac{569.1}{79.5}
\][/tex]
[tex]\[
A \approx 7.16
\][/tex]
Therefore, the game system will become worthless (i.e., have a retail price of [tex]\( \$0 \)[/tex]) at approximately 7.16 years old.
### Summary
- The linear regression equation for the given data is:
[tex]\[
R = -79.5A + 569.1
\][/tex]
- The predicted retail price for a game system that is 6 years old is [tex]\( \$92.10 \)[/tex].
- The game system will be worthless at approximately 7.16 years of age.
