91Ó°ÊÓ

Nordin \(^{49}\) obtained the following data relating output and total final cost for an electric utility in Iowa. (Instead of using the original data set for 541 eight-hour shifts, we have just given the array means.) $$\begin{array}{|l|llllllll}\hline x & 25 & 28 & 33 & 38 & 41 & 45 & 50 & 53 \\\\\hline y & 24 & 24 & 26 & 28 & 27 & 31 & 34 & 38 \\\\\hline x & 57 & 63 & 66 & 71 & 74 & 79 & 83 & 89 \\\\\hline y & 39 & 42 & 43 & 51 & 53 & 52 & 54 & 52 \\\\\hline\end{array} $$ Here \(x\) is the output as a percent of capacity, and \(y\) is final total cost in dollars (multiplied by a constant not given to avoid showing the exact level of the costs.) Use cubic regression to find the best-fitting cubic to the data and the correlation coefficient. Graph. Also find the correlation coefficients associated with both linear and quadratic regression, and compare them to that found for cubic regression.

Step by step solution

01

Set Up the Data Points

Set up the data points for the regression analysis. You have pairs of values: \( (x, y) \) where \( x \) represents output and \( y \) represents final total cost. These are the points given in the table.

02

Perform Cubic Regression

Use statistical software or a calculator that supports polynomial regression to fit a cubic polynomial to the data.You will obtain a cubic equation of the form \( y = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants determined by the regression.

03

Calculate the Correlation Coefficient for Cubic Regression

Once you have the cubic regression model, calculate the correlation coefficient \( R^2 \), which indicates how well the cubic polynomial fits the data. This value is usually provided by the regression software.

04

Perform Linear Regression

Fit a linear regression line to the data points. The linear equation will take the form \( y = mx + b \), where \( m \) and \( b \) are the slope and intercept, respectively.

05

Calculate the Correlation Coefficient for Linear Regression

Calculate the correlation coefficient \( R^2 \) for the linear regression model. Compare this value to the \( R^2 \) obtained from the cubic regression to see which model better fits the data.

06

Perform Quadratic Regression

Fit a quadratic polynomial to the data points. The quadratic equation will be of the form \( y = ax^2 + bx + c \).

07

Calculate the Correlation Coefficient for Quadratic Regression

Calculate the \( R^2 \) for the quadratic regression model. This value will be compared to the cubic and linear \( R^2 \) values to determine which model best captures the relationship.

08

Graph the Data and Regressions

Plot the original data points and overlay the best-fit lines for the linear, quadratic, and cubic regressions. Visually assess which regression line fits the data best.

09

Compare the Correlation Coefficients

Compare all three correlation coefficients (linear, quadratic, and cubic). The regression model with the highest \( R^2 \) value is considered the best fit for the data.

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Key Concepts

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

Polynomial Regression

Polynomial regression is an extension of linear regression, which is used when the relationship between the independent variable \( x \) and the dependent variable \( y \) is not a straight line. Instead, it can be curved or complex. Polynomial regression fits data to an \( n\)-degree polynomial function, such as quadratic (degree 2), cubic (degree 3), or higher. This method is particularly useful when the data trends are better represented by curves rather than straight lines.

• The polynomial equation is generally expressed in the form: \( y = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \).
• The coefficients \( a_n, a_{n-1}, ... \) are determined using methods like the least squares regression, ensuring that the polynomial closely fits the data points.
• It's important to choose the right degree; too high might overfit, too low might oversimplify.

Choosing polynomial regression models wisely is crucial for accurately capturing the complexities in the data.

Correlation Coefficient

The correlation coefficient, often represented by \( R \) or \( R^2 \), measures the strength and direction of a linear relationship between two variables. When extended to polynomial regression, it helps evaluate how well the polynomial fit represents the data.

• \( R \) values range from -1 to 1, where 1 means a perfect positive correlation, 0 indicates no correlation, and -1 means a perfect negative correlation.
• \( R^2 \) provides information on the proportion of the variance in the dependent variable that is predictable from the independent variable(s).
• An \( R^2 \) of 1 indicates that the regression predictions perfectly fit the data.

In the context of comparing regression models, including linear, quadratic, and cubic, the model with the highest \( R^2 \) value is typically considered to have the best fit.

Linear Regression

Linear regression is the simplest form of regression analysis, which aims to model the relationship between two variables by fitting a linear equation to the observed data. This method assumes a straight-line relationship between the variables.

• The linear equation is represented as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• It is ideal when the data shows a clear linear trend but not suitable for data with curves or more complex patterns.
• The simplicity of linear regression makes it easy to implement and interpret, offering insights with minimal computational power.

While linear regression is informative, it may not always provide the most accurate predictions if the data exhibits non-linear characteristics.

Quadratic Regression

Quadratic regression is suitable for data with a parabolic trend, meaning it has a curve which can be described by a quadratic polynomial, i.e., a polynomial of degree 2. This type of regression fits the data to a curve, often creating a U-shaped or inverted U-shaped path.

• The quadratic equation is commonly expressed as \( y = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants.
• Quadratic regression is particularly useful when there are hints of acceleration or deceleration in the rate of change of the dependent variable.
• Just like higher degree polynomial regressions, it's crucial to ensure that the quadratic model is a better fit for the data than a simple linear model.

Quadratic regression strikes a balance between simplicity and flexibility, making it a good choice for many real-world data sets that exhibit non-linear trends.