We appreciate your visit to Run a regression analysis on the following bivariate set of data with tex y tex as the response variable tex begin array c c hline. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
Sure! Let's walk through solving this problem step-by-step.
### 1. Find the Correlation Coefficient, [tex]\( r \)[/tex]
The correlation coefficient [tex]\( r \)[/tex] measures the strength and direction of a linear relationship between two variables. A value close to 1 indicates a strong positive correlation.
According to the calculations:
- The correlation coefficient [tex]\( r \)[/tex] is approximately [tex]\( 0.908 \)[/tex].
### 2. Calculate the Proportion of Variation Explained, [tex]\( R^2 \)[/tex]
The coefficient of determination, [tex]\( R^2 \)[/tex], explains the proportion of the variance in the response variable that can be predicted from the explanatory variable.
[tex]\[ R^2 = r^2 \][/tex]
[tex]\[ R^2 = (0.908)^2 \][/tex]
[tex]\[ R^2 \approx 0.824 \][/tex]
To convert [tex]\( R^2 \)[/tex] into a percentage, multiply by 100:
[tex]\[ R^2 \approx 82.4\% \][/tex]
### 3. Determine the Least-Squares Regression Line
The equation of the least-squares regression line is given as:
[tex]\[ \hat{y} = mx + c \][/tex]
Where:
- [tex]\( m \)[/tex] is the slope of the regression line.
- [tex]\( c \)[/tex] is the y-intercept of the regression line.
Given:
- Slope [tex]\( m \)[/tex] is approximately [tex]\( 1.963 \)[/tex].
- Intercept [tex]\( c \)[/tex] is approximately [tex]\( -115.577 \)[/tex].
Thus, the regression line is:
[tex]\[ \hat{y} = 1.963x - 115.577 \][/tex]
### 4. Predict the Response Variable for [tex]\( x = 66.9 \)[/tex]
To find the predicted value of [tex]\( y \)[/tex] for a given [tex]\( x = 66.9 \)[/tex], substitute [tex]\( x \)[/tex] into the regression equation:
[tex]\[ \hat{y} = 1.963(66.9) - 115.577 \][/tex]
Calculating this gives:
[tex]\[ \hat{y} \approx 15.8 \][/tex]
So, the predicted value for [tex]\( y \)[/tex] when [tex]\( x = 66.9 \)[/tex] is approximately [tex]\( 15.8 \)[/tex].
### Summary
- Correlation coefficient, [tex]\( r \)[/tex]: 0.908
- Proportion of the variation explained, [tex]\( R^2 \)[/tex]: 82.4%
- Least-squares regression line: [tex]\( \hat{y} = 1.963x - 115.577 \)[/tex]
- Predicted [tex]\( y \)[/tex] value for [tex]\( x = 66.9 \)[/tex]: 15.8
### 1. Find the Correlation Coefficient, [tex]\( r \)[/tex]
The correlation coefficient [tex]\( r \)[/tex] measures the strength and direction of a linear relationship between two variables. A value close to 1 indicates a strong positive correlation.
According to the calculations:
- The correlation coefficient [tex]\( r \)[/tex] is approximately [tex]\( 0.908 \)[/tex].
### 2. Calculate the Proportion of Variation Explained, [tex]\( R^2 \)[/tex]
The coefficient of determination, [tex]\( R^2 \)[/tex], explains the proportion of the variance in the response variable that can be predicted from the explanatory variable.
[tex]\[ R^2 = r^2 \][/tex]
[tex]\[ R^2 = (0.908)^2 \][/tex]
[tex]\[ R^2 \approx 0.824 \][/tex]
To convert [tex]\( R^2 \)[/tex] into a percentage, multiply by 100:
[tex]\[ R^2 \approx 82.4\% \][/tex]
### 3. Determine the Least-Squares Regression Line
The equation of the least-squares regression line is given as:
[tex]\[ \hat{y} = mx + c \][/tex]
Where:
- [tex]\( m \)[/tex] is the slope of the regression line.
- [tex]\( c \)[/tex] is the y-intercept of the regression line.
Given:
- Slope [tex]\( m \)[/tex] is approximately [tex]\( 1.963 \)[/tex].
- Intercept [tex]\( c \)[/tex] is approximately [tex]\( -115.577 \)[/tex].
Thus, the regression line is:
[tex]\[ \hat{y} = 1.963x - 115.577 \][/tex]
### 4. Predict the Response Variable for [tex]\( x = 66.9 \)[/tex]
To find the predicted value of [tex]\( y \)[/tex] for a given [tex]\( x = 66.9 \)[/tex], substitute [tex]\( x \)[/tex] into the regression equation:
[tex]\[ \hat{y} = 1.963(66.9) - 115.577 \][/tex]
Calculating this gives:
[tex]\[ \hat{y} \approx 15.8 \][/tex]
So, the predicted value for [tex]\( y \)[/tex] when [tex]\( x = 66.9 \)[/tex] is approximately [tex]\( 15.8 \)[/tex].
### Summary
- Correlation coefficient, [tex]\( r \)[/tex]: 0.908
- Proportion of the variation explained, [tex]\( R^2 \)[/tex]: 82.4%
- Least-squares regression line: [tex]\( \hat{y} = 1.963x - 115.577 \)[/tex]
- Predicted [tex]\( y \)[/tex] value for [tex]\( x = 66.9 \)[/tex]: 15.8
Thanks for taking the time to read Run a regression analysis on the following bivariate set of data with tex y tex as the response variable tex begin array c c hline. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
