During new product development, companies often involve “lead users,” i.e., creative individuals who are on the leading edge of an important market trend. Creativity and Innovation Management (Feb. 2008) published an article on identifying the social network characteristics of lead users of children’s computer games. Data were collected for n = 326 children and the following variables measured: lead-user rating ( y , measured on a 5-point scale), gender (

$$x _ { 1 } = 1$$

if female, 0 if male), age

$$\left( x _ { 2 } , \text { years } \right)$$

degree of centrality (

$$x _ { 3 }$$

, measured as the number of direct ties to other peers in the network), and betweenness centrality (

$$x_4$$

, measured as the number of shortest paths between peers). A first-order model for y was fit to the data, yielding the following least squares prediction equation:

$$\hat { y } = 3.58 + .01 x _ { 1 } - .06 x _ { 2 } - .01 x _ { 3 } + .42 x _ { 4 }$$

Give a practical interpretation of the estimate of

$$\beta _ { 4 }$$

in the model.

During new product development, companies often involve “lead users,” i.e., creative individuals who are on the leading edge of an important market trend. Creativity and Innovation Management (Feb. 2008) published an article on identifying the social network characteristics of lead users of children’s computer games. Data were collected for n = 326 children and the following variables measured: lead-user rating ( y , measured on a 5-point scale), gender (

$$x _ { 1 } = 1$$

if female, 0 if male), age

$$\left( x _ { 2 } , \text { years } \right)$$

degree of centrality (

$$x _ { 3 }$$

, measured as the number of direct ties to other peers in the network), and betweenness centrality (

$$x_4$$

, measured as the number of shortest paths between peers). A first-order model for y was fit to the data, yielding the following least squares prediction equation:

$$\hat { y } = 3.58 + .01 x _ { 1 } - .06 x _ { 2 } - .01 x _ { 3 } + .42 x _ { 4 }$$

Give two properties of the errors of prediction that result from using the method of least squares to obtain the parameter estimates.

Accuracy of software effort estimates. Refer to the Journal of Empirical Software Engineering (Vol. 9, 2004) study of the accuracy of new software effort estimates, Exercise 12.114 (p. 757). Recall that stepwise regression was used to develop a model for the relative error in estimating effort $(y)$ as a function of company role of estimator $\left(x_1=1\right.$ if developer, 0 if project leader $)$ and previous accuracy $\left(x_8=1\right.$ if more than $20 \%$ accurate, 0 if less than $20 \%$ accurate). The stepwise regression yielded the prediction equation $\hat{y}=.12-.28 x_1+.27 x_8$. The researcher is concerned that the sign of the estimated $\beta$ multiplied by $x_1$ is the opposite from what is expected. (The researcher expects a project leader to have a smaller relative error of estimation than a developer.) Give at least one reason why this phenomenon occurred.

Aluminum scraps that are recycled into alloys are classified into three categories: soft-drink cans, pots and pans, and automobile crank chambers. A study of how these three materials affect the metal elements present in aluminum alloys was published in Advances in Applied Physics (Vol. 1, 2013). Data on 126 production runs at an aluminum plant were used to model the percentage (y) of various elements ( e.g., silver, boron, iron) that make up the aluminum alloy. Three independent variables were used in the model: $x_1$ = proportion of aluminum scraps from cans,

$$\begin{matrix} \text{Element (y)} & \ {\beta_{0}} & \ {\beta_{1}} & \ {\beta_{2}} & \ {\beta_{3}} & \text{F p-value} & \ {R^2}\\ \text{Silver} & \text{.0020} & \text{-.0000141} & \text{-.0000123} & \text{- .0000357} & \text{.049} & \text{.075}\\ \text{ } & \text{(.001)} & \text{(.015)} & \text{(.035)} & \text{(.009)} & \text{ } & \text{ }\\ \text{Iron} & \text{-5.958} & \text{.079} & \text{.084} & \text{.054} & \text{<.001} & \text{.783}\\ \text{ } & \text{(<.001)} & \text{(< .001)} & \text{( < .001)} & \text{(< .001)} & \text{ } & \text{ }\\ \end{matrix}$$

