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Lab 3: Linear Regression, Probability, and Sampling Stats 10: Introduction to Statistical Reasoning Spring 2022 All rights reserved, Adam Chaffee, Michael Tsiang, and Maria Cha, 2017-2022

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Lab 3: Linear Regression, Probability, and Sampling
Stats 10: Introduction to Statistical Reasoning
Spring 2022
All rights reserved, Adam Chaffee, Michael Tsiang, and Maria Cha, 2017-2022.
Do not post, share, or distribute anywhere or with anyone without explicit permission.
Objectives
1. Understand linear regression in R and verify linear regression assumptions
2. Plotting time series to analyze trends
3. Use R for sampling and simulation
4. Calculate theory-based probabilities for normal and binomial distributions
Collaboration Policy
In lab you are encouraged to work in pairs or small groups to discuss the concepts on the
assignments. However, DO NOT copy each other’s work as this constitutes cheating. The work
you submit must be entirely your own. If you have a question in lab, feel free to reach out to
other groups or talk to your TA if you get stuck.
Linear Regression in R
You learned about linear regression in lecture. Now, we will learn how to code it in lab. We will
be using the following new commands:
• lm() will create an output containing the equation of the regression line, correlation
coefficient, p-values, residuals, and much more. We typically create an object to store the
results of the lm() function (see example below).
• abline() is a command to generate a regression line on a plot of the form y = a + bx. It
requires arguments for a and b, or linear model results (see example below).
Try running an example of the code on the following page by loading and using NCbirths below.
We will discuss what each line does in section.
## Run the linear model of weight against Mom's age and print a summary
linear_model <- lm(NCbirths$weight ~ NCbirths$Mage)
summary(linear_model)
## Create a plot of the data, and draw the regression line using abline
plot(NCbirths$weight ~ NCbirths$Mage, xlab = "Mom Age", ylab = "Weight",
main = "Regression of Weight on Mother's Age")
abline(linear_model, col = "red", lwd = 2)
## Create a plot of the residuals to assess regression assumptions
plot(linear_model$residuals ~ NCbirths$Mage, main = "Residuals plot")
## Add a line of y = 0 to help visualize the residuals
abline(a = 0, b = 0, col = "red", lwd = 2)
Exercise 1
We revisit the college graduate’s data obtained from American Community Survey 2010-2012
Public Use Microdata Series. Download the data ‘recent-grads.csv’ from CCLE Week 3 and read
it into R. When you read in the data, name your object “grads”. To remove the missing values,
add the line below:
grads <- na.omit(grads)
a. Run a linear regression of the number of full-time employed (Full_time) against the number of
male graduates (Men). Here, treat ‘Full_time’ as the response variable. Use the summary
function just like in the example above and paste the output into your report.
b. Plot the Full_time and Men, then use the abline() function to overlay the regression line onto
the data.
c. In a separate plot, plot the residuals of the regression from (a), and again use the abline()
function to overlay a horizontal line.
Parts d-h can be answered by hand, using a calculator, or any R functions of your choice.
d. Based on the output from (a), what is the equation of the linear regression line?
e. Imagine we have a new data point. We find out some major with 75,000 male graduates
(Men). What would we expect the number of full-time employed to be for the major?
f. Imagine two majors (A and B) for which we only observe the number of male graduates.
Major A has 1,000 more male graduates than major B. How much higher would we expect the
number of full-time employed to be in major A compared to major B?
g. Report the R-squared value and explain in words what it means in context.
h. Comment what you observe from the residual plot, and discuss about the goodness of fit of the
model.
Exercise 2
Our next data set is what is known as a time series, or data in time. It contains the measurements
via satellite imagery of sea ice extent in millions of square kilometers for each month from 1990
to 2011. Please download the “sea_ice” data from CCLE and read it into R. If you have your
working directory properly set, you can use the line below:
ice <- read.csv("sea_ice.csv", