91Ó°ÊÓ

The electric power consumed each month by a chemical plant is thought to be related to the average ambient temperature \(\left(x_{1}\right),\) the number of days in the month \(\left(x_{2}\right),\) the average product purity \(\left(x_{3}\right),\) and the tons of product produced \(\left(x_{4}\right) .\) The past year's historical data are available and are presented below. $$ \begin{array}{ccccc} y & x_{1} & x_{2} & x_{3} & x_{4} \\ 240 & 25 & 24 & 91 & 100 \\ 236 & 31 & 21 & 90 & 95 \\ 270 & 45 & 24 & 88 & 110 \\ 274 & 60 & 25 & 87 & 88 \\ 301 & 65 & 25 & 91 & 94 \\ 316 & 72 & 26 & 94 & 99 \\ 300 & 80 & 25 & 87 & 97 \\ 296 & 84 & 25 & 86 & 96 \\ 267 & 75 & 24 & 88 & 110 \\ 276 & 60 & 25 & 91 & 105 \\ 288 & 50 & 25 & 90 & 100 \\ 261 & 38 & 23 & 89 & 98 \end{array} $$ a. Fit a multiple linear regression model to these data. b. Estimate \(\sigma^{2}\). c. Compute the standard errors of the regression coefficients. Are all of the model parameters estimated with the same precision? Why or why not? d. Predict power consumption for a month in which \(x_{1}=75^{\circ} \mathrm{F}, x_{2}=24\) days, \(x_{2}=90 \%,\) and \(x_{4}=98\) tons.

Step by step solution

01

Organize the data for regression analysis

First, extract the data from the table and prepare it for linear regression analysis. The data consists of the dependent variable, power consumption \((y)\), and the independent variables: average ambient temperature \((x_1)\), number of days in the month \((x_2)\), average product purity \((x_3)\), and tons of product produced \((x_4)\).

02

Formulate the multiple linear regression model

Set up the multiple linear regression model as: \[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \epsilon \]where \(y\) is the power consumption, \(x_1, x_2, x_3, \) and \(x_4\) are the predictors, \(\beta_0\) is the intercept, \(\beta_1, \beta_2, \beta_3, \) and \(\beta_4\) are the coefficients of the predictors, and \(\epsilon\) is the error term.

03

Fit the regression model to the data

Using a statistical software or calculator, input the data to perform multiple linear regression analysis. The software will provide estimated coefficients for \(\beta_0, \beta_1, \beta_2, \beta_3, \) and \(\beta_4\).

04

Estimate the error variance \(\sigma^2\)

After fitting the model, compute the error variance \(\sigma^2\) using:\[ \sigma^2 = \frac{SSE}{n-p} \]where \(SSE\) is the sum of squared errors from the regression output, \(n\) is the number of observations, and \(p\) is the number of parameters including the intercept.

05

Compute standard errors of the regression coefficients

Calculate the standard errors for each estimated coefficient \(\beta_i\) using the variance-covariance matrix provided by the regression analysis:\[ SE(\beta_i) = \sqrt{\text{Var}(\beta_i)} \]These values indicate the precision of the parameter estimates.

06

Analyze precision of model parameters

Compare the standard errors to evaluate whether all parameters are estimated with the same precision. Smaller standard errors indicate more precise estimates.

07

Predict the power consumption for a given scenario

Substitute the given values \(x_1 = 75\), \(x_2 = 24\), \(x_3 = 90\), and \(x_4 = 98\) into the regression equation:\[ \hat{y} = \beta_0 + \beta_1 \cdot 75 + \beta_2 \cdot 24 + \beta_3 \cdot 90 + \beta_4 \cdot 98 \]Compute the predicted power consumption based on the estimated coefficients from Step 3.

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Key Concepts

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

Error Variance

In multiple linear regression, understanding error variance is crucial. Error variance refers to the variability or spread of the error terms (\(\epsilon\)) in a regression model. This is critical because we want to understand how closely our model's predictions are to the actual observed data.

Error variance is calculated using:\[\sigma^2 = \frac{SSE}{n-p}\]where \(SSE\) is the sum of squared errors, \(n\) is the number of observations, and \(p\) is the number of parameters in the model. This formula helps give us a sense of how well our model fits the data.

Error variance allows us to measure the "noise" in the data that isn't explained by our model. A smaller error variance signals that the model captures the data trends well. Conversely, a larger error variance indicates more unexplained variability, suggesting the model might need improvement or additional variables.

Standard Error

The standard error is an essential statistical concept used to measure the precision of the estimated coefficients in a regression model. Each estimated coefficient, like \(\beta_1\), has a corresponding standard error. It helps us understand how much variation exists from the true coefficient value.

The standard error for a particular parameter estimate, \(SE(\beta_i)\), can be calculated using the variance-covariance matrix. The formula for the standard error is:\[SE(\beta_i) = \sqrt{\text{Var}(\beta_i)}\] Smaller standard errors are preferable and indicate that the coefficient estimate is precise and reliable. Larger standard errors suggest more uncertainty and less reliability in the coefficient estimate.

When comparing coefficients, smaller standard errors imply that the parameter estimates are consistently closer to the true population parameters. Evaluating standard errors helps in assessing the precision of your model's parameter estimates.

Regression Model

A regression model serves as a mathematical tool that describes the relationship between a dependent variable and one or more independent variables. In the context of multiple linear regression, the model takes the following form:
\[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \epsilon\] Here, \(y\) represents the dependent variable, such as power consumption. The terms \(x_1, x_2, x_3,\) and \(x_4\) are independent variables, such as ambient temperature and number of days in the month. The coefficients \(\beta_0, \beta_1, \beta_2, \beta_3, \beta_4\) illustrate the impact or weight each independent variable has on the dependent variable.

The error term,\(\epsilon\), accounts for the randomness or unobserved factors affecting the dependent variable.

This model is helpful to predict outcomes, identify relationships, and make decisions based on the interaction of multiple variables. It forms the foundation of many statistical analyses in various fields of research.

Parameter Estimation

Parameter estimation involves the process of using data to determine the values of parameters in a regression model. In our model, we estimate parameters such as\(\beta_0\) (intercept) and \(\beta_1, \beta_2, \beta_3, \beta_4\) (slopes). These values help us understand the relationship dynamics between the dependent and independent variables.

Estimating these parameters often requires statistical software or calculations that fit the model to minimize differences between predicted and observed data. This approach, commonly known as "least squares," finds the best parameter values that lead to the minimum sum of squared differences between actual and predicted values.

Understanding parameter estimation is significant because it directly affects the predictive capacity of the regression model. Accurate parameter estimates indicate a strong model, while poor estimates may lead to incorrect predictions or interpretations.