Lab 1

Author

Your name here

This is an R markdown script. You can use it to execute code as well as produce a pdf output.

This is your first lab exercise. Please turn in a rendered pdf at the end of the lab period to receive credit for lab attendance.

Recall that you can execute R code as follows:

1 + 1

[1] 2

Load the data

Go to the course website data page and download a copy of the grades dataset. (It’s towards the bottom.) Load the dataset into a dataframe called grades_df.

Each row of the dataset is a student. What are the columns in this dataset? Which classes and semesters does this dataframe cover? Restrict the dataset to contain only data for 151A in the fall 2024 semester.

Plot the data

The course grade is in the grade column. Plot a histogram of the course grades with ggplot. Plot a vertical red line at the passing point, which is 60%. What do you notice?

The grade on the final exam is in the final column. Make a plot with the final on the x-axis and the course grade on the y-axis with ggplot. What do you notice?

Compute a regression line

Compute an ordinary least squares (OLS) regression line to pass through the points between the final exam and course grade. Recall from earlier classes that the OLS regression line is given by $y = a + b x$ where

\[ \begin{aligned} a ={}& \bar{Y} - b \bar{X}\\ b ={}& \frac{n \sum X_i Y_i - \sum X_i \sum Y_i}{n \sum X_i^2 - (\sum X_i)^2} = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \end{aligned} \] Write code to compute and plot the line in these steps:

Define variables for $n$, $X$, and $Y$
Compute $\bar{Y}$ and $\bar{X}$
Compute b
Compute a
Plot the line on your above graph (hint: in ggplot use geom_abline)

Does the line look correct?

Make a regression function

Now, define a function that takes in an $X$ and $Y$ and returns a list with the intercept and slope of a regression line. The name of your function should be a verb. Check that it gives the same answer as above.

Plot with and without outliers

Using your regression function, plot the regression lines with and without the observations that failed the class. Plot the two lines in different colors with a plot legend. How does the line change?

Compare with `lm`

Now, compare the results of your regression function with the output of lm to make sure you get the same answer.

Compute the residuals

Now, using your regression function, compute the residuals $e = y - (a + b x)$, and plot a histogram. Do they look like they are Gaussian?

Meaning

How (if at all) does the relationship between the final exam score and course grade in this previous class inform your decisions about the course you are about to take?

--- title: "Lab 1" author: 'Your name here' format: html: code-download: true code-tools: true ---  ```{r setup, include = FALSE} library(tidyverse) knitr::opts_chunk$set(echo = TRUE) ``` This is an R markdown script. You can use it to execute code as well as produce a pdf output. This is your first lab exercise. Please turn in a rendered pdf at the end of the lab period to receive credit for lab attendance. Recall that you can execute R code as follows: ```{r} 1 + 1 ``` # Load the data Go to the course website [data page](https://stat151a.berkeley.edu/spring-2026/datasets/data.html) and download a copy of the grades dataset. (It's towards the bottom.) Load the dataset into a dataframe called `grades_df`. Each row of the dataset is a student. What are the columns in this dataset? Which classes and semesters does this dataframe cover? Restrict the dataset to contain only data for 151A in the fall 2024 semester. ```{r} ``` # Plot the data The course grade is in the `grade` column. Plot a histogram of the course grades with `ggplot`. Plot a vertical red line at the passing point, which is 60%. What do you notice? ```{r} ``` The grade on the final exam is in the `final` column. Make a plot with the final on the x-axis and the course grade on the y-axis with `ggplot`. What do you notice? ```{r} ``` # Compute a regression line Compute an ordinary least squares (OLS) regression line to pass through the points between the final exam and course grade. Recall from earlier classes that the OLS regression line is given by $y = a + b x$ where $$ \begin{aligned} a ={}& \bar{Y} - b \bar{X}\\ b ={}& \frac{n \sum X_i Y_i - \sum X_i \sum Y_i}{n \sum X_i^2 - (\sum X_i)^2} = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \end{aligned} $$ Write code to compute and plot the line in these steps: 1. Define variables for $n$, $X$, and $Y$ 1. Compute $\bar{Y}$ and $\bar{X}$ 1. Compute `b` 1. Compute `a` 1. Plot the line on your above graph (hint: in `ggplot` use `geom_abline`) Does the line look correct? ```{r} ``` ```{r} ``` # Make a regression function Now, define a function that takes in an $X$ and $Y$ and returns a list with the intercept and slope of a regression line. The name of your function should be a verb. Check that it gives the same answer as above. ```{r} ``` # Plot with and without outliers Using your regression function, plot the regression lines with and without the observations that failed the class. Plot the two lines in different colors with a plot legend. How does the line change? ```{r} ``` # Compare with `lm` Now, compare the results of your regression function with the output of `lm` to make sure you get the same answer. ```{r} ``` # Compute the residuals Now, using your regression function, compute the residuals $e = y - (a + b x)$, and plot a histogram. Do they look like they are Gaussian? ```{r} ``` # Meaning How (if at all) does the relationship between the final exam score and course grade in this previous class inform your decisions about the course you are about to take?