A student would like to know if the time spent on studying and number of lectures attended for an exam is related to the exam scores. The student now collects data on the number of lectures attended in the same period. The information obtained is tabulated and the output from a multiple regression analysis is shown below: State the estimated multiple regression equation.

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BlackTom AI · Accepted Answer

Let’s parse the question carefully to identify what is being estimated in the multiple regression output and how it should be written.
Option 1: 'Y=24.10+0.58X_1+5.24X_2, where Y is the exam score, X_1 is hours spent studying and X_2 is the number of lectures attended'. This uses Y (not hat Y) as the dependent variable, which is inconsistent with regression output where the predicted value is denoted by Ŷ (hat Y). The interpretation of coefficients would be the same, but the notation should reflect the estimated (predicted) value, not the observed Y.
Option 2: '\hat{Y}=24.10+0.58X_1+5.24X_2, where\hat{Y} is the estimated exam score, X_1 is hours spent studying and X_2 is the number of lectures attended'. This aligns with standard regression notation: Ŷ denotes the predicted (fitted) exam score, the intercept is 24.10, the coefficient for Hours spent studying (X_1) is 0.58, and the coefficient for Number of lectures attended (X_2) is 5.24. The description correctly identifies Ŷ as the estimated score and X_1, X_2 as the explanatory variables.
Option 3: 'Y=24.10+0.58X_1+5.24X_2, where Y is the estimated exam score, X_1 is hours spent studying and X_2 is the number of lectures attended'. Here Y is labeled as the estimated score, which is inconsistent with the standard notation that uses Ŷ for predicted values. Moreover, the same issue as in Option 1 arises regarding the dependent variable notation.
Option 4: '\hat{Y}=24.10+0.58X_1+5.24X_2, where\hat{Y} is the exam score, X_1 is hours spent studying and X_2 is the number of lectures attended'. This correctly uses Ŷ for the estimated score, but the trailing phrase 'where\hat{Y} is the exam score' is slightly inaccurate because Ŷ represents the predicted value of the exam score, not the actual observed score. The essential meaning is close, but the description should specify that Ŷ is the estimated (predicted) score, not just 'the exam score'.
In summary, the key factors are: proper use of Ŷ to denote the predicted value, correct intercept and coefficients, and precise variable descriptions. The option that faithfully mirrors the standard regression notation and matches the given estimated equation is the one that uses Ŷ for the predicted score and correctly defines X_1 and X_2 as hours studied and lectures attended, respectively.

ETX5900 - S2 2025 Take Home Quiz Week 10 (due on 17th of October)

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