Answer :
To assess the relationship between sodium intake and systolic blood pressure, a linear regression analysis was performed. Below is a detailed breakdown of the summary output and ANOVA results provided by the software.
### Regression Analysis Summary
1. Multiple R (Correlation Coefficient): 0.9425
- This value represents the strength and direction of the linear relationship between sodium intake and systolic blood pressure. A value of 0.9425 suggests a very strong positive correlation.
2. R Square (Coefficient of Determination): 0.8882
- This indicates that approximately 88.82% of the variability in systolic blood pressure can be explained by the variability in sodium intake. This is a very high proportion, indicating a strong predictive power of the model.
3. Adjusted R Square: 0.8770
- The adjusted R square adjusts the R square value for the number of predictors in the model, providing a more accurate measure when multiple predictors are used. Since we only have one predictor, it is close to the R square.
4. Standard Error: 0.1129
- This measures the typical distance that the observed values fall from the regression line. A lower standard error indicates that the model's predictions are close to the actual data points.
5. Observations: 12
- This is the number of data points used in the analysis. Here, we have data for 12 individuals.
### ANOVA (Analysis of Variance)
The ANOVA section helps us understand the overall significance of the regression model.
1. Regression
- SS (Sum of Squares): 1.01256
- MS (Mean Square): 1.01256
- F-Statistic: 79.457
- Significance F: 5E-06
- The F-statistic with its corresponding p-value (5E-06) is highly significant, indicating that the model provides a better fit than a model with no predictors.
2. Residual
- SS: 0.12744 (implied from the total SS and regression SS)
- The residual sum of squares measures the variation that is not explained by the model.
### Coefficients and Significance
The table of coefficients provides the values required to describe the regression line, namely the intercept and the slope (coefficient) for sodium intake.
1. Intercept: 4.1356
- Standard Error: 0.3230
- t Stat: 12.8040
- P-value: 0.0000
- 95% Confidence Interval: [3.4159, 4.8553]
- The intercept is highly significant with a p-value of 0 (which is less than any common level of significance like 0.05). This means the intercept is statistically different from 0.
2. Sodium BP Relationship (Coefficient): 0.0169
- Standard Error: 0.0019
- t Stat: 8.9139
- P-value: 0.0000
- 95% Confidence Interval: [0.0127, 0.0212]
- The coefficient for sodium is also statistically significant with a p-value of 0, suggesting that for every unit increase in sodium intake, systolic blood pressure increases on average by 0.0169 units.
### Conclusion
The regression model developed is highly significant with a very strong positive correlation between sodium intake and systolic blood pressure. Nearly 88.82% of the variability in blood pressure can be explained by sodium intake. Both the intercept and the slope are statistically significant, reinforcing the reliability of the model for predicting systolic blood pressure based on sodium intake.
### Regression Analysis Summary
1. Multiple R (Correlation Coefficient): 0.9425
- This value represents the strength and direction of the linear relationship between sodium intake and systolic blood pressure. A value of 0.9425 suggests a very strong positive correlation.
2. R Square (Coefficient of Determination): 0.8882
- This indicates that approximately 88.82% of the variability in systolic blood pressure can be explained by the variability in sodium intake. This is a very high proportion, indicating a strong predictive power of the model.
3. Adjusted R Square: 0.8770
- The adjusted R square adjusts the R square value for the number of predictors in the model, providing a more accurate measure when multiple predictors are used. Since we only have one predictor, it is close to the R square.
4. Standard Error: 0.1129
- This measures the typical distance that the observed values fall from the regression line. A lower standard error indicates that the model's predictions are close to the actual data points.
5. Observations: 12
- This is the number of data points used in the analysis. Here, we have data for 12 individuals.
### ANOVA (Analysis of Variance)
The ANOVA section helps us understand the overall significance of the regression model.
1. Regression
- SS (Sum of Squares): 1.01256
- MS (Mean Square): 1.01256
- F-Statistic: 79.457
- Significance F: 5E-06
- The F-statistic with its corresponding p-value (5E-06) is highly significant, indicating that the model provides a better fit than a model with no predictors.
2. Residual
- SS: 0.12744 (implied from the total SS and regression SS)
- The residual sum of squares measures the variation that is not explained by the model.
### Coefficients and Significance
The table of coefficients provides the values required to describe the regression line, namely the intercept and the slope (coefficient) for sodium intake.
1. Intercept: 4.1356
- Standard Error: 0.3230
- t Stat: 12.8040
- P-value: 0.0000
- 95% Confidence Interval: [3.4159, 4.8553]
- The intercept is highly significant with a p-value of 0 (which is less than any common level of significance like 0.05). This means the intercept is statistically different from 0.
2. Sodium BP Relationship (Coefficient): 0.0169
- Standard Error: 0.0019
- t Stat: 8.9139
- P-value: 0.0000
- 95% Confidence Interval: [0.0127, 0.0212]
- The coefficient for sodium is also statistically significant with a p-value of 0, suggesting that for every unit increase in sodium intake, systolic blood pressure increases on average by 0.0169 units.
### Conclusion
The regression model developed is highly significant with a very strong positive correlation between sodium intake and systolic blood pressure. Nearly 88.82% of the variability in blood pressure can be explained by sodium intake. Both the intercept and the slope are statistically significant, reinforcing the reliability of the model for predicting systolic blood pressure based on sodium intake.
