Draw a scatter plot of the variable normexam against the variable standLRT from the data set Exam{mlmRev}. Superimpose on it another scatter plot using the means of the same two variables by school. You can start by editing this rmarkdown template in RStudio before submitting the output to moodle.

In this section, we load the exam scores data set, activate the help page for it, and examine the first 6 lines of the data frame object.

```
# install a contributed package
#install.packages("mlmRev")
```

```
# load the package to working directory
library(mlmRev)
```

```
# load the data from the package
data(Exam, package="mlmRev")
```

```
# invoke help document
?Exam
```

```
# view first 6 lines
head(Exam)
```

```
school normexam schgend schavg vr intake standLRT sex type student
1 1 0.26132 mixed 0.16618 mid 50% bottom 25% 0.61906 F Mxd 143
2 1 0.13407 mixed 0.16618 mid 50% mid 50% 0.20580 F Mxd 145
3 1 -1.72388 mixed 0.16618 mid 50% top 25% -1.36458 M Mxd 142
4 1 0.96759 mixed 0.16618 mid 50% mid 50% 0.20580 F Mxd 141
5 1 0.54434 mixed 0.16618 mid 50% mid 50% 0.37111 F Mxd 138
6 1 1.73490 mixed 0.16618 mid 50% bottom 25% 2.18944 M Mxd 155
```

We first compute the correlation coefficient between exam scores and LRT scores using all students in the data set. We then compute the mean exam scores and mean LRT scores by school. The correlation coefficient between mean school exam scores and mean school LRT scores is then calculated.

`with(Exam, cor(normexam, standLRT))`

`[1] 0.59165`

```
# compute the means by school
mLRT <- with(Exam, tapply(standLRT, school, mean))
mexam <- with(Exam, tapply(normexam, school, mean))
```

`cor(mexam, mLRT)`

`[1] 0.69241`

We draw a scatter diagram of the exam scores against the LRT scores and add the regression line. Next we superimpose on the scatter plot the mean school exam scores and mean school LRT scores (in color cyan) and add the regression line based on the mean school scores.

```
with(Exam, plot(normexam ~ standLRT,
bty = 'n',
cex = 0.5,
xlab = 'Standardized LR test score',
ylab = 'Normalized exam score'))
grid()
with(Exam, abline(lm(normexam ~ standLRT)))
points(mLRT, mexam, pch=16, col=5)
abline(lm(mexam ~ mLRT), col=5)
```