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9.3 Baby weights, Part III. We considered the variables smoke and parity, one at a time, in modeling birth weights of babies in Exercises 9.1 and \(9.2 .\) A more realistic approach to modeling infant weights is to consider all possibly related variables at once. Other variables of interest include length of pregnancy in days (gestation), mother's age in years (age), mother's height in inches (height), and mother's pregnancy weight in pounds (weight). Below are three observations from this data set. $$ \begin{array}{rccccccc} \hline & \text { bwt } & \text { gestation } & \text { parity } & \text { age } & \text { height } & \text { weight } & \text { smoke } \\ \hline 1 & 120 & 284 & 0 & 27 & 62 & 100 & 0 \\ 2 & 113 & 282 & 0 & 33 & 64 & 135 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1236 & 117 & 297 & 0 & 38 & 65 & 129 & 0 \\ \hline \end{array} $$ The summary table below shows the results of a regression model for predicting the average birth weight of babies based on all of the variables included in the data set. $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -80.41 & 14.35 & -5.60 & 0.0000 \\ \text { gestation } & 0.44 & 0.03 & 15.26 & 0.0000 \\ \text { parity } & -3.33 & 1.13 & -2.95 & 0.0033 \\ \text { age } & -0.01 & 0.09 & -0.10 & 0.9170 \\ \text { height } & 1.15 & 0.21 & 5.63 & 0.0000 \\ \text { weight } & 0.05 & 0.03 & 1.99 & 0.0471 \\ \text { smoke } & -8.40 & 0.95 & -8.81 & 0.0000 \\ \hline \end{array} $$ (a) Write the equation of the regression model that includes all of the variables. (b) Interpret the slopes of gestation and age in this context. (c) The coefficient for parity is different than in the linear model shown in Exercise 9.2 . Why might there be a difference? (d) Calculate the residual for the first observation in the data set. (e) The variance of the residuals is \(249.28,\) and the variance of the birth weights of all babies in the data set is 332.57. Calculate the \(R^{2}\) and the adjusted \(R^{2}\). Note that there are 1,236 observations in the data set.

A survey of 55 Duke University students asked about their GPA, number of hours they study at night, number of nights they go out, and their gender. Summary output of the regression model is shown below. Note that male is coded as \(1 .\) $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & 3.45 & 0.35 & 9.85 & 0.00 \\ \text { studyweek } & 0.00 & 0.00 & 0.27 & 0.79 \\ \text { sleepnight } & 0.01 & 0.05 & 0.11 & 0.91 \\ \text { outnight } & 0.05 & 0.05 & 1.01 & 0.32 \\ \text { gender } & -0.08 & 0.12 & -0.68 & 0.50 \\ \hline \end{array} $$ (a) Calculate a \(95 \%\) confidence interval for the coefficient of gender in the model, and interpret it in the context of the data. (b) Would you expect a \(95 \%\) confidence interval for the slope of the remaining variables to include \(0 ?\) Explain

Logistic regression fact checking. Determine which of the following statements are true and false. For each statement that is false, explain why it is false. (a) Suppose we consider the first two observations based on a logistic regression model, where the first variable in observation 1 takes a value of \(x_{1}=6\) and observation 2 has \(x_{1}=4\). Suppose we realized we made an error for these two observations, and the first observation was actually \(x_{1}=7\) (instead of 6 ) and the second observation actually had \(x_{1}=5\) (instead of \(\left.4\right) .\) Then the predicted probability from the logistic regression model would increase the same amount for each observation after we correct these variables. (b) When using a logistic regression model, it is impossible for the model to predict a probability that is negative or a probability that is greater than \(1 .\) (c) Because logistic regression predicts probabilities of outcomes, observations used to build a logistic regression model need not be independent. (d) When fitting logistic regression, we typically complete model selection using adjusted \(R^{2}\).