$x_2$ = proportion of aluminum scraps from pots/pans, and $x_3$ = proportion of aluminum scraps from crank chambers. The first-order model, $E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}$, was fit to the data for several elements. The estimates of the model parameters (p-values in parentheses) for silver and iron are shown in the table above. Based on the parameter estimates, sketch the relationship between percentage of iron (y) and proportion of aluminum scraps from cans ($x_1$). Conduct a test to determine if this relationship is statistically significant at $\alpha=.05$

Aluminum scraps that are recycled into alloys are classified into three categories: soft-drink cans, pots and pans, and automobile crank chambers. A study of how these three materials affect the metal elements present in aluminum alloys was published in Advances in Applied Physics (Vol. 1, 2013). Data on 126 production runs at an aluminum plant were used to model the percentage (y) of various elements ( e.g., silver, boron, iron) that make up the aluminum alloy. Three independent variables were used in the model: $x_1$ = proportion of aluminum scraps from cans,

$$\begin{matrix} \text{Element (y)} & \ {\beta_{0}} & \ {\beta_{1}} & \ {\beta_{2}} & \ {\beta_{3}} & \text{F p-value} & \ {R^2}\\ \text{Silver} & \text{.0020} & \text{-.0000141} & \text{-.0000123} & \text{- .0000357} & \text{.049} & \text{.075}\\ \text{ } & \text{(.001)} & \text{(.015)} & \text{(.035)} & \text{(.009)} & \text{ } & \text{ }\\ \text{Iron} & \text{-5.958} & \text{.079} & \text{.084} & \text{.054} & \text{<.001} & \text{.783}\\ \text{ } & \text{(<.001)} & \text{(< .001)} & \text{( < .001)} & \text{(< .001)} & \text{ } & \text{ }\\ \end{matrix}$$

$x_2$ = proportion of aluminum scraps from pots/pans, and $x_3$ = proportion of aluminum scraps from crank chambers. The first-order model, $E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}$, was fit to the data for several elements. The estimates of the model parameters (p-values in parentheses) for silver and iron are shown in the table above. Based on the parameter estimates, sketch the relationship between percentage of silver (y) and proportion of aluminum scraps from cans ($x_1$). Conduct a test to determine if this relationship is statistically significant at $\alpha=.05$.

Refer to the National Semiconductor study of manufactured silicon wafer integrated circuit chips, Exercise $12.63$ (p. 749). Recall that the failure times of the microchips (in hours) were decided at different solder temperatures (degrees Celsius). The data are represented in the table below.

b. Calculate the residual for a microchip manufactured at a temperature of $149^{\circ} \mathrm{C}$.

Aluminum scraps that are recycled into alloys are classified into three categories: soft-drink cans, pots and pans, and automobile crank chambers. A study of how these three materials affect the metal elements present in aluminum alloys was published in Advances in Applied Physics (Vol. 1, 2013). Data on 126 production runs at an aluminum plant were used to model the percentage (y) of various elements ( e.g., silver, boron, iron) that make up the aluminum alloy. Three independent variables were used in the model: $x_1$ = proportion of aluminum scraps from cans,

$$\begin{matrix} \text{Element (y)} & \ {\beta_{0}} & \ {\beta_{1}} & \ {\beta_{2}} & \ {\beta_{3}} & \text{F p-value} & \ {R^2}\\ \text{Silver} & \text{.0020} & \text{-.0000141} & \text{-.0000123} & \text{- .0000357} & \text{.049} & \text{.075}\\ \text{ } & \text{(.001)} & \text{(.015)} & \text{(.035)} & \text{(.009)} & \text{ } & \text{ }\\ \text{Iron} & \text{-5.958} & \text{.079} & \text{.084} & \text{.054} & \text{<.001} & \text{.783}\\ \text{ } & \text{(<.001)} & \text{(< .001)} & \text{( < .001)} & \text{(< .001)} & \text{ } & \text{ }\\ \end{matrix}$$

$x_2$ = proportion of aluminum scraps from pots/pans, and $x_3$ = proportion of aluminum scraps from crank chambers. The first-order model, $E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}$, was fit to the data for several elements. The estimates of the model parameters (p-values in parentheses) for silver and iron are shown in the table above. Is the overall model statistically useful (at $\alpha=.05$) for predicting the percentage of iron in the alloy? If so, give a practical interpretation of $R^2$.

Aluminum scraps that are recycled into alloys are classified into three categories: soft-drink cans, pots and pans, and automobile crank chambers. A study of how these three materials affect the metal elements present in aluminum alloys was published in Advances in Applied Physics (Vol. 1, 2013). Data on 126 production runs at an aluminum plant were used to model the percentage (y) of various elements ( e.g., silver, boron, iron) that make up the aluminum alloy. Three independent variables were used in the model: $x_1$ = proportion of aluminum scraps from cans,

$$\begin{matrix} \text{Element (y)} & \ {\beta_{0}} & \ {\beta_{1}} & \ {\beta_{2}} & \ {\beta_{3}} & \text{F p-value} & \ {R^2}\\ \text{Silver} & \text{.0020} & \text{-.0000141} & \text{-.0000123} & \text{- .0000357} & \text{.049} & \text{.075}\\ \text{ } & \text{(.001)} & \text{(.015)} & \text{(.035)} & \text{(.009)} & \text{ } & \text{ }\\ \text{Iron} & \text{-5.958} & \text{.079} & \text{.084} & \text{.054} & \text{<.001} & \text{.783}\\ \text{ } & \text{(<.001)} & \text{(< .001)} & \text{( < .001)} & \text{(< .001)} & \text{ } & \text{ }\\ \end{matrix}$$

$x_2$ = proportion of aluminum scraps from pots/pans, and $x_3$ = proportion of aluminum scraps from crank chambers. The first-order model, $E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}$, was fit to the data for several elements. The estimates of the model parameters (p-values in parentheses) for silver and iron are shown in the table above. Is the overall model statistically useful (at $\alpha=.05$) for predicting the percentage of silver in the alloy? If so, give a practical interpretation of $R^2$.

Refer to the Body Image: An International Journal of Research (March 2010) analysis of the influence of reality TV shows on one's desire to undergo cosmetic surgery, Exercise $12.17$ (p. 725). Simulated data for the analysis are saved in the file. In Exercise 12.17, you fit the first order model, $E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{4}$, where $y=$ desire to have cosmetic surgery, $x_{1}$ is a dummy variable for gender, $x_{2}=$ level of self-esteem, $x_{3}=$ level of body satisfaction, and $x_{4}=$ impression of reality TV.

b. Execute a complete residual analysis for the model. Do you detect any violations of the assumptions? If so, what modifications to the model do you suggest?

Reality TV and cosmetic surgery. Refer to the Body Image: An International Journal of Research (March 2010) study of the influence of reality TV shows on one’s desire to undergo cosmetic surgery, Exercise 12.17 (p. 725). Simulated data for the study are saved in the file. In Exercise 12.17, you fit the first-order model, $E_1y_2 =\beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_4$, where $y$ = desire to have cosmetic surgery, $x_1$ is a dummy variable for gender, $x_2$ = level of self-esteem, $x_3$ = level of body satisfaction, and $x_4$ = impression of reality TV.

a. Review the data for multicollinearity. If you notice multicollinearity, what modifications to the model do you suggest?

Refer to the Journal of Applied Psychology (Jan. 2011) study of the determinants of task performance. In addition to $x_1$ = conscientiousness score and $x_2$ = {1 if highly complex job, 0 if not}, the researchers also used $x_3$ = emotional stability score, $x_4$ = organizational citizenship behavior score, and $x_5$ = counterproductive work behavior score to model y = task performance score. One of their concerns is the level of multicollinearity in the data. A matrix of correlations for all possible pairs of independent variables follows. Based on this information, do you detect a moderate or high level of multicollinearity? If so, what are your recommendations?

$$\begin{matrix} \text{ } & \ {x_1} & \ {x_2} & \ {x_3} & \ {x_4}\\ \ {Conscientiousness \ (x_1)} & \text{ } & \text{ } & \text{ } & \text{ }\\ \ {Job \ Complexity \ (x_2)} & \text{.13} & \text{ } & \text{ } & \text{ }\\ \ {Emotional \ Stability \ (x_3)} & \text{.62} & \text{.14} & \text{ } & \text{ }\\ \ {Organizational \ Citizenship \ (x_4)} & \text{.24} & \text{.03} & \text{.24} & \text{ }\\ \ {Counterproductive \ Work \ (x_5)} & \text{-.23} & \text{-.23} & \text{-.02} & \text{-.62}\\ \end{matrix}$$

Some states permit only licensed firms to sell funeral goods (e.g., caskets, urns) to the consumer, while other states have no restrictions. States with casket sales restrictions are being challenged in court to lift these monopolistic restrictions. A paper in the Journal of Law and Economics (Feb. 2008) used multiple regression to investigate the impact of lifting casket sales restrictions on the cost of a funeral. Data collected for a sample of 1,437 funerals were used to fit the model. A simpler version of the model estimated by the researchers is $E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}$, where y is the price (in dollars) of a direct burial, $x_1$ = {1 if funeral home is in a restricted state, 0 if not} , and x2 = { 1 if price includes a basic wooden casket, 0 if no casket}. The estimated equation (with standard errors in parentheses) is

$$\begin{matrix} \ {\hat{y}} & \text{ =} & \text{1,432} & \text{+} & \ {793x_1} & \text{-} & \ {252x_2} & \text{+} & \ {261x_1x_2,} & \ {R^2} & \text{=} & \text{.78}\\ \text{ } & \text{ } & \text{ } & \text{ } & \text{(70)} & \text{ } & \text{(134)} & \text{ } & \text{(109)} & \text{ } & \text{ } & \text{ }\\ \end{matrix}$$

The data also include a direct burial funeral with a basic wooden casket at a funeral home in a restricted state that costs $2,500. Again, assume that the standard deviation of the model is$50. Is this data value an outlier?

Some states permit only licensed firms to sell funeral goods (e.g., caskets, urns) to the consumer, while other states have no restrictions. States with casket sales restrictions are being challenged in court to lift these monopolistic restrictions. A paper in the Journal of Law and Economics (Feb. 2008) used multiple regression to investigate the impact of lifting casket sales restrictions on the cost of a funeral. Data collected for a sample of 1,437 funerals were used to fit the model. A simpler version of the model estimated by the researchers is $E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}$, where y is the price (in dollars) of a direct burial, $x_1$ = {1 if funeral home is in a restricted state, 0 if not} , and x2 = { 1 if price includes a basic wooden casket, 0 if no casket}. The estimated equation (with standard errors in parentheses) is

$$\begin{matrix} \ {\hat{y}} & \text{ =} & \text{1,432} & \text{+} & \ {793x_1} & \text{-} & \ {252x_2} & \text{+} & \ {261x_1x_2,} & \ {R^2} & \text{=} & \text{.78}\\ \text{ } & \text{ } & \text{ } & \text{ } & \text{(70)} & \text{ } & \text{(134)} & \text{ } & \text{(109)} & \text{ } & \text{ } & \text{ }\\ \end{matrix}$$

The data include a direct burial funeral with a basic wooden casket at a funeral home in a restricted state that costs $2,200. Assuming the standard deviation of the model is$50, is this data value an outlier?

Define extrapolation.