The electric power consumed each month by a chemical plant...

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 in the following table: $$ \begin{array}{ccccc} \hline y & x_{1} & x_{2} & x_{3} & x_{4} \\ \hline 240 & 25 & 24 & 91 & 100 \\ 236 & 31 & 21 & 90 & 95 \\ 270 & 45 & 24 & 88 & 110 \\ 274 & 60 & 25 & 87 & 88 \\ 301 & 65 & 25 & 91 & 94 \\ 316 & 72 & 26 & 94 & 99 \\ 300 & 80 & 25 & 87 & 97 \\ 296 & 84 & 25 & 86 & 96 \\ 267 & 75 & 24 & 88 & 110 \\ 276 & 60 & 25 & 91 & 105 \\ 288 & 50 & 25 & 90 & 100 \\ 261 & 38 & 23 & 89 & 98 \\ \hline \end{array} $$(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, $x_{3}=90 \%,$ and $x_{4}=98$ tons.

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 in the following table:
$$ \begin{array}{ccccc} \hline y & x_{1} & x_{2} & x_{3} & x_{4} \\ \hline 240 & 25 & 24 & 91 & 100 \\ 236 & 31 & 21 & 90 & 95 \\ 270 & 45 & 24 & 88 & 110 \\ 274 & 60 & 25 & 87 & 88 \\ 301 & 65 & 25 & 91 & 94 \\ 316 & 72 & 26 & 94 & 99 \\ 300 & 80 & 25 & 87 & 97 \\ 296 & 84 & 25 & 86 & 96 \\ 267 & 75 & 24 & 88 & 110 \\ 276 & 60 & 25 & 91 & 105 \\ 288 & 50 & 25 & 90 & 100 \\ 261 & 38 & 23 & 89 & 98 \\ \hline \end{array} $$(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, $x_{3}=90 \%,$ and $x_{4}=98$ tons.

Key Concepts

Least Squares Estimation

Least squares estimation is the standard approach used in fitting linear regression models. It works by finding the set of parameters that minimizes the sum of the squared differences between the observed values and the values predicted by the model, essentially reducing the overall error in the model fit. This method yields the best linear unbiased estimator under certain assumptions.

Multiple Linear Regression

Multiple linear regression is a statistical technique that models the linear relationship between a dependent variable and several independent variables. It involves fitting a linear equation to observed data in order to predict the outcome variable based on predictor variables. This is achieved by estimating coefficients that show the change in the outcome for a unit change in each predictor while holding others constant.

Error Variance Estimation

Error variance estimation involves calculating the variance of the error terms (often denoted as ?²) which reflects the variability in the dependent variable that is not explained by the independent variables in the model. This is generally estimated using the residual sum of squares divided by the degrees of freedom, providing important insight into the model’s accuracy and consistency.

Standard Errors of Regression Coefficients

The standard errors of regression coefficients measure the precision with which the true population parameters are estimated from the sample data. They are derived from the variance-covariance matrix of the parameter estimates and depend on both the error variance and the design of the predictor variables. Differences in the standard errors indicate that some parameters are estimated with greater precision than others due to factors such as scale, variability, and correlation among the predictors.

Prediction in Regression

Prediction in regression involves using the fitted model to estimate the dependent variable for new observations of the predictor variables. Alongside the point prediction, it is important to quantify the uncertainty of the prediction, often through confidence or prediction intervals, which accounts for both the variability in the data and the uncertainty in the model parameter estimates.

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