Suppose a doctor measures the height, x, and head circumference, y, of 8 children and obtains the data below. The correlation coefficient is 0.879 and the least squares regression line is y=0.216x +11.540 Complete parts (a) and (b) below Height, x 27.75 24.5 26.75 25.75 27.75 26 25 25.75 26.75 27 26.75 27 D Head Circumference, y 17.6 16.9 17.1 16.9 17.5 172 172 174 17.4 174 174 (a) Compute the coefficient of determination, R2 R²=% (Round to one decimal place as needed.). (b) Interpret the coefficient of determination and comment on the adequacy of the linear model Approximately % of the variation in is explained by the least-squares regression model. According to the residual plot, the linear model appears to be (Round to one decimal place as needed)
Suppose a doctor measures the height, x, and head circumference, y, of 8 children and obtains the data below. The correlation coefficient is 0.879 and the least squares regression line is y=0.216x +11.540 Complete parts (a) and (b) below Height, x 27.75 24.5 26.75 25.75 27.75 26 25 25.75 26.75 27 26.75 27 D Head Circumference, y 17.6 16.9 17.1 16.9 17.5 172 172 174 17.4 174 174 (a) Compute the coefficient of determination, R2 R²=% (Round to one decimal place as needed.). (b) Interpret the coefficient of determination and comment on the adequacy of the linear model Approximately % of the variation in is explained by the least-squares regression model. According to the residual plot, the linear model appears to be (Round to one decimal place as needed)
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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Chapter1: Starting With Matlab
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![### Regression and Correlation Analysis
**Example Problem:**
Suppose a doctor measures the height, \( x \), and head circumference, \( y \), of 8 children and obtains the data below. The correlation coefficient is 0.879 and the least squares regression line is \( \hat{y} = 0.216x + 11.540 \). Complete parts (a) and (b) below.
**Data:**
| Height (\( x \)) | Head Circumference (\( y \)) |
|------------------|------------------------------|
| 27.75 | 17.6 |
| 24.5 | 16.9 |
| 26.75 | 17.1 |
| 25.75 | 16.9 |
| 27.5 | 17.5 |
| 26.25 | 17.2 |
| 25.75 | 17.4 |
| 26.75 | 17.4 |
| 27 | 17.4 |
**Questions:**
(a) Compute the coefficient of determination, \( R^2 \).
\[ R^2 = \_\_\_% \] (Round to one decimal place as needed.)
(b) Interpret the coefficient of determination and comment on the adequacy of the linear model.
Approximately \(\_\_\_%\) of the variation in \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_) is explained by the least-squares regression model. According to the residual plot, the linear model appears to be \_\_\_\_\_\_\_\_\_\_\_\_\_\_. (Round to one decimal place as needed.)
**Explanation:**
1. **Coefficient of Determination, \( R^2 \):**
The \( R^2 \) value represents the proportion of the variance in the dependent variable (head circumference, \( y \)) that is predictable from the independent variable (height, \( x \)). It ranges from 0 to 1, where:
- 0 means the model does not explain any of the variability in the response data around its mean,
- 1 means the model explains all the variability in the response data around its mean.
2. **Interpretation:**
Percentages derived from \( R^2 \):
- The closer \( R^2 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd86b4514-aa18-4e84-8d65-86fc6c642f07%2F4360d574-256a-4675-a8e2-403dbe9daf05%2Frg93syg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Regression and Correlation Analysis
**Example Problem:**
Suppose a doctor measures the height, \( x \), and head circumference, \( y \), of 8 children and obtains the data below. The correlation coefficient is 0.879 and the least squares regression line is \( \hat{y} = 0.216x + 11.540 \). Complete parts (a) and (b) below.
**Data:**
| Height (\( x \)) | Head Circumference (\( y \)) |
|------------------|------------------------------|
| 27.75 | 17.6 |
| 24.5 | 16.9 |
| 26.75 | 17.1 |
| 25.75 | 16.9 |
| 27.5 | 17.5 |
| 26.25 | 17.2 |
| 25.75 | 17.4 |
| 26.75 | 17.4 |
| 27 | 17.4 |
**Questions:**
(a) Compute the coefficient of determination, \( R^2 \).
\[ R^2 = \_\_\_% \] (Round to one decimal place as needed.)
(b) Interpret the coefficient of determination and comment on the adequacy of the linear model.
Approximately \(\_\_\_%\) of the variation in \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_) is explained by the least-squares regression model. According to the residual plot, the linear model appears to be \_\_\_\_\_\_\_\_\_\_\_\_\_\_. (Round to one decimal place as needed.)
**Explanation:**
1. **Coefficient of Determination, \( R^2 \):**
The \( R^2 \) value represents the proportion of the variance in the dependent variable (head circumference, \( y \)) that is predictable from the independent variable (height, \( x \)). It ranges from 0 to 1, where:
- 0 means the model does not explain any of the variability in the response data around its mean,
- 1 means the model explains all the variability in the response data around its mean.
2. **Interpretation:**
Percentages derived from \( R^2 \):
- The closer \( R^2 \
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