91Ó°ÊÓ

For the data set $$ \begin{array}{ccccc} \boldsymbol{x}_{1} & \boldsymbol{x}_{2} & \boldsymbol{x}_{3} & \boldsymbol{x}_{4} & \boldsymbol{y} \\ \hline 47.3 & 0.9 & 4 & 76 & 105.5 \\ \hline 53.1 & 0.8 & 6 & 55 & 113.8 \\ \hline 56.7 & 0.8 & 4 & 65 & 115.2 \\ \hline 48.8 & 0.5 & 7 & 67 & 118.9 \\ \hline 42.7 & 1.1 & 7 & 74 & 148.9 \\ \hline 44.3 & 1.1 & 6 & 76 & 120.2 \\ \hline 44.5 & 0.7 & 8 & 68 & 121.6 \\ \hline 37.7 & 0.7 & 7 & 79 & 140.0 \\ \hline 36.9 & 1.0 & 5 & 73 & 141.5 \\ \hline 28.1 & 1.8 & 6 & 68 & 141.9 \\ \hline 32.0 & 0.8 & 8 & 81 & 152.8 \\ \hline 34.7 & 0.8 & 10 & 68 & 156.5 \\\\\hline \end{array} $$ (a) Construct a correlation matrix between \(x_{1}, x_{2}, x_{3}, x_{4},\) and \(y\). Is there any evidence that multicollinearity may be a problem? (b) Determine the multiple regression line using all the explanatory variables listed. Does the \(F\) -test indicate that we should reject \(H_{0}: \beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0 ?\) Which explanatory variables have slope coefficients that are not significantly different from zero? (c) Remove the explanatory variable with the highest \(P\) -value from the model and recompute the regression model. Does the \(F\) -test still indicate that the model is significant? Remove any additional explanatory variables on the basis of the \(P\) -value of the slope coefficient. Then compute the model with the variable removed. (d) Draw residual plots and a box plot of the residuals to assess the adequacy of the model. (e) Use the final model constructed in part (c) to predict the value of \(y\) if \(x_{1}=44.3, x_{2}=1.1, x_{3}=7,\) and \(x_{4}=69 .\) (f) Draw a normal probability plot of the residuals. Is it reasonable to construct confidence and prediction intervals? (g) Construct \(95 \%\) confidence and prediction intervals if \(x_{1}=44.3, x_{2}=1.1, x_{3}=7,\) and \(x_{4}=69 .\)

Step by step solution

01

- Construct the correlation matrix

Calculate the correlation coefficients between each pair of variables: \(x_1, x_2, x_3, x_4,\) and \(y\). The Pearson correlation coefficient formula is given by: \[ r_{xy} = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2}} \]

02

- Analyze multicollinearity

Examine the correlation matrix. If any pair of explanatory variables has a correlation coefficient above 0.8, multicollinearity might be a concern.

03

- Determine the multiple regression line

Fit the multiple regression model using the formula \(y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \epsilon\). Calculate the coefficients \(\beta\) using software or statistical tools.

04

- Perform the F-test

Conduct an F-test to determine if the model is significant. The null hypothesis \(H_0\) states that all regression coefficients are zero: \(\beta_1 = \beta_2 = \beta_3 = \beta_4 = 0\). If the p-value of the F-test is less than 0.05, reject \(H_0\).

05

- Evaluate the slope coefficients

Find the p-values associated with each explanatory variable’s slope coefficient. If any p-value is greater than 0.05, the associated variable's coefficient is not significantly different from zero.

06

- Refine the model

Remove the explanatory variable with the highest p-value and refit the regression model. Repeat the process until all remaining variables have p-values less than 0.05.

07

- Assess model adequacy with residuals

Draw residual plots (residuals vs. fitted values, residuals vs. each predictor) and a box plot of the residuals to check for constant variance, independence, and normal distribution of residuals.

08

- Predict the value of y

Using the final regression model from Step 6, predict the value of \(y\) when \(x_1 = 44.3, x_2 = 1.1, x_3 = 7, \) and \(x_4 = 69\) by plugging these values into the regression equation.

09

- Draw a normal probability plot

Generate a normal probability plot of the residuals to check if they follow a normal distribution, which validates the assumptions needed for reliable inference.

10

- Construct confidence and prediction intervals

Using the final regression model, compute the 95% confidence and prediction intervals for \(y\) when \(x_1 = 44.3, x_2 = 1.1, x_3 = 7, \) and \(x_4 = 69\). The formula for the confidence interval is given by: \[ \hat{y} \pm t_{\alpha/2, n-p-1} \cdot SE(\hat{y})\] and for the prediction interval: \[ \hat{y} \pm t_{\alpha/2, n-p-1} \cdot \sqrt{MSE + SE(\hat{y})^2} \]

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Key Concepts

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

Correlation Matrix

To start analyzing multiple regression, we need to understand the relationship between the variables. This is where the correlation matrix comes in. It shows the Pearson correlation coefficients for each pair of variables. For the data set, calculate the correlation coefficients using the formula \( r_{xy} = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2}} \). The correlation matrix will tell you how strongly the variables are related on a scale from -1 to 1. Positive values indicate a positive relationship, where increases in one variable relate to increases in another. Conversely, negative values indicate an inverse relationship. In a large correlation matrix, values close to +1 or -1 highlight strong relationships, which can sometimes be problematic in multiple regression.

Multicollinearity

Multicollinearity is a situation where two or more explanatory variables in a regression model are highly correlated, leading to unreliable coefficient estimates. To check for multicollinearity, examine the correlation matrix. If any pair of variables has a correlation coefficient above 0.8, this indicates potential multicollinearity. High multicollinearity can inflate the variance of the coefficient estimates, making it hard to determine the individual effect of each variable. You can address multicollinearity by either removing one of the correlated variables, combining them into a single predictor, or using techniques like Principal Component Analysis (PCA).

F-test

In multiple regression analysis, the F-test checks the overall significance of the model. It tests the null hypothesis \( H_0: \beta_1 = \beta_2 = \beta_3 = \beta_4 = 0 \), meaning that none of the explanatory variables are related to the response variable. To perform the F-test, calculate the F-statistic and compare its p-value with a significance level (typically 0.05). If the p-value is less than 0.05, there is evidence to reject \( H_0 \), indicating that the model explains a significant portion of the variance in the response variable. This suggests that at least one predictor is meaningful.

P-value

Each variable in your regression model has an associated p-value. This p-value assesses the null hypothesis that the variable’s regression coefficient is equal to zero (no effect). If a p-value for a variable is less than 0.05, it suggests that the variable significantly contributes to the model. To ensure a reliable model, exclude any variable with a p-value greater than 0.05. When refining your model, iteratively remove the variable with the highest p-value, recompute the model, and check the p-values of the remaining variables until all are below the significance threshold.

Residual Analysis

Residuals are the differences between the observed and predicted values. Analyzing these residuals helps assess the adequacy of your regression model. Plot residuals against fitted values and each predictor to check for constant variance and independence. Additionally, create a box plot to spot any outliers. Ideally, residuals should be randomly distributed without patterns, indicating a good fit. Lastly, generate a normal probability plot to see if residuals follow a normal distribution. Proper residual analysis ensures the underlying assumptions of the regression model are met, making the model reliable.