Answer Happy

We ran multiple linear regression analysis against students’
test score data.
The basic idea is to estimate SAT scores based on sample exams
of several subjects (Math, Physics, English, German, and
Music).
The results are shown in Model 1.
When interpreting the results, we were a bit confused because
Physics and German variables’ coefficients were negative.
We excluded Physics variable and tested it again.
This time, the outcome was much more
understandable.
What was the potential problem with Model 1 case?
Model 1: sat = math + physics + english + german + music
t Coefficients : Estimate Std. Error t value Pr>t] (Intercept) -28.25344 1.52769-18.49 0.00000000000013 *** math 1.02369 0.03024 33.85 < 0.0000000000000002 *** physics -0.13107 0.03162 -4.15 0.00055 *** english 0.54094 0.02939 18.41 0.00000000000014 *** german -0.05214 0.04294 -1.21 0.23948 music 0.18924 0.00552 34.27 < 0.0000000000000002 *** + --- Signif. codes: O ****' 0.001 ***' 0.01 '*' 0.05 '.' 0.1'' 1- Residual standard error: 0.32 on 19 degrees of freedom- Multiple R-squared: Adjusted R-squared: 1 + F-statistic: 2.12e+04 on 5 and 19 DF, p-value: <0.0000000000000002- 19 + Model 2: sat = math + english + german + music Coefficients: Estimate Std. Error t value Pr(>1t1 (Intercept) -34.24069 0.66916 -51.17 < 0.0000000000000002 **** math 0.89928 0.00497 180.80 < 0.0000000000000002 *** english 0.43398 0.01892 22.94 0.00000000000000078 *** german 0.11522 0.01965 5.86 0.00000978071071254 *** music 0.20592 0.00509 40.48 < 0.0000000000000002 *** + --- Signif. codes: 0 ****' 0.001 "**' 0.01 6 6 *' 0.05 '.' 0.1 !1+ Residual standard error: 0.431 on 20 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 + F-statistic: 1.46e+04 on 4 and 20 DF, p-value: <0.0000000000000002