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Chapter 13: Problem 29

Because dolphins (and other large marine mammals) are considered to be the top predators in the marine food chain, the heavy metal concentrations in striped dolphins were measured as part of a marine pollution study. The concentration of mercury, the heavy metal reported in this study, is expected to differ in males and females because the mercury in a female is apparently transferred to her offspring during gestation and nursing. This study involved 28 males between the ages of .21 and 39.5 years, and 17 females between the ages of .80 and 34.5 years. For the data in the table, $$ \begin{aligned} x_{1}=& \text { Age of the dolphin (in years) } \\ x_{2}=&\left\\{\begin{array}{ll} 0 & \text { if female } \\ 1 & \text { if male } \end{array}\right.\\\ y=& \text { Mercury concentration (in } \\ & \text { micrograms/gram) in the liver } \end{aligned} $$ $$ \begin{array}{rrl|lll} y & x_{1} & x_{2} & y & x_{1} & x_{2} \\ \hline 1.70 & .21 & 1 & 481.00 & 22.50 & 1 \\ 1.72 & .33 & 1 & 485.00 & 24.50 & 1 \\ 8.80 & 2.00 & 1 & 221.00 & 24.50 & 1 \\ 5.90 & 2.20 & 1 & 406.00 & 25.50 & 1 \\ 101.00 & 8.50 & 1 & 252.00 & 26.50 & 1 \\ 85.40 & 11.50 & 1 & 329.00 & 26.50 & 1 \\ 118.00 & 11.50 & 1 & 316.00 & 26.50 & 1 \\ 183.00 & 13.50 & 1 & 445.00 & 26.50 & 1 \\ 168.00 & 16.50 & 1 & 278.00 & 27.50 & 1 \\ 218.00 & 16.50 & 1 & 286.00 & 28.50 & 1 \\ 180.00 & 17.50 & 1 & 315.00 & 29.50 & 1 \\ 264.00 & 20.50 & 1 & & & \end{array} $$ $$ \begin{array}{rrl|lll} y & x_{1} & x_{2} & y & x_{1} & x_{2} \\ \hline 241.00 & 31.50 & 1 & 142.00 & 17.50 & 0 \\ 397.00 & 31.50 & 1 & 180.00 & 17.50 & 0 \\ 209.00 & 36.50 & 1 & 174.00 & 18.50 & 0 \\ 314.00 & 37.50 & 1 & 247.00 & 19.50 & 0 \\ 318.00 & 39.50 & 1 & 223.00 & 21.50 & 0 \\ 2.50 & .80 & 0 & 167.00 & 21.50 & 0 \\ 9.35 & 1.58 & 0 & 157.00 & 25.50 & 0 \\ 4.01 & 1.75 & 0 & 177.00 & 25.50 & 0 \\ 29.80 & 5.50 & 0 & 475.00 & 32.50 & 0 \\ 45.30 & 7.50 & 0 & 342.00 & 34.50 & 0 \\ 101.00 & 8.05 & 0 & & & \\ 135.00 & 11.50 & 0 & & & \end{array} $$ a. Write a second-order model relating \(y\) to \(x_{1}\) and \(x_{2}\). Allow for curvature in the relationship between age and mercury concentration, and allow for an interaction between gender and age. Use a computer software package to perform the multiple regression analysis. Refer to the printout to answer these questions. b. Comment on the fit of the model, using relevant statistics from the printout. c. What is the prediction equation for predicting the mercury concentration in a female dolphin as a function of her age? d. What is the prediction equation for predicting the mercury concentration in a male dolphin as a function of his age? e. Does the quadratic term in the prediction equation for females contribute significantly to the prediction of the mercury concentration in a female dolphin? f. Are there any other important conclusions that you feel were not considered regarding the fitted prediction equation?

Short Answer

Expert verified
Answer: The main goal of the analysis is to examine the relationship between mercury concentration in dolphins and their age and gender, and to provide predictive equations for both genders while allowing for curvature and interaction between variables.

