Update 04-ch4.Rmd

04-ch4.Rmd

@@ -70,7 +70,7 @@ The following code reproduces Figure 4.1 from the textbook.
 
 ```{r , echo=TRUE, fig.align='center', cache=TRUE}
 # create a scatterplot of the data
-plot(TestScore ~ STR,ylab="Test Score")
+plot(TestScore ~ STR,ylab="Test Score",pch=20)
 
 # add the systematic relationship to the plot
 abline(a = 713, b = -3)
@@ -744,7 +744,7 @@ curve(dnorm(x,
             -2, 
             sqrt(var_b0)), 
       add = T, 
-      col = "darkred")
+      col = "darkred",lwd=2)
 
 # plot histograms of beta_hat_1 
 hist(fit[, 2],
@@ -758,7 +758,7 @@ curve(dnorm(x,
             3.5, 
             sqrt(var_b1)), 
       add = T, 
-      col = "darkred")
+      col = "darkred",lwd=2)
 ```
 
 
@@ -820,24 +820,7 @@ and
 $$Cov(X,Y)=4.$$
 
 Formally, this is written down as
-
-\begin{equation}
-  \begin{pmatrix}
-    X \\
-    Y \\
-  \end{pmatrix}
-  \overset{i.i.d.}{\sim} \ \mathcal{N} 
-  \left[
-    \begin{pmatrix}
-      5 \\
-      5 \\
-    \end{pmatrix}, \ 
-    \begin{pmatrix}
-      5 & 4 \\
-      4 & 5 \\
-    \end{pmatrix}
-  \right]. \tag{4.3}
-\end{equation}
+$$\begin{pmatrix} X \\ Y  \end{pmatrix}\overset{i.i.d.}{\sim} \ \mathcal{N}\left[\begin{pmatrix} 5 \\ 5  \end{pmatrix}, \begin{pmatrix} 5 & 4 \\ 4 & 5 \end{pmatrix} \right].\tag{4.3} $$
 
 To carry out the random sampling, we make use of the function `r ttcode("mvrnorm()")` from the package `r ttcode("MASS")` [@R-MASS] which allows to draw random samples from multivariate normal distributions, see `?mvtnorm`. Next, we use `r ttcode("subset()")` to split the sample into two subsets such that the first set, `r ttcode("set1")`, consists of observations that fulfill the condition $\lvert X - \overline{X} \rvert > 1$ and the second set, `r ttcode("set2")`, includes the remainder of the sample. We then plot both sets and use different colors to distinguish the observations.
 
@@ -871,6 +854,12 @@ plot(set1,
 points(set2, 
        col = "steelblue", 
        pch = 19)
+legend("topleft", 
+       legend = c("Set1", 
+                  "Set2"),
+       cex = 1,
+       pch = 19,
+       col = c("black","steelblue"))
 ```
 
 
@@ -887,8 +876,15 @@ plot(set1, xlab = "X", ylab = "Y", pch = 19)
 points(set2, col = "steelblue", pch = 19)
 
 # add both lines to the plot
-abline(lm.set1, col = "green")
-abline(lm.set2, col = "red")
+abline(lm.set1, col = "black",lwd=2)
+abline(lm.set2, col = "steelblue",lwd=2)
+legend("bottomright", 
+       legend = c("Set1", 
+                  "Set2"),
+       cex = 1,
+       lwd=2,
+       col = c("black","steelblue"))
+
 ```