Using your favorite statistics software package, you generate a scatter plot with a regression equation and correlation coefficient. The regression equation is reported as y=-11.36z+ 94.45 and the r=-0.729. What proportion of the variation in y can be explained by the variation in the values of x? r² = 186 Report answer as a percentage accurate to one decimal place.
Using your favorite statistics software package, you generate a scatter plot with a regression equation and correlation coefficient. The regression equation is reported as y=-11.36z+ 94.45 and the r=-0.729. What proportion of the variation in y can be explained by the variation in the values of x? r² = 186 Report answer as a percentage accurate to one decimal place.
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![### Understanding Regression Analysis in Statistics
**Question**
Using your favorite statistics software package, you generate a scatter plot with a regression equation and correlation coefficient. The regression equation is reported as:
\[ y = -11.36x + 94.45 \]
and the \( r = -0.729 \).
What proportion of the variation in \( y \) can be explained by the variation in the values of \( x \)?
\[ r^2 = \boxed{\hspace{30pt}} \% \]
Report answer as a percentage accurate to one decimal place.
**Explanation of Elements Involved:**
- **Regression Equation**: This equation describes the relationship between the dependent variable \( y \) and the independent variable \( x \). In this case, the coefficient \(-11.36\) indicates that for each unit increase in \( x \), \( y \) decreases by 11.36 units. The constant term \( 94.45 \) is the intercept, representing the value of \( y \) when \( x = 0 \).
- **Correlation Coefficient (\( r \))**: The value \(-0.729\) measures the strength and direction of the linear relationship between \( x \) and \( y \). A negative value indicates an inverse relationship, meaning as \( x \) increases, \( y \) typically decreases.
- **Proportion of Variation Explained (\( r^2 \))**: To find the proportion of the variation in \( y \) explained by \( x \), square the correlation coefficient and convert it to a percentage. \( r^2 = (-0.729)^2 = 0.531841 \), which means about 53.18% of the variation in \( y \) can be explained by the variation in \( x \).
Fill in the appropriate blank on your worksheet or online form:
\[ r^2 = \boxed{53.2} \% \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F402afc63-9cac-4d1f-a06a-b4853df74aca%2F6256e82c-2102-4fb4-8d9c-1c856b3a123e%2Fvmlw6e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Regression Analysis in Statistics
**Question**
Using your favorite statistics software package, you generate a scatter plot with a regression equation and correlation coefficient. The regression equation is reported as:
\[ y = -11.36x + 94.45 \]
and the \( r = -0.729 \).
What proportion of the variation in \( y \) can be explained by the variation in the values of \( x \)?
\[ r^2 = \boxed{\hspace{30pt}} \% \]
Report answer as a percentage accurate to one decimal place.
**Explanation of Elements Involved:**
- **Regression Equation**: This equation describes the relationship between the dependent variable \( y \) and the independent variable \( x \). In this case, the coefficient \(-11.36\) indicates that for each unit increase in \( x \), \( y \) decreases by 11.36 units. The constant term \( 94.45 \) is the intercept, representing the value of \( y \) when \( x = 0 \).
- **Correlation Coefficient (\( r \))**: The value \(-0.729\) measures the strength and direction of the linear relationship between \( x \) and \( y \). A negative value indicates an inverse relationship, meaning as \( x \) increases, \( y \) typically decreases.
- **Proportion of Variation Explained (\( r^2 \))**: To find the proportion of the variation in \( y \) explained by \( x \), square the correlation coefficient and convert it to a percentage. \( r^2 = (-0.729)^2 = 0.531841 \), which means about 53.18% of the variation in \( y \) can be explained by the variation in \( x \).
Fill in the appropriate blank on your worksheet or online form:
\[ r^2 = \boxed{53.2} \% \]
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