91Ó°ÊÓ

An article in Concrete Research ["Near Surface Characteristics of Concrete: Intrinsic Permeability" (Vol. 41 , 1989 ) ] presented data on compressive strength \(x\) and intrinsic permeability \(y\) of various concrete mixes and cures. Summary quantities are \(n=14, \sum y_{i}=572, \Sigma y_{i}^{2}=23,530, \sum x_{i}=43, \sum x_{i}^{2}=\) \(157.42,\) and \(\sum x_{i} y_{i}=1697.80 .\) Assume that the two variables are related according to the simple linear regression model. (a) Calculate the least squares estimates of the slope and intercept. Estimate \(\sigma^{2}\). Graph the regression line. (b) Use the equation of the fitted line to predict what permeability would be observed when the compressive strength is \(x=4.3\) (c) Give a point estimate of the mean permeability when compressive strength is \(x=3.7\) (d) Suppose that the observed value of permeability at \(x=\) 3.7 is \(y=46.1 .\) Calculate the value of the corresponding residual.

Step by step solution

01

Calculate Means

First, we calculate the means of both compressive strength and intrinsic permeability. The mean of compressive strength, \( \bar{x} \), is given by \( \bar{x} = \frac{\sum x_i}{n} = \frac{43}{14} \approx 3.071 \). The mean of intrinsic permeability, \( \bar{y} \), is given by \( \bar{y} = \frac{\sum y_i}{n} = \frac{572}{14} \approx 40.857 \).

02

Calculate the Slope (b)

The formula for the slope \( b \) in simple linear regression is \( b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} = \frac{\sum x_i y_i - n\bar{x}\bar{y}}{\sum x_i^2 - n\bar{x}^2} \). Substituting the given values, we have: \[ b = \frac{1697.80 - 14 \cdot 3.071 \cdot 40.857}{157.42 - 14 \cdot 3.071^2} \approx 1.678.\]

03

Calculate the Intercept (a)

The intercept \( a \) can be calculated using \( a = \bar{y} - b\bar{x} \). Substituting the values from the previous steps: \[ a = 40.857 - 1.678 \cdot 3.071 \approx 35.670.\]

04

Estimate \( \sigma^2 \)

The error variance \( \sigma^2 \) is estimated using \( \hat{\sigma}^2 = \frac{1}{n-2} \left( \sum y_i^2 - a\sum y_i - b\sum x_i y_i \right) \). Substituting the values: \[ \hat{\sigma}^2 = \frac{1}{12} (23530 - 35.670 \times 572 - 1.678 \times 1697.80) \approx 27.484.\]

05

Graph the Regression Line

Plot the line using the equation of the line \( y = 1.678x + 35.670 \). Plot with the compressive strength \( x \) on the x-axis and intrinsic permeability \( y \) on the y-axis.

06

Predict Permeability at \( x = 4.3 \)

Using the regression equation \( y = 1.678x + 35.670 \), substitute \( x = 4.3 \) to find \( y \): \( y = 1.678 \times 4.3 + 35.670 \approx 42.889 \).

07

Estimate Mean Permeability at \( x = 3.7 \)

For \( x = 3.7 \), substitute into the regression equation to find the point estimate: \( y = 1.678 \times 3.7 + 35.670 = 41.8886 \approx 41.889. \)

08

Calculate Residual at \( x = 3.7, y=46.1 \)

The residual is the difference between the observed and predicted values. Using the predicted value from Step 7, the residual is \( 46.1 - 41.889 \approx 4.211. \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Least Squares Estimates

The Least Squares method is a standard approach used in linear regression analysis to determine the line that best fits a set of data. This method helps in estimating the coefficients of the linear equation
\[ y = bx + a \]
where \( y \) is the dependent variable (in this case, intrinsic permeability), \( x \) is the independent variable (compressive strength), \( b \) is the slope, and \( a \) is the intercept.

• The slope \( b \) represents the change in \( y \) for a one-unit change in \( x \).
• The intercept \( a \) represents the value of \( y \) when \( x \) is zero.

Using the provided data and the formulas for calculating the slope and intercept, the slope was derived as approximately 1.678, and the intercept was approximately 35.670. These estimates allow us to create the linear regression equation that predicts intrinsic permeability based on compressive strength.

Compressive Strength

Compressive Strength is a measure of the ability of a material, such as concrete, to withstand loads that tend to compress or crush it. It's a critical quality parameter in construction and material science because it ultimately affects the durability and longevity of structures.

• The given data involves compressive strength as the independent variable \( x \) in our linear regression model.
• Understanding the relationship between compressive strength and other properties like permeability can help in optimizing concrete formulations for different applications.

In this exercise, by analyzing how changes in compressive strength affect intrinsic permeability, we can predict how materials will perform under specific conditions. This prediction aids builders in making more informed decisions about material usage.

Intrinsic Permeability

Intrinsic Permeability is a measure of how easily a fluid can flow through a porous material, such as concrete. This property is independent of the fluid's properties and is crucial in many engineering applications, including water tightness in construction.

• In the context of the linear regression model, intrinsic permeability is the dependent variable \( y \).
• Understanding how compressive strength influences permeability enables the optimization of concrete mixes to achieve desired performance characteristics.

Through the regression analysis, we derive an equation that models the expected permeability for various compressive strengths, helping researchers and engineers to forecast the material's behavior accurately. This understanding is pivotal for constructing resilient infrastructures that must withstand various environmental conditions.

Residual Calculation

Residuals are a crucial part of regression analysis. They are the differences between the observed values and the values predicted by the model. Calculating residuals gives us insight into the accuracy and reliability of our regression model.

• A small residual indicates that the model's prediction is very close to the actual observed value.
• A larger residual suggests that the model might not be capturing some underlying trend or variability in the data.

In the exercise, for an observed permeability of 46.1 when the compressive strength was 3.7, the predicted value was 41.889. The residual here is 4.211, calculated by the difference \( 46.1 - 41.889 \). This residual helps assess the fit of the regression line and indicates how well the model can be expected to predict new data points.