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Chapter 12: Problem 7

The article "Some Field Experience in the Use of an Accelerated Method in Estimating 28-Day Strength of Concrete" (J. of Amer: Concrete Institute, 1969: 895) considered regressing \(y=28\)-day standard-cured strength (psi) against \(x=\) accelerated strength (psi). Suppose the equation of the true regression line is \(y=1800+1.3 x\). a. What is the expected value of 28 -day strength when accelerated strength \(=2500\) ? b. By how much can we expect 28-day strength to change when accelerated strength increases by 1 psi? c. Answer part (b) for an increase of 100 psi. d. Answer part (b) for a decrease of 100 psi.

Short Answer

Expert verified
a. Expected value is 5050 psi. b. Increase by 1.3 psi. c. Increase by 130 psi. d. Decrease by 130 psi.

Step by step solution

01

Understand the Regression Equation

The true regression line is given by the equation \( y = 1800 + 1.3x \), where \( y \) is the 28-day standard-cured strength and \( x \) is the accelerated strength. The coefficient of \( x \) (which is 1.3 in this case) tells us how much \( y \) is expected to change with a one unit increase in \( x \). The constant 1800 is the intercept and represents the expected value of \( y \) when \( x = 0 \).
02

Calculate Expected Value for 2500 psi

The expected value of the 28-day strength when accelerated strength \( x = 2500 \) psi can be calculated by substituting 2500 into the regression equation: \( y = 1800 + 1.3(2500) \).Calculate: \( y = 1800 + 3250 = 5050 \).Therefore, the expected 28-day strength is 5050 psi.
03

Change in 28-day Strength per 1 psi Increase

The coefficient of \( x \) in the regression equation, which is 1.3, represents the expected change in the 28-day strength for a 1 psi increase in accelerated strength. Thus, for every 1 psi increase in accelerated strength, the 28-day strength is expected to increase by 1.3 psi.
04

Change in 28-day Strength for 100 psi Increase

To find the expected change in 28-day strength for an increase of 100 psi in accelerated strength, multiply the change per 1 psi by 100: \( 1.3 \times 100 = 130 \).Thus, for a 100 psi increase in accelerated strength, the 28-day strength is expected to increase by 130 psi.
05

Change in 28-day Strength for 100 psi Decrease

For a decrease of 100 psi in accelerated strength, the change in 28-day strength is simply the negative of the change for an increase: \( -1.3 \times 100 = -130 \).Hence, a 100 psi decrease in accelerated strength will reduce the 28-day strength by 130 psi.

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Key Concepts

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

Understanding the Regression Equation
The regression equation is a mathematical formula that predicts the relationship between two variables. In this context, the equation \( y = 1800 + 1.3x \) helps us predict the 28-day standard-cured strength of concrete based on its accelerated strength. Here, \( y \) is the predicted 28-day strength, and \( x \) is the known accelerated strength.
  • The number 1800 is known as the intercept. This value shows the expected 28-day strength when the accelerated strength \( x \) is zero.
  • The coefficient 1.3 gives the rate at which the 28-day strength changes for each unit increase in accelerated strength.
This equation allows engineers and scientists to forecast concrete performance, ensuring that structures are built safely and efficiently.
Calculating the Expected Value
The expected value in regression indicates what the outcome variable (in this case, the 28-day concrete strength) should be given a specific input (the accelerated strength). To find this value for an accelerated strength of 2500 psi, you substitute \( x = 2500 \) into the regression equation.
Following the equation: \[y = 1800 + 1.3(2500) = 1800 + 3250 = 5050\]Therefore, when the accelerated strength is 2500 psi, the expected 28-day strength is 5050 psi.

The expected value simplifies complex data, providing a single figure that summarizes potential outcomes, making predictions easier to understand and apply.
Strength Prediction with Changes in psi
Predicting changes in 28-day strength based on changes in accelerated strength requires understanding the coefficient of \(x\). In this regression equation, \(1.3\) is the coefficient of \(x\).
  • If accelerated strength increases by \(1\) psi, the 28-day strength is expected to increase by \(1.3\) psi.
  • For a \(100\) psi increase, multiply the change per \(1\) psi by \(100\): \(1.3 \times 100 = 130\). So, the strength is expected to rise by \(130\) psi.
  • Conversely, if accelerated strength decreases by \(100\) psi, the 28-day strength would reduce by \(130\) psi, calculated as \(-1.3 \times 100 = -130\).
Understanding these calculations helps builders and engineers adjust testing parameters and anticipate how concrete will behave in real-world conditions.
Interpreting Coefficients in Regression
Coefficients in a regression model serve as indicators of the relationship's strength and direction between variables. In the equation \(y = 1800 + 1.3x\), the coefficients tell us two things:
  • The intercept \(1800\) suggests what the concrete strength would theoretically be if there was no accelerated strength at all. While in practice, \(x = 0\) might not be possible, it serves as a baseline.
  • The slope \(1.3\) (or the coefficient of \(x\)) highlights that for every extra psi increase in accelerated strength, the 28-day strength increases by \(1.3\) psi. If this number were negative, it would indicate a decrease.
Understanding how to interpret coefficients allows practitioners to make informed decisions based on the model, optimizing processes and materials accordingly.

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