Step by step solution

01

Writing the Second-Order Model

The given task is to write a second-order model that relates the mercury concentration in dolphins (\(y\)) to their age (\(x_1\)) and gender (\(x_2\)), considering curvature and interaction between age and gender. Therefore, the second-order model can be written as: $$ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1^2 + \beta_4 x_1 x_2 + \epsilon $$ where \(\beta_i\) are the regression coefficients and \(\epsilon\) is the error term.
02

Perform Multiple Regression Analysis

Using a computer software package like R, Python, Excel or any other statistical software, perform multiple regression analysis using the given data. Obtain the printout with the results, including the estimated regression coefficients, R-squared, adjusted R-squared and p-values.
03

Analyzing the Fit of the Model

Examine the printout obtained from the regression analysis to check the model's fit. Comment on the R-squared value, adjusted R-squared value, and the p-values of each predictor to assess the model's significance and contribution of each variable to the model.
04

Write Prediction Equations for Female and Male Dolphins

From the printout, and considering coefficients values, the prediction equation for female dolphins (\(x_2 = 0\)) can be written as: $$ y_f = \beta_0 + \beta_1 x_1 + \beta_3 x_1^2 $$ Similarly, the prediction equation for male dolphins (\(x_2 = 1\)) can be written as: $$ y_m = (\beta_0+\beta_2) + (\beta_1 + \beta_4) x_1 + \beta_3 x_1^2 $$ Evaluate the coefficients from the software printout and write the equations.
05

Assessing Significance of the Quadratic Term in Female's Prediction Equation

To assess the significance of the quadratic term in the prediction equation for females, check the p-value of the \(\beta_3\) coefficient (which represents the quadratic term) in the software printout. If the p-value is less than the chosen significance level (e.g., 0.05), then the quadratic term contributes significantly to the prediction of the mercury concentration in a female dolphin.
06

Assess Other Important Conclusions

Evaluate the printout further to identify any potential patterns, outlier effects, or areas where the model could be improved by adding or removing predictors. Check residuals, leverage, and influence measures to identify any influential points that may impact the model's predictions. Also, make sure to address any other scientific or biological factors that might affect the results, such as dolphin's habitat, feeding habits, or any other environmental factors.

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Most popular questions from this chapter

Refer to Exercise 13.12 . A command in the MINITAB regression menu provides output in which \(R^{2}\) and \(R^{2}\) (adj) are calculated for all possible subsets of the four independent variables. The printout is provided here. a. If you had to compare these models and choose the best one, which model would you choose? Explain. b. Comment on the usefulness of the model you chose in part a. Is your model valuable in predicting a taste score based on the chosen predictor variables?

A quality control engineer is interested in predicting the strength of particle board \(y\) as a function of the size of the particles \(x_{1}\) and two types of bonding compounds. If the basic response is expected to be a quadratic function of particle size, write a linear model that incorporates the qualitative variable "bonding compound" into the predictor equation.

Suppose \(E(y)\) is related to two predictor variables \(x_{1}\) and \(x_{2}\) by the equation $$ E(y)=3+x_{1}-2 x_{2}+x_{1} x_{2} $$ a. Graph the relationship between \(E(y)\) and \(x_{1}\) when \(x_{2}=0 .\) Repeat for \(x_{2}=2\) and for \(x_{2}=-2\) b. Repeat the instructions of part a for the model $$ E(y)=3+x_{1}-2 x_{2} $$ c. Note that the equation for part a is exactly the same as the equation in part \(b\) except that we have added the term \(x_{1} x_{2}\). How does the addition of the \(x_{1} x_{2}\) term affect the graphs of the three lines? d. What flexibility is added to the first-order model \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}\) by the addition of the term \(\beta_{3} x_{1} x_{2}\), using the model \(E(y)=\beta_{0}+\beta_{1} x_{1}+\) \(\beta_{2} x_{2}+\beta_{3} x_{1} x_{2} ?\)

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?

The tuna fish data from Exercise 11.16 were analyzed as a completely randomized design with four treatments. However, we could also view the experimental design as a \(2 \times 2\) factorial experiment with unequal replications. The data are shown below. \(^{9}\) The data can be analyzed using the model $$ y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}+\epsilon $$ where \(x_{1}=0\) if oil, 1 if water \(x_{2}=0\) if light tuna, 1 if white tuna a. Show how you would enter the data into a computer spreadsheet, entering the data into columns for \(y, x_{1}, x_{2},\) and \(x_{1} x_{2}\). b. The printout generated by MINITAB is shown below. What is the least-squares prediction equation? c. Is there an interaction between type of tuna and type of packing liquid? d. Which, if any, of the main effects (type of tuna and type of packing liquid) contribute significant information for the prediction of \(y ?\) e. How well does the model fit the data? Explain.
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