Postgraduate medical program
A top-ranked medical school has a 5-year postgraduate medical program designed for
students seeking to become fully-qualified doctors. As part of the application process,
applicants to the program are required to take an entrance exam. Applicants who are
successfully admitted to the program will also take a general exam at the end of each year
of the program. A random sample of 400 students that completed the program was selected
and each student’s scores in each of the six exams (Entrance, Year1, Year2, Year3, Year4
and Year5) were collated into a data set. In addition to the exam scores, the data set
includes a variable (Biology) that indicates whether each student majored in biology in
their undergraduate degree (1 if they did and 0 otherwise). The data are stored in the file
AssignmentData.RData in the data frame Q3.df. The director of the postgraduate medical
program would like to investigate any relationships that may exist between the entrance
exam scores and the scores in the exams taken during the program.
For this question, you will be analysing the scores for the entrance exam (Entrance) and the
exam taken at the end of year 5 of the program (Year5).
(a) [3 marks] Create a scatter plot of the end of year 5 exam scores against the en- trance
exam scores. Make sure to give your plot an appropriate title and appropriate labels for the
x and y axes. Describe the relationship between these two variables.
Answer parts (b) to (d) without using the lm function or any other R function designed to
fit, analyse or interpret regression models.
2. (b) [4 marks] Fit a simple linear regression model with the end of year 5 exam score
as the dependent variable and the entrance exam score as the independent variable.
Write down the estimated regression model.
3. (c) [6 marks] Discuss whether the assumptions for a simple linear regression model
hold for the model you fitted in part (b), making sure to provide clear justifica- tions
for your answer. Note that you will need to use R to manually calculate the residuals.
4. (d) [4 marks] For the model you fitted in part (b), test whether the coefficient
param- eter for entrance exam score is less than 0.1. Clearly state your hypotheses
and use a significance level of α = 5%. Do not use any R functions that are designed
to perform hypothesis tests.
The remaining parts of this question can be answered using the lm function.
(e) [2 marks] Create the scatter plot of part (a) again, but now colour all the points
corresponding to students who majored in biology in their undergraduate degree in red
and all the other points in black. Describe the relationship between the two variables for
students who majored in biology in their undergraduate degree.
(f) [3 marks] Considering only the students who majored in biology in their undergraduate degree, fit a simple linear regression model with the end of year 5 exam score as
the dependent variable and the entrance exam score as the independent variable. Write
down the estimated regression model. You may find it useful to create a new data frame
that only contains the relevant data.
(g) [5 marks] Discuss whether the assumptions for a simple linear regression model hold
for the model you fitted in part (f), making sure to provide clear justifications for your
answer.
(h) [4 marks] Using the model fitted in part (f), calculate a 95% prediction interval for the
end of year 5 exam score for a student that scored 95 on their entrance exam. Comment on
the accuracy of this prediction interval. Do not use any R functions that are designed to
calculate any predictions, confidence intervals or predictions intervals.
