1. Corrosion of still reinforcing bars is the most important durability problem for reinforced concrete structures. Carbonation of concrete results from a chemical reaction that lowers the pH value enough to to initiate corrosion. The table below shows a sample dependence of the strength in MPa (y) on the carbonation depth in mm (x) taken from a particular building. 8.0 15.0 16.5 20.0 20.0 27.5 30.0 30.0 35.0 22.8 27.2 23.7 17.1 21.5 18.6 16.1 23.4 13.4 38.0 40.0 45.0 50.0 50.0 55.0 55.0 59.0 65.0 19.5 12.4 13.2 11.4 10.3 14.1 9.7 12.0 6.8 a) Fit a linear regression model to the data. What is the estimated expected value of the strength if the carbonation depth is 25.0 mm? b) Do the data suggest that the regression is significant? (Use the t-test). Assume that a = 0.01. What is the corresponding P-value? c) Use the analysis of variance approach to test the significance of regression. Find the P-value. Compare with problem b). d) Suppose we test one more sample with carbonation depth of 25.0 mm and find the strength to be 12.2 MPa? Is it consistent with our model (using 95% confidence level)? Hint: find the PI for a new observation and see whether 12.2 belongs to that PI. e) Find a 95% confidence interval on the expected value of the strength if the carbonation depth is equal to 25.0 mm. What is the corresponding interval if the depth of carbonation is 35.0 mm? In which case are we more uncertain about the expected strength? f) Assess the adequacy of the model by analyzing the residuals: do a normal plot, residual histogram, and appropriate residual plots. What are your conclusions? Are there any outliers? g) Test for the adequacy of the model using the lack-of-fit test. Set the significance level at 0.1. Is the linear model adequate?
1. Corrosion of still reinforcing bars is the most important durability problem for reinforced concrete structures. Carbonation of concrete results from a chemical reaction that lowers the pH value enough to to initiate corrosion. The table below shows a sample dependence of the strength in MPa (y) on the carbonation depth in mm (x) taken from a particular building. 8.0 15.0 16.5 20.0 20.0 27.5 30.0 30.0 35.0 22.8 27.2 23.7 17.1 21.5 18.6 16.1 23.4 13.4 38.0 40.0 45.0 50.0 50.0 55.0 55.0 59.0 65.0 19.5 12.4 13.2 11.4 10.3 14.1 9.7 12.0 6.8 a) Fit a linear regression model to the data. What is the estimated expected value of the strength if the carbonation depth is 25.0 mm? b) Do the data suggest that the regression is significant? (Use the t-test). Assume that a = 0.01. What is the corresponding P-value? c) Use the analysis of variance approach to test the significance of regression. Find the P-value. Compare with problem b). d) Suppose we test one more sample with carbonation depth of 25.0 mm and find the strength to be 12.2 MPa? Is it consistent with our model (using 95% confidence level)? Hint: find the PI for a new observation and see whether 12.2 belongs to that PI. e) Find a 95% confidence interval on the expected value of the strength if the carbonation depth is equal to 25.0 mm. What is the corresponding interval if the depth of carbonation is 35.0 mm? In which case are we more uncertain about the expected strength? f) Assess the adequacy of the model by analyzing the residuals: do a normal plot, residual histogram, and appropriate residual plots. What are your conclusions? Are there any outliers? g) Test for the adequacy of the model using the lack-of-fit test. Set the significance level at 0.1. Is the linear model adequate?
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Author:Amos Gilat
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Chapter1: Starting With Matlab
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Transcribed Image Text:### Corrosion in Steel Reinforcement: An Analysis of Concrete Carbonation
**Problem Statement:**
Corrosion of steel reinforcing bars is a critical issue affecting the durability of reinforced concrete structures. Carbonation of concrete, a chemical reaction, reduces the pH value enough to initiate corrosion. The table below presents data showing the strength in MPa (\(y\)) relative to the carbonation depth in mm (\(x\)), gathered from a particular building.
#### Data Table
| \( x \) (mm) | 8.0 | 15.0 | 16.5 | 20.0 | 20.0 | 27.5 | 30.0 | 30.0 | 35.0 |
|--------------|-----|------|------|------|------|------|------|------|------|
| \( y \) (MPa)| 22.8 | 27.2 | 23.7 | 17.1 | 21.5 | 18.6 | 16.1 | 23.4 | 13.4 |
| \( x \) (mm) | 38.0 | 40.0 | 45.0 | 50.0 | 50.0 | 55.0 | 55.0 | 59.0 | 65.0 |
|--------------|-----|------|------|------|------|------|------|------|------|
| \( y \) (MPa)| 19.5 | 12.4 | 13.2 | 11.4 | 10.3 | 14.1 | 9.7 | 12.0 | 6.8 |
**Questions:**
a) **Fit a Linear Regression Model**:
- Fit a linear regression model to the data.
- Estimate the expected strength value if the carbonation depth is \(25.0 \text{ mm}\).
b) **Test Regression Significance**:
- Use the t-test to determine if the regression is significant.
- Assume the significance level \(\alpha = 0.01\).
- Report the corresponding P-value.
c) **Analyze Variance**:
- Utilize the analysis of variance (ANOVA) to test the significance of the regression.
- Compare the P-value with that found in part b.
d) **Prediction Interval**:
- Test a new sample
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