91Ó°ÊÓ

In an investigation to determine the relationship between the degree of metal corrosion and the length of time the metal is exposed to the action of soil acids, the percentage of corrosion and exposure time were measured weekly. $$ \begin{array}{l|llllllll} y & 0.1 & 0.3 & 0.5 & 0.8 & 1.2 & 1.8 & 2.5 & 3.4 \\ \hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \end{array} $$ The data were fitted using the quadratic model, $E(y)=\beta_{0}+\beta_{1} x+\beta_{2} x^{2},$ with the following results. $$ \begin{array}{lr} {\text { SUMMARY OUTPUT }} \\ \hline {\text { Regression Statistics }} \\ \hline \text { Multiple R } & 0.9993 \\ \text { R Square } & 0.9985 \\ \text { Adjusted R Square } & 0.9979 \\ \text { Standard Error } & 0.0530 \\ \text { Observations } & 8 \end{array} $$ $$ \begin{aligned} &\text { ANOVA }\\\ &\begin{array}{lrrrrr} \hline & d f & \mathrm{SS} & M S & F & \text { Significance } F \\ \hline \text { Regression } & 2 & 9.421 & 4.710 & 1676.610 & 0.000 \\ \text { Residual } & 5 & 0.014 & 0.003 & & \\ \text { Total } & {7} & 9.435 & & & \\ \hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \\ \hline \text { Intercept } & 0.196 & 0.074 & 2.656 & 0.045 & \\ \text { x } & -0.100 & 0.038 & -2.652 & 0.045 & \\ \text { x-Sq } & 0.062 & 0.004 & 15.138 & 0.000 & \\ \hline \end{array} \end{aligned} $$ a. What percentage of the total variation is explained by the quadratic regression of \(y\) on \(x ?\) b. Is the regression on \(x\) and \(x^{2}\) significant at the \(\alpha=.05\) level of significance? c. Is the linear regression coefficient significant when \(x^{2}\) is in the model? d. Is the quadratic regression coefficient significant when \(x\) is in the model? e. The data were fitted to a linear model without the quadratic term with the results that follow. What can you say about the contribution of the quadratic term when it is included i \(\mathrm{n}\) the model? $$ \begin{array}{lc} \text { SUMMARY OUTPUT } & \\ \hline {\text { Regression Statistics }} \\ \hline \text { Multiple R } & 0.9645 \\ \text { R Square } & 0.9303 \\ \text { Adjusted R Square } & 0.9187 \\ \text { Standard Error } & 0.3311 \\ \text { Observations } & 8 \\ \hline \end{array} $$ $$ \begin{aligned} &\text { ANOVA }\\\ &\begin{array}{lccccc} \hline & d f & S S & M S & F & & \text { Significance } F \\ \hline \text { Regression } & 1 & 8.777 & 8.777 & 80.052 & 0.000 \\ \text { Residual } & 6 & 0.658 & 0.110 & & & \\ \text { Total } & 7 & 9.435 & & & & \\ \hline & & {\text { Standard }} \\ & \text { Coefficients } & \text { Error } & t \text { Stat } & P \text { -value } \\ \hline \text { Intercept } & -0.732 & 0.258 & -2.838 & 0.030 \\ x & 0.457 & 0.051 & 8.947 & 0.000 \\ \hline \end{array} \end{aligned} $$ f. The plot of the residuals from the linear regression model in part e shows a specific pattern. What is the term in the model that seems to be missing?

Step by step solution

01

Find the R Square value from the quadratic regression summary output.

We have the R Square value from the quadratic model's summary output as: R Square = 0.9985

02

Calculate the percentage of total variation explained.

By multiplying R Square value by 100, we get the percentage of total variation explained by the quadratic regression of y on x. Percentage of explained variation = R Square * 100 = 0.9985 * 100 = 99.85% So, 99.85% of the total variation is explained by the quadratic regression of y on x. #b. Is the regression on x and x^2 significant at the α=.05 level of significance?#

03

Find the p-value for the quadratic regression model F-statistic.

From the summary output of the quadratic regression, the Significance F value is: Significance F = 0.000

04

Check if the p-value is less than α=0.05.

Since the p-value (0.000) is less than α=0.05, we can conclude that the regression on x and x^2 is significant at the α=0.05 level of significance. #c. Is the linear regression coefficient significant when x^2 is in the model?#

05

Find the p-value for the linear coefficient x in the quadratic model.

From the ANOVA table of the quadratic model, we have the P-value (t Stat) for x: P-value for x = 0.045

06

Check if the p-value is less than α=0.05.

Since the p-value (0.045) is less than α=0.05, we can conclude that the linear regression coefficient x is significant when x^2 is in the model. #d. Is the quadratic regression coefficient significant when x is in the model?#

07

Find the p-value for the quadratic coefficient x^2 in the quadratic model.

From the ANOVA table of the quadratic model, we have the P-value (t Stat) for x^2: P-value for x^2 = 0.000

08

Check if the p-value is less than α=0.05.

Since the p-value (0.000) is less than α=0.05, we can conclude that the quadratic regression coefficient x^2 is significant when x is in the model. #e. What can you say about the contribution of the quadratic term when it is included in the model?#

09

Compare R Square values of quadratic and linear models.

From the summary outputs, we have the R Square values for the quadratic and linear models: R Square (quadratic) = 0.9985 R Square (linear) = 0.9303

10

Interpret the difference in R Square values.

The higher R Square value for the quadratic model (0.9985) compared to the linear model (0.9303) indicates that the quadratic term has a significant contribution when it is included in the model. The quadratic model explains more of the variation in the data compared to the linear model. #f. What is the term in the model that seems to be missing?#

11

Observe the plot of residuals and quadratic term.

The plot of the residuals from the linear regression model is not given in the given problem. However, the exercise's wording suggests that there is a specific pattern observed in the residuals from the linear regression model.

12

Identify the missing term based on the pattern.

Based on the given information and the question, the pattern observed in the residuals from the linear regression model indicates that there's a missing quadratic term in the linear regression model. This missing term is x^2, which is present in the quadratic regression model, and it captures the nonlinear relationship between metal corrosion and exposure time.

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