Your report should include the R code that you use. For calculations, please add enough explanation to help the reader understand what you did and why. Recall that you are permitted to collaborate, or to use any internet resources, but you must list all sources that influenced your report. As usual, a statement of Sources is required to get credit for the homework.

Q1. Making and interpreting an F test

We consider again the data on freshman GPA, ACT exam scores and percentile ranking of each student within their high school for 705 students at a large state university. In addition, we now consider the year (1996 to 2000) in which the student entered college.

download.file(destfile="gpa.txt",
  url="https://ionides.github.io/401f18/hw/hw10/gpa.txt")
gpa <- read.table("gpa.txt",header=T)
head(gpa)
##   ID  GPA High_School ACT Year
## 1  1 0.98          61  20 1996
## 2  2 1.13          84  20 1996
## 3  3 1.25          74  19 1996
## 4  4 1.32          95  23 1996
## 5  5 1.48          77  28 1996
## 6  6 1.57          47  23 1996

The predictive relationship between freshman GPA and the admission scores may, or may not, be stable over time. Intercepts and/or slopes of fitted lines may be approximately constant, or may show trends through time. The director of admissions is interested in what ways, if any, there is evidence for changes larger than can be explained by chance variation. Let the null hypothesis, \(H_0\), be a probability model for which freshman GPA is modeled to depend linearly on ACT score and high school ranking. Let \(H_a\) be the probability model where \(H_0\) is extended to include year as a factor, as fitted in R by

lm_gpa <- lm(GPA~High_School+ACT+factor(Year),data=gpa)

  1. Write out the null and alternative hypotheses by completely specifying the probability models. Sometimes, to save time and space, we don’t write the complete details of each model. Here, you are asked to write out the details in full. Looking at model.matrix(lm_gpa) may help you to understand the model that R has fitted. You can write the models in matrix, subscript or double subscript form.

  2. Interpret the results in summary(lm_gpa) in the context of whether there is evidence that some year, or years, may have substantially different intercepts. Why is there no row in the results table for factor(Year)1996?

  3. Explain which numbers in anova(lm_gpa) are relevant for constructing and interpreting an F test to test \(H_0\) against \(H_a\). Explain how to calculate the sample test statistic for this test. Specify the distribution of the model-generated test statistic under \(H_0\), and saying how the resulting p-value is calculated. What do you conclude?


Q2. Data manipulation and visualization exercises.

If at all possible, do these by yourself. It is okay to discuss the exercises with your peers or to come to office hours after a reasonable amount of time working by yourself, experimenting by trial and error, consulting R help pages, and collecting information from the internet. One source to reinforce R skills appropriate for STATS 401 is rstatistics.net/r-tutorial-exercise-for-beginners.

  1. Obtain the last 10 lines of gpa by two methods: (i) using matrix indexing; (2) via the tail() function.

  2. Write R code to count the number of students in the dataset gpa who were admitted in 1997. Hint: sum() applied to a logical vector counts the number of TRUE values.

sum(c(TRUE,TRUE,FALSE,TRUE))
## [1] 3

  1. Write R code to find the proportion of the students in gpa who are both above the 50th percentile in their high school and have an ACT score greater than 15.
    Hint: there are various ways to do this. Using an “and” operation & on suitably constructed logical vectors is perhaps the simplest and best, and is something you should be able to do.

  2. Make a scatterplot of high school percentile rank against ACT score, with points colored black, blue, red, green and cyan according to the year of admission.
    Hint: one way to get started on this is to create a vector my_color <- rep("black",length=nrow(gpa)) and then modify the entries of my_color corresponding to the different years. For example, you can create a logical vector gpa$Year==1997 to pick out the rows with admission year 1997. The corresponding entries of my_color can then be modified to take the value "blue". Finally, you can use plot(...,col=my_color) to color your scatterplot.

  3. Make 5 separate scatterplots of high school percentile rank against ACT score for each of the years 1996, 1997, 1998, 1999, 2000. Use par(mfrow=c(3,2)) to combine the scatterplots into a single figure. Use the xlim and ylim arguments to plot() to give all five scatterplots the same axis scales: this is helpful to do since otherwise the differing scales on each plot make them hard to compare.
    Hints: the xlim argument expects a vector of length 2 giving the lower and upper limits of the x-axis. Conveniently, range(gpa$ACT) produces a vector of length 2 giving the minimum and maximum values in the vector gpa$ACT. An internet search on R plot mfrow example will provide examples using par(mfrow=...) to combine several plots into one figure.
    Note: For this particular plot, ggplot() will fairly easily give you a nicer version if you have learned previously how to use it. However, learning to use base R graphics (as you are asked to do here) remains a useful skill.

  4. Comment on the relative strengths and weaknesses of the two types of plot in (d) and (e) above as a graphical tool for inspecting the admissions data from different years.

License: This material is provided under an MIT license