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Researchers developed a model to explain the age gap between husbands and wives at first marriage. The model is below: $$ \hat{y}=0.0321 x_{1}+0.9848 x_{2}+0.5391 x_{3}-0.000145 x_{4}^{2}+3.8483 $$ where y: Age gap at first marriage (male - female) \(x_{1}:\) Percent of children aged 10 to 14 involved in child labor \(x_{2}:\) Indicator variable where 1 is an African country, 0 otherwise \(x_{3}:\) Percent of the population that is Muslim \(x_{4}:\) Percent of the population that is literate Source: Xu Zhang and Solomon W. Polachek, State University of New York at Binghamton "The Husband- Wife Age Gap at First Marriage: A Cross-Country Analysis" (a) Use the model to predict the age gap at first marriage for an African country where the percent of children aged 10 to 14 who are involved in child labor is \(12,\) the percent of the population that is Muslim is \(30,\) and the percent of the population that is literate is \(75 .\) (b) What would be the mean difference in age gap between an African country and a non-African country? (c) Interpret the coefficient of "percent of children aged 10 to 14 involved in child labor." (d) The coefficient of determination for this model is 0.593 . Interpret this result. (e) The \(P\) -value for the test \(H_{1}: \beta_{1} \neq 0\) versus \(H_{1}: \beta_{1} \neq 0\) is \(0.008 .\) What would you conclude about this test?

Putting It Together: Purchasing Diamonds The value of a diamond is determined by the four C's: carat weight, color, clarity, and cut. Carat weight is the standard measure for the size of a diamond. Generally, the more a diamond weighs, the more valuable it will be. The Gemological Institute of America (GIA) determines the color of diamonds using a 22 -grade scale from D (almost clear white) to \(Z\) (light yellow). Colorless diamonds are generally considered the most desirable. The clarity of a diamond refers to how "free" the diamond is of imperfections and is determined using an 11 -grade scale: flawless (FL), internally flawless (IF), very, very slightly imperfect (VVS1, VVS2), very slightly imperfect (VS1, VS2), slightly imperfect (SI1,SI2), and imperfect (I1, I2, I3). The cut of a diamond refers to the diamond's proportions and finish. Put simply, the better the diamond's cut is, the better it reflects and refracts light, which makes it more beautiful and thus more valuable. The cut of a diamond is rated using a five-grade scale: Excellent, Very Good, Good, Fair, and Poor. Finally, the shape of a diamond (which is not one of the four C's) refers to its basic form: round, oval, pear-shaped, marquis, and so on. A novice might confuse shape with cut, so be careful not to confuse the two. Go to www.pearsonhighered.com/sullivanstats to obtain the data file \(14_{-} 6_{-} 8\) using the file format of your choice for the version of the text you are using. The data represent a random sample of 40 unmounted, round-shaped diamonds. Use the data to answer the questions that follow: (a) Determine the level of measurement for each variable. (i) Carat weight (iv) Cut (ii) Color (v) Price (iii) Clarity (vi) Shape (b) Construct a correlation matrix. To do so, first convert the variables color, clarity, and cut to numeric values as follows: Color: \(\mathrm{D}=1, \mathrm{E}=2, \mathrm{~F}=3, \mathrm{G}=4, \mathrm{H}=5, \mathrm{I}=6, \mathrm{~J}=7\) Clarity: \(\mathrm{FL}=1, \mathrm{IF}=2, \mathrm{VVS} 1=3, \mathrm{VVS} 2=4, \mathrm{VS} 1=5\) \(\mathrm{VS} 2=6, \mathrm{SI} 1=7, \mathrm{SI} 2=8\) Cut: Excellent \(=1,\) Very Good \(=2,\) Good \(=3\) If price is to be the response variable in our model, is there reason to be concerned about multicollinearity? Explain. (c) Find the "best" model for predicting the price of a diamond. (d) Draw residual plots, a boxplot of the residuals, and a normal probability plot of the residuals to assess the adequacy of the "best" model. (e) For the "best" model, interpret each regression coefficient. (f) Determine and interpret \(R^{2}\) and the adjusted \(R^{2}\). (g) Predict the mean price of a round-shaped diamond with the following characteristics: 0.85 carat, E, VVS1, Excellent. (h) Construct a \(95 \%\) confidence interval for the mean price found in part (g). (i) Predict the price of an individual round-shaped diamond with the following characteristics: 0.85 carat, E, VVS1 Excellent. (j) Construct a \(95 \%\) prediction interval for the price found in \(\operatorname{part}(\mathrm{i})\) (k) Explain why the predictions in parts \((\mathrm{g})\) and (i) are the same, yet the intervals in parts \((\mathrm{h})\) and \((\mathrm{j})\) are different.

