Answer :
Certainly! Let's work through the problem step-by-step.
### Part a: Determine the correlation coefficient and linear regression equation
For this part, the provided information gives us the correlation coefficient and the linear regression equation:
- Correlation Coefficient (r): This is a measure of how well the data fits a linear relationship, denoted as [tex]\( r = 0.987 \)[/tex]. This value, close to 1, indicates a strong positive linear relationship between square footage and monthly rent.
- Linear Regression Equation: This equation is used to estimate the square footage ([tex]\( y \)[/tex]) based on a given monthly rent ([tex]\( x \)[/tex]). The equation is:
[tex]\[
y = 0.56x - 149.07
\][/tex]
Here, [tex]\( 0.56 \)[/tex] is the slope of the line, indicating how much the square footage is expected to increase for each additional dollar of rent. The intercept is [tex]\(-149.07\)[/tex], which is the estimated square footage when the rent is \[tex]$0 (though this isn't practically meaningful in this context, it's part of the mathematical model).
### Part b: Estimate the square footage for a \$[/tex]1,710 apartment
To determine the square footage you might expect when the monthly rent is \$1,710, you use the linear regression equation. Substitute [tex]\( x = 1710 \)[/tex] into the equation:
[tex]\[
y = 0.56 \times 1710 - 149.07
\][/tex]
First, calculate the term [tex]\( 0.56 \times 1710 \)[/tex]:
[tex]\[
0.56 \times 1710 = 957.6
\][/tex]
Next, subtract 149.07 from 957.6 to find the estimated square footage:
[tex]\[
957.6 - 149.07 = 808.53
\][/tex]
Therefore, you can expect the square footage of the apartment to be approximately 809 square feet, rounding to the nearest whole number for practical purposes.
I hope this helps you understand how to use the correlation coefficient and linear regression equation to make predictions about square footage based on monthly rent! If you have any more questions, feel free to ask.
### Part a: Determine the correlation coefficient and linear regression equation
For this part, the provided information gives us the correlation coefficient and the linear regression equation:
- Correlation Coefficient (r): This is a measure of how well the data fits a linear relationship, denoted as [tex]\( r = 0.987 \)[/tex]. This value, close to 1, indicates a strong positive linear relationship between square footage and monthly rent.
- Linear Regression Equation: This equation is used to estimate the square footage ([tex]\( y \)[/tex]) based on a given monthly rent ([tex]\( x \)[/tex]). The equation is:
[tex]\[
y = 0.56x - 149.07
\][/tex]
Here, [tex]\( 0.56 \)[/tex] is the slope of the line, indicating how much the square footage is expected to increase for each additional dollar of rent. The intercept is [tex]\(-149.07\)[/tex], which is the estimated square footage when the rent is \[tex]$0 (though this isn't practically meaningful in this context, it's part of the mathematical model).
### Part b: Estimate the square footage for a \$[/tex]1,710 apartment
To determine the square footage you might expect when the monthly rent is \$1,710, you use the linear regression equation. Substitute [tex]\( x = 1710 \)[/tex] into the equation:
[tex]\[
y = 0.56 \times 1710 - 149.07
\][/tex]
First, calculate the term [tex]\( 0.56 \times 1710 \)[/tex]:
[tex]\[
0.56 \times 1710 = 957.6
\][/tex]
Next, subtract 149.07 from 957.6 to find the estimated square footage:
[tex]\[
957.6 - 149.07 = 808.53
\][/tex]
Therefore, you can expect the square footage of the apartment to be approximately 809 square feet, rounding to the nearest whole number for practical purposes.
I hope this helps you understand how to use the correlation coefficient and linear regression equation to make predictions about square footage based on monthly rent! If you have any more questions, feel free to ask.
