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Answer :
Let's solve the problem step-by-step.
### Part (a): Find the equation of the least-squares line
To find the equation of the least-squares line, we first need to determine the slope [tex]\( m \)[/tex] and the intercept [tex]\( b \)[/tex] of the line. These are found using linear regression techniques based on the provided data of odometer readings and corresponding retail values.
The general form of the least-squares line equation is:
[tex]\[ \hat{y} = mx + b \][/tex]
Based on the data provided, the slope [tex]\( m \)[/tex] is approximately [tex]\(-0.1768\)[/tex], and the intercept [tex]\( b \)[/tex] is approximately [tex]\(54790.3205\)[/tex].
Thus, the equation of the least-squares line is:
[tex]\[ \hat{y} = -0.1768x + 54790.3205 \][/tex]
### Part (b): Predict the retail price for a 2020 Corvette with an odometer reading of 28,000 miles
We can use the equation from part (a) to predict the retail value when the odometer reads 28,000 miles. Substitute [tex]\( x = 28000 \)[/tex] into the equation:
[tex]\[ \hat{y} = -0.1768 \times 28000 + 54790.3205 \][/tex]
This calculation results in a retail value of approximately [tex]\(49800\)[/tex] dollars when rounded to the nearest $100.
### Part (c): Find the linear correlation coefficient
The linear correlation coefficient, denoted as [tex]\( r \)[/tex], measures the strength and direction of a linear relationship between two variables. For this data, the linear correlation coefficient is approximately [tex]\(-0.9987\)[/tex].
### Part (d): Significance of the negative linear correlation coefficient
A negative linear correlation coefficient means that there is an inverse relationship between the two variables. As the odometer reading increases, the retail value decreases. This negative correlation indicates that higher mileage on the car leads to a reduced retail price.
In conclusion, the least-squares equation, retail value prediction for an odometer reading of 28,000, and the linear correlation coefficient provide insights into how the car's mileage affects its value.
### Part (a): Find the equation of the least-squares line
To find the equation of the least-squares line, we first need to determine the slope [tex]\( m \)[/tex] and the intercept [tex]\( b \)[/tex] of the line. These are found using linear regression techniques based on the provided data of odometer readings and corresponding retail values.
The general form of the least-squares line equation is:
[tex]\[ \hat{y} = mx + b \][/tex]
Based on the data provided, the slope [tex]\( m \)[/tex] is approximately [tex]\(-0.1768\)[/tex], and the intercept [tex]\( b \)[/tex] is approximately [tex]\(54790.3205\)[/tex].
Thus, the equation of the least-squares line is:
[tex]\[ \hat{y} = -0.1768x + 54790.3205 \][/tex]
### Part (b): Predict the retail price for a 2020 Corvette with an odometer reading of 28,000 miles
We can use the equation from part (a) to predict the retail value when the odometer reads 28,000 miles. Substitute [tex]\( x = 28000 \)[/tex] into the equation:
[tex]\[ \hat{y} = -0.1768 \times 28000 + 54790.3205 \][/tex]
This calculation results in a retail value of approximately [tex]\(49800\)[/tex] dollars when rounded to the nearest $100.
### Part (c): Find the linear correlation coefficient
The linear correlation coefficient, denoted as [tex]\( r \)[/tex], measures the strength and direction of a linear relationship between two variables. For this data, the linear correlation coefficient is approximately [tex]\(-0.9987\)[/tex].
### Part (d): Significance of the negative linear correlation coefficient
A negative linear correlation coefficient means that there is an inverse relationship between the two variables. As the odometer reading increases, the retail value decreases. This negative correlation indicates that higher mileage on the car leads to a reduced retail price.
In conclusion, the least-squares equation, retail value prediction for an odometer reading of 28,000, and the linear correlation coefficient provide insights into how the car's mileage affects its value.
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