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An engineer at a semiconductor company wants to model the relationship between the device HFE \((y)\) and three parameters: Emitter-RS \(\left(x_{1}\right),\) Base-RS \(\left(x_{2}\right),\) and Emitter-toBase \(\operatorname{RS}\left(x_{3}\right)\). The data are shown in the Table \(\mathrm{E} 12-5\) (a) Fit a multiple linear regression model to the data. (b) Estimate \(\sigma^{2}\). (c) Find the standard errors \(\operatorname{se}\left(\hat{\beta}_{j}\right)\). Are all of the model parameters estimated with the same precision? Justify your answer. (d) Predict HFE when \(x_{1}=14.5, x_{2}=220,\) and \(x_{3}=5.0 .\) $$ \begin{array}{cccc} \hline \begin{array}{c} x_{1} \\ \text { Emitter-RS } \end{array} & \begin{array}{c} x_{2} \\ \text { Base-RS } \end{array} & \begin{array}{c} x_{3} \\ \text { E-B-RS } \end{array} & \begin{array}{c} y \\ \text { HFE-1M-5V } \end{array} \\ \hline 14.620 & 226.00 & 7.000 & 128.40 \\ 15.630 & 220.00 & 3.375 & 52.62 \\ 14.620 & 217.40 & 6.375 & 113.90 \\ 15.000 & 220.00 & 6.000 & 98.01 \\ 14.500 & 226.50 & 7.625 & 139.90 \\ 15.250 & 224.10 & 6.000 & 102.60 \\ 16.120 & 220.50 & 3.375 & 48.14 \\ 15.130 & 223.50 & 6.125 & 109.60 \\ 15.500 & 217.60 & 5.000 & 82.68 \mathrm{~b} \\ 15.130 & 228.50 & 6.625 & 112.60 \\ 15.500 & 230.20 & 5.750 & 97.52 \\ 16.120 & 226.50 & 3.750 & 59.06 \\ 15.130 & 226.60 & 6.125 & 111.80 \\ 15.630 & 225.60 & 5.375 & 89.09 \\ 15.380 & 229.70 & 5.875 & 101.00 \\ 14.380 & 234.00 & 8.875 & 171.90 \\ 15.500 & 230.00 & 4.000 & 66.80 \\ 14.250 & 224.30 & 8.000 & 157.10 \\ 14.500 & 240.50 & 10.870 & 208.40 \\ 14.620 & 223.70 & 7.375 & 133.40 \\ \hline \end{array} $$

Step by step solution

01

Organize the Data

First, collect the data from Table E 12-5 and organize it into a format suitable for analysis. You have independent variables \(x_1\), \(x_2\), \(x_3\), and the dependent variable \(y\). List them out corresponding to each observation.

02

Fit a Multiple Linear Regression Model

To fit a multiple linear regression model, you will use the equation:\[y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \epsilon\]Using statistical software or a calculator, calculate the coefficients \(\beta_0\), \(\beta_1\), \(\beta_2\), and \(\beta_3\). The software will provide you with a fitted linear model of the form \(\hat{y} = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3\).

03

Estimate \(\sigma^2\)

The estimation of \(\sigma^2\), the variance of the error terms, is given by \(\sigma^2 = \frac{SSE}{n - p}\), where \(SSE\) is the sum of squared errors, \(n\) is the number of observations, and \(p\) is the number of predictors plus one. Calculate \(SSE\) and substitute \(n\) and \(p\) into the formula to find \(\sigma^2\).

04

Calculate Standard Errors

The standard errors of the coefficients \(\hat{\beta}_j\) are calculated using the formula:\[se(\hat{\beta}_j) = \sqrt{c_{jj} \cdot \hat{\sigma}^2}\]where \(c_{jj}\) are diagonal elements of the covariance matrix of the parameters. Use statistical software to output these standard errors and assess whether they are equal or not.

05

Predict \(y\) for New Data Points

Using the fitted model from Step 2 and substituting \(x_1 = 14.5\), \(x_2 = 220\), and \(x_3 = 5.0\), predict the value of \(y\). Substitute these new \(x\) values into the regression equation to compute the predicted \(y\).

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Key Concepts

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

Error Variance Estimation

When working with multiple linear regression, understanding and estimating error variance is crucial. The error variance, denoted by \( \sigma^2 \), represents the variance of the errors or deviations of observed values from the predicted values in your regression model.
The formula for estimating \( \sigma^2 \) is \( \sigma^2 = \frac{SSE}{n - p} \), where:

• \( SSE \) is the sum of squared errors, which measures how much variation in the dependent variable \( y \) that the model fails to capture.
• \( n \) is the total number of observations or data points.
• \( p \) is the number of predictors in the model plus one (for the intercept \( \beta_0 \)).

To compute \( SSE \), each actual data point in \( y \) is compared to its corresponding predicted value from the regression model. These differences are squared and summed together. Dividing the \( SSE \) by \( (n - p) \) gives an average error variance, adjusting for the complexity of the model. A smaller \( \sigma^2 \) indicates a better fit, as the errors (or disturbances) are relatively small compared to the predictions.

Standard Error Calculation

In regression analysis, calculating the standard errors of the model coefficients is key to understanding the precision of the estimations. The standard error of a coefficient \( \hat{\beta}_j \) provides insight into the accuracy of our estimated coefficients for the regression equation.
The formula used is:\[ se(\hat{\beta}_j) = \sqrt{c_{jj} \cdot \hat{\sigma}^2} \]where:

• \( c_{jj} \) are diagonal elements of the covariance matrix \( C \) of the parameters. This matrix arises from the estimation process and reflects estimated variances and covariance of the coefficients.
• \( \hat{\sigma}^2 \) is the estimated error variance, as discussed above.

By computing \( se(\hat{\beta}_j) \), we get a measure of dispersion around each coefficient estimate. The smaller the standard error, the more precise is the estimated parameter. This precision is crucial when interpreting results and making predictions, as it affects the confidence intervals around the estimated coefficients.

Predictive Modeling

Predictive modeling involves using the regression equation obtained from multiple linear regression to predict new outcomes for given values of the independent variables. This process leverages the relationship identified in the fitted model to forecast future or unseen data.
For instance, in our scenario, if an engineer wants to predict the device HFE for given values \( x_1 = 14.5 \), \( x_2 = 220 \), and \( x_3 = 5.0 \), they would substitute these values into the regression equation:\[ \hat{y} = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 \]Once the coefficients \( \beta_0, \beta_1, \beta_2, \text{and} \ \beta_3 \) are estimated through the model fitting process, we simply plug in the \( x \) values to determine \( \hat{y} \), the predicted outcome.
This approach is powerful for making predictions in real-world scenarios, as it simplifies decision-making. However, ensuring that model assumptions hold and the model is appropriately validated with available data is essential to maintain prediction accuracy.