You have been asked by your younger sister to help her with a science fair project. During the previous years she already studied why objects float and there also was the inevitable volcano project. Having learned regression techniques recently, you suggest that she investigate the weight-height relationship of 4^th to 6^th graders. Her presentation topic will be to explain how people at carnivals predict weight. You collect data for roughly 100 boys and girls between the ages of nine and twelve and estimate for her the following relationship:

\begin{array} { c } \widehat { \text { Weight } } = 45.59 + 4.32 \times \text { Height } ^ { 4 } , R ^ { 2 } = 0.55 , \text { SER } = 15.69 \\( 3.81 ) \quad ( 0.46 )\end{array}

where Weight is in pounds, and Height4 is inches above 4 feet.
(a)Interpret the results.
(b)You remember from the medical literature that females in the adult population are, on average, shorter than males and weigh less. You also seem to have heard that females, controlling for height, are supposed to weigh less than males. To see if this relationship holds for children, you add a binary variable (DFY)that takes on the value one for girls and is zero otherwise. You estimate the following regression function:

\begin{array} { l } \widehat { \text { Weight } } = 36.27 + 17.33 \times \text { DFY } + 5.32 \times \text { Height } 4 - 1.83 \times ( \text { DFY } \times \text { Height } 4 ) \\\quad\quad\quad\quad( 5.99 ) \quad ( 7.36 ) \quad\quad\quad(0.80)\quad\quad\quad\quad(0.90)\\\end{array}

R ^ { 2 } =0.58 , S E R = 15.41

Are the signs on the new coefficients as expected? Are the new coefficients individually statistically significant? Write down and sketch the regression function for boys and girls separately.
(c)The medical literature provides you with the following information for median height and weight of nine- to twelve-year-olds:
Median Height and Weight for Children, Age 9-12

\begin{array} { | l | c | c | c | c | } \hline & \text { Boys' Weight } & \text { Boys' Height } & \text { Girls' Weight } & \text { Girls' Height } \\\hline \text { 9-year-old } & 60 & 52 & 60 & 49 \\\hline \text { 10-year-old } & 70 & 54 & 70 & 52 \\\hline \text { 11-year-old } & 77 & 56 & 80 & 57 \\\hline \text { 12-year-old } & 87 & 58.5 & 92 & 60 \\\hline\end{array}

Insert two height/weight measures each for boys and girls and see how accurate your predictions are.
(d)The F-statistic for testing that the intercept and slope for boys and girls are identical is 2.92. Find the critical values at the 5% and 1% level, and make a decision. Allowing for a different intercept with an identical slope results in a t-statistic for DFY of (-0.35). Having identical intercepts but different slopes gives a t-statistic on (DFYHeight4)of (-0.35)also. Does this affect your previous conclusion?
(e)Assume that you also wanted to test if the relationship changes by age. Briefly outline how you would specify the regression including the gender binary variable and an age binary variable (Older)that takes on a value of one for eleven to twelve year olds and is zero otherwise. Indicate in a table of two rows and two columns how the estimated relationship would vary between younger girls, older girls, younger boys, and older boys.

The Interpretation of the Slope Coefficient in the Model Yi

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The Interpretation of the Slope Coefficient in the Model Y_i

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