91Ó°ÊÓ

The following data is representative of that reported in the article "An Experimental Correlation of Oxides of Nitrogen Emissions from Power Boilers Based on Field Data" (J. of Engr: for Power; July 1973: 165-170), with \(x=\) burner-area liberation rate \(\left(\mathrm{MBtu} / \mathrm{hr}^{-\mathrm{ft}^{2}}\right)\) and \(y=\mathrm{NO}_{x}\) emission rate (ppm): $$ \begin{array}{l|rrrrrrr} x & 100 & 125 & 125 & 150 & 150 & 200 & 200 \\ \hline y & 150 & 140 & 180 & 210 & 190 & 320 & 280 \\ x & 250 & 250 & 300 & 300 & 350 & 400 & 400 \\ \hline y & 400 & 430 & 440 & 390 & 600 & 610 & 670 \end{array} $$ a. Assuming that the simple linear regression model is valid, obtain the least squares estimate of the true regression line. b. What is the estimate of expected \(\mathrm{NO}_{x}\) emission rate when burner area liberation rate equals 225 ? c. Estimate the amount by which you expect \(\mathrm{NO}_{x}\) emission rate to change when burner area liberation rate is decreased by 50 . d. Would you use the estimated regression line to predict emission rate for a liberation rate of 500 ? Why or why not?

Step by step solution

01

Organize the Data

We have two sets of data: the independent variable \(x\) is the burner-area liberation rate, and the dependent variable \(y\) is the \(\text{NO}_x\) emission rate. We'll organize these as coordinate points \((x, y)\) for further calculations.

02

Calculate Averages

Calculate the average \(\bar{x}\) and \(\bar{y}\) of the data sets. For example, for \(x\), sum all \(x\) values and divide by the number of data points (14), and do the same for \(y\).

03

Calculate Slope (\(b_1\))

Determine the slope \(b_1\) using the formula: \[ b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \].Plug in the values of \(x_i\), \(y_i\), \(\bar{x}\), and \(\bar{y}\) to find \(b_1\).

04

Calculate Intercept (\(b_0\))

Use the formula for the intercept \(b_0\):\[ b_0 = \bar{y} - b_1 \bar{x} \].Substitute \(\bar{x}\), \(\bar{y}\), and \(b_1\) into this formula to find \(b_0\).

05

Write the Regression Line Equation

With \(b_0\) and \(b_1\) calculated, the regression line equation is: \[ \hat{y} = b_0 + b_1x \].This equation represents the least squares estimate of the true regression line.

06

Estimate NOx Emission Rate at x = 225

Substitute \(x = 225\) into the regression equation:\[ \hat{y} = b_0 + b_1 \cdot 225 \].This will give the estimated \(\text{NO}_x\) emission rate.

07

Find Change in NOx Emission Rate for Δ x = -50

The change in emission rate for a change \(\Delta x = -50\) is given by:\[ \Delta \hat{y} = b_1 \cdot (-50) \].Calculate this by substituting the slope \(b_1\).

08

Evaluate Using the Model at x = 500

Consider the validity of predicting at \(x = 500\): since 500 is outside the range of the existing data (100 to 400), predictions in this region may be unreliable, as the linear model may not be valid.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

least squares estimate

The least squares estimate is a method widely used in linear regression for finding the line of best fit. The primary goal of this method is to minimize the sum of squared differences between the observed values and those predicted by the line. In the context of the exercise, we calculate this by first determining the slope \(b_1\) and the y-intercept \(b_0\).
To start, the formula for the slope \(b_1\) is \[ b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]. This helps in deciding how steep the line should be to best fit the data points.
Then, the intercept is calculated using \(b_0 = \bar{y} - b_1 \bar{x}\), ensuring the line crosses the y-axis at the appropriate point.
Finally, with these parameters, the regression line equation becomes \( \hat{y} = b_0 + b_1x \). This equation is the least squares estimate, providing the best linear approximation to the observed data points.

NOx emission rate

The NOx emission rate, denoted as \(y\), measures the concentration of nitrogen oxides in parts per million (ppm). This is an important environmental metric as NOx gases contribute to air pollution and respiratory problems.
In this exercise, we are particularly interested in how the NOx emission rate changes with respect to the burner-area liberation rate \(x\). More simply, we want to see how the operation rate, or intensity, of a burner influences the emission of these gases.
The linear relationship we model will help in estimating NOx emissions at untested levels of burner activity, crucial for predictions in environmental management and policy-making decisions.

burner-area liberation rate

The burner-area liberation rate \(x\) refers to the energy released per unit area from a burner's surface, measured in million British thermal units per hour square foot (MBtu/hr-ft\(^2\)).
This rate indicates the intensity of fuel consumption and energy release, acting as the independent variable in our regression model. In essence, it shows how intensely the burner is operating at any given time.
In terms of the regression model, the liberation rate \(x\) influences the NOx emission rate \(y\), helping to establish an operational understanding between energy use and environmental impact.

prediction validity

Prediction validity refers to the reliability and applicability of using the regression model for estimates outside the observed data range. In linear regression, this is a crucial concept—models are best used within the range they were developed.
For instance, in the exercise data, the burner-area liberation rate ranges from 100 to 400. Making predictions for rates significantly outside this, such as at \(x = 500\), can be risky. This is because the model's accuracy and linear assumption may not hold.
When developing and using linear models, always remember that predictions are most valid within the scope of your data. Venturing beyond may lead to errors or invalid predictions.