Question: Chance in its Winter 2001 volume published a study of students who paid a private tutor (or coach) to help them improve their Scholastic Assessment Test (SAT) scores. Multiple regression was used to estimate the effect of coaching on SAT-Mathematics scores. Data on 3,492 students ( 573 of whom were coached) were used to fit the model \(\mu_{y}=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}\), where \(y\) corresponds to the SAT-Mathematics score, \(x_{1}=\) score on PSAT, and \(x_{2}=\{1\) if student was coached, 0 if not \(\}\).

1. Which scenario (coaching or no coaching) corresponds to the base level?
   \(\begin{array}{ll}\text { COACHING } & \text { NO COACHING }\end{array}\)
2. The estimate of \(\beta_{2}\) in the model was 19, with a standard error of $3 .$ Use this information to form a \(95 \%\) confidence interval for \(\beta_{2}\) assuming that \(t^{*}=1.962 .\) Interpret the interval.
3. Based on the above confidence interval, what can you say about the effect of coaching on SATMath scores? Explain your answer.
4. Assume the variable \(x_{2}\) was coded oppositely, i.e. \(x_{2}=\{0\) if student was coached, 1 if not \(\}\), how would the estimate of \(\beta_{2}\) change? What would be the value of the standard error? Does it matter in which way the dummy/indicator variable is coded? Briefly explain.
5. Assume that we find a significant interaction term between \(x_{1}\) the score on PSAT and \(x_{2}\) the indicator/dummy variable for coaching. The updated estimated least squares regression line is now given by
   \[\widehat{y}=850+1.5 x_{1}+19 x_{2}+4 x_{1} x_{2}\]
   What is the change in the predicted SAT Mathematics score for every additional point on the PSAT test for a student that received coaching? Assume that we use the coding of the dummy variable as described in the introduction of the question.
6. How does the previous answer change for a student that did not receive coaching?
7. A friend not having taken this statistics class but who is familiar with multiple linear regression asks you what is meant by the phrase an interaction between \(x_{1}\) the score on PSAT and \(x_{2}\) the indicator/dummy variable for coaching exists. Provide an explanation to your friend.

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