library(tidyverse)

Supervisor Performance Data \(( \ p = 6 \ )\)

Consider the Supervisor Performance Data in Table 3.3

table3.3 <- read_tsv("Table3.3.txt")


Estimate the regression coefficients vector \(\hat\beta\).

The value for the estimated regression coefficients are listed below each of their associated variables below:

fit3.3 <- lm(Y~., data = table3.3)
round(coefficients(fit3.3), 4)
## (Intercept)          X1          X2          X3          X4          X5 
##     10.7871      0.6132     -0.0731      0.3203      0.0817      0.0384 
##          X6 
##     -0.2171


Verify that \(\small \sum\limits_{i=1}^{n} \hat y_i = \sum\limits_{i=1}^{n} y_i\)

sum1 <- sum(fit3.3$fitted.values)
sum2 <- sum(table3.3$Y)

\(\sum\limits_{i=1}^{n} \hat y_i =\) 1939 and \(\sum\limits_{i=1}^{n} y_i =\) 1939

Predicting Final Exam Scores \((\ p = 2 \ )\)

Predict student scores on the final exam using student their scores on previous two exams (Table 3.10)

table3.10 <- read_tsv("Table3.10.txt")


Fit All Possible Models (with intercept)

fit_P1 <- lm(`F` ~ P1, data = table3.10)
coefP1 <- round(coefficients(fit_P1), 4)

fit_P2 <- lm(`F` ~ P2, data = table3.10)
coefP2 <- round(coefficients(fit_P2), 4)

fit_P1P2 <- lm(`F` ~., data = table3.10)
coefP1P2 <- round(coefficients(fit_P1P2), 4)

\(\underline{\text{Model 1:}} \hspace{0.25cm} \hat F =\) -22.3424 \(+\) 1.2605 \(P_1\)
\(\underline{\text{Model 2:}} \hspace{0.25cm} \hat F =\) -1.8535 \(+\) 1.0043 \(P_2\)
\(\underline{\text{Model 3:}} \hspace{0.25cm} \hat F =\) -14.5005 \(+\) 0.4883 \(P_1\) \(+\) 0.672 \(P_2\)


Test whether \(\beta_0 = 0\) in each of the three models.

I will use t-test hypothesis test for each model where \(H_0: \hat\beta_0 = 0\) and \(H_A: \hat\beta_0 \neq 0\)

Model 1 and Model 2 both have \(20\) degrees of freedom (d.o.f) and Model 3 has \(19\) d.o.f.
Using a significance level, \(\alpha = 0.05\), then the critical t-values for a two-tailed test are below:

\(t_{crit \ \left (\frac{\alpha}{2}, \ 20 \right )} = -2.086 \hspace{1.5cm} t_{crit \ \left (\frac{\alpha}{2}, \ 19 \right )} = -2.093\)

# Saving Model Summaries
sumP1 <- summary(fit_P1)
sumP2 <- summary(fit_P2)
sumP1P2 <- summary(fit_P1P2)
# Standard Errors
seP1B0 <- sumP1$coefficients[1,2]
seP2B0 <- sumP2$coefficients[1,2]
seP1P2B0 <- sumP1P2$coefficients[1,2]

\(t^* = \frac{\hat\beta_0}{s.e.(\hat\beta_0)}\)

\(\underline{\text{Model 1:}} \hspace{0.25cm} t^* = \frac{-22.342}{11.564} = -1.932\)
\(\underline{\text{Model 2:}} \hspace{0.25cm} t^* = \frac{-1.854}{7.562} = -0.245\)
\(\underline{\text{Model 3:}} \hspace{0.25cm} t^* = \frac{-14.501}{9.236} = -1.57\)

For all models \(|t^*| < |t_{crit}|\). As a result, we fail to reject the null hypothesis.


Which Predictor is Better? \(P_1\) or \(P_2\)? (Quick Model Selection)

To determine which predictor is better, I will select the predictor that yields the highest \(R^2\) values in SLR (Model 1 and Model 2) and RTO (new models).

# RTO: F ~ P1
fit_0P1 <- lm(`F` ~ 0 + P1, data = table3.10)
coef0P1 <- round(coefficients(fit_0P1), 4)
sum0P1 <- summary(fit_0P1)
# RTO: F ~ P2
fit_0P2 <- lm(`F` ~ 0 + P2, data = table3.10)
coef0P2 <- round(coefficients(fit_0P2), 4)
sum0P2 <- summary(fit_0P2)
# sumP1$r.squared
# sumP2$r.squared
# sum0P1$r.squared
# sum0P2$r.squared

Recall Previously Fitted Models:
\(\underline{\text{Model 1:}} \hspace{0.25cm} \hat F =\) -22.3424 \(+\) 1.2605 \(P_1\)
\(\underline{\text{Model 2:}} \hspace{0.25cm} \hat F =\) -1.8535 \(+\) 1.0043 \(P_2\)

New Fitted RTO Models:
\(\underline{\text{Model 1.1:}} \hspace{0.25cm} \hat F =\) 0.9913 \(P_1\)
\(\underline{\text{Model 2.1:}} \hspace{0.25cm} \hat F =\) 0.9822 \(P_2\)

\(\underline{\text{Model 1:}} \hspace{0.25cm} R^2 = 0.8023\)
\(\underline{\text{Model 2:}} \hspace{0.25cm} R^2 = 0.8600\)
\(\underline{\text{Model 1.1:}} \hspace{0.25cm} R^2 = 0.9959\)
\(\underline{\text{Model 2.1:}} \hspace{0.25cm} R^2 = 0.9975\)

\(\underline{\textbf{Conclusion:}} \hspace{0.25cm} P_2\) is a better predictor of \(F\) as it yields higher \(R^2\) than \(P_1\) in both of their respective RTO and SLR models.


Which Model would you use to predict \(F\)? (Quick Model Selection)

Which of the three models with intercepts would you use to predict the final examination scores for a student who scored 78 and 85 on the first and second preliminary examinations, respectively? What is your prediction in this case? 

# sumP1$adj.r.squared
# sumP2$adj.r.squared
# sumP1P2$adj.r.squared

Since Model 3 has two predictors, I will consider adjusted \(R^2\) values:

\(\underline{\text{Model 1:}} \hspace{0.25cm} R_{\textit{adj}}^2 = 0.7924\)
\(\underline{\text{Model 2:}} \hspace{0.25cm} R_{\textit{adj}}^2 = 0.8530\)
\(\underline{\text{Model 3:}} \hspace{0.25cm} R_{\textit{adj}}^2 = 0.8744\)

\(\underline{\textbf{Conclusion:}} \hspace{0.25cm}\) I would use Model 3: \(\hat F =\) -14.5005 + 0.4883 \(P_1\) + 0.672 \(P_2\)

If \(P_1 = 78\) and \(P_2 = 85\), Model 3 predicts \(F = 80.71\).