TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Bura)/Übungen 2020W/HW11.4

Regression (part 1)

A lecture was evaluated. In the file Evaluation.Rdata you find data of ${\textstyle n=25}$ students. Acquired were first the points achieved in the associated exercises (between 0 and 200 pos- sible), and second the result of the exam (in %). Can the result of the exam be explained by the points achieved in the exercises?

• a) Plot the result of the exam (${\textstyle y_{i}}$) against the exercise points (${\textstyle x_{i}}$). Do you observe a relation?
• b) Compute the intercept ${\textstyle b_{0}}$ and the slope ${\textstyle b_{1}}$ of the regression line (without lm()) and plot the regression line. Comment on the meaning of the slope.
• c) Would you say that the relation is causal? This means, that e.g., many points in the exercises are the reason for a good exam? (Think about the extreme case of a student who throughout cheated, copying all the correct solutions from colleagues, but never spent no time with any material...)

Lösungsvorschlag von Friday

--Friday Sa 30 Jan 2021 19:30:02 CET

1a)

load("Evaluation.Rdata")
x <- Evaluation$Uebungspunkte
y <- Evaluation$Klausurergebnis
plot(Evaluation,
    main="Evaluation",
    ylab="Exam",
    xlab="Exercise Points",
)

There seams to exist a realtion between the exercies points and the exam results. However, as the points are quite scattered and not even close to a straight line, one could asume that the error is quite large.

1b)

r <- cor(x,y)
sx <- sd(x)
sy <- sd(y)
mx <- mean(x)
my <- mean(y)
b1 <- r * (sy / sx)     # 0.2323655
b0 <- my - b1 * mx      # 42.93522
abline(b0, b1, col="coral1", lwd=1)

The positive slope for this dataset indicates that students that score better on their exercises also do better on their exams. In more detail, for every point you score on your exercises your exam improves by around ${\textstyle 0.2324}$ points.

1c)

Ofcourse, students who spend more time with their homework (asuming the do it themself) will have more practice and therefore are more likely to score better on an exam.

However, I would argue that that the relation is not causal as the two practises require quite a different set of skills. For an exercise you need to learn new concepts from the lecture/scripts to solve problems you haven’t solved before, for which you have quite a lot of time and can ask other people for help. For exams you need to solve familiar problems with a timelimit and compleatly on your own.

Moreover, lets asume the example of an exchange-student. While they can be quite good at the exercises as they can ask other students in the case the problems are not clear or maybe they do spend just more time understanding the problems. On an exam they neither have the option ask anyone nor the time to think long about each problem.