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 \(x_{4}=69 .\)

A researcher wants to determine a model that can be used to predict the 28 -day strength of a concrete mixture. The following data represent the 28 -day and 7 -day strength (in pounds per square inch) of a certain type of concrete along with the concrete's slump. Slump is a measure of the uniformity of the concrete, with a higher slump indicating a less uniform mixture. $$ \begin{array}{ccc} \text { Slump (inches) } & \text { 7-Day psi } & \text { 28-Day psi } \\ \hline 4.5 & 2330 & 4025 \\ \hline 4.25 & 2640 & 4535 \\\ \hline 3 & 3360 & 4985 \\ \hline 4 & 1770 & 3890 \\ \hline 3.75 & 2590 & 3810 \\ \hline 2.5 & 3080 & 4685 \\ \hline 4 & 2050 & 3765 \\ \hline 5 & 2220 & 3350 \\ \hline 4.5 & 2240 & 3610 \\ \hline 5 & 2510 & 3875 \\ \hline 2.5 & 2250 & 4475 \end{array} $$ (a) Construct a correlation matrix between slump, 7 -day psi, and 28 -day psi. Is there any reason to be concerned with multicollinearity based on the correlation matrix? (b) Find the least-squares regression equation \(\hat{y}=b_{0}+b_{1} x_{1}+b_{2} x_{2},\) where \(x_{1}\) is slump, \(x_{2}\) is 7 -day strength, and \(y\) is the response variable, 28 -day strength. (c) Draw residual plots and a boxplot of the residuals to assess the adequacy of the model. (d) Interpret the regression coefficients for the least-squares regression equation. (e) Determine and interpret \(R^{2}\) and the adjusted \(R^{2}\). (f) Test \(H_{0}: \beta_{1}=\beta_{2}=0\) versus \(H_{1}:\) at least one of the \(\beta_{1} \neq 0\) at the \(\alpha=0.05\) level of significance. (g) Test the hypotheses \(H_{0}: \beta_{1}=0\) versus \(H_{1}: \beta_{1} \neq 0\) and \(H_{0}: \beta_{2}=0\) versus \(H_{1}: \beta_{2} \neq 0\) at the \(\alpha=0.05\) level of significance. (h) Predict the mean 28 -day strength of all concrete for which slump is 3.5 inches and 7 -day strength is 2450 psi. (i) Predict the 28 -day strength of a specific sample of concrete for which slump is 3.5 inches and 7 -day strength is 2450 psi. (j) Construct \(95 \%\) confidence and prediction intervals for concrete for which slump is 3.5 inches and 7 -day strength is 2450 psi. Interpret the results.

Kepler's Law of Planetary Motion The time it takes for a planet to complete its orbit around the sun is called the planet's sidereal year. Johann Kepler studied the relation between the sidereal year of a planet and its distance from the sun in 1618 . The following data show the distances that the planets are from the sun and their sidereal years. $$ \begin{array}{lcc} \text { Planet } & \begin{array}{l} \text { Distance from Sun, } x \\ \text { (millions of miles) } \end{array} & \text { Sidereal Year, } \boldsymbol{y} \\ \hline \text { Mercury } & 36 & 0.24 \\ \hline \text { Venus } & 67 & 0.62 \\ \hline \text { Earth } & 93 & 1.00 \\ \hline \text { Mars } & 142 & 1.88 \\ \hline \text { Jupiter } & 483 & 11.9 \\ \hline \text { Saturn } & 887 & 29.5 \\ \hline \text { Uranus } & 1785 & 84.0 \\ \hline \text { Neptune } & 2797 & 165.0 \\ \hline \text { Pluto* } & 3675 & 248.0 \end{array} $$ (a) Determine the least-squares regression equation, treating distance from the sun as the explanatory variable. (b) A normal probability plot of the residuals indicates that the residuals are approximately normally distributed. Test whether a linear relation exists between distance from the sun and sidereal year. (c) Draw a scatter diagram, treating distance from the sun as the explanatory variable. (d) Plot the residuals against the explanatory variable, distance from the sun. (e) Does a linear model seem appropriate based on the scatter diagram and residual plot? (Hint: See Section 4.3.) (f) What is the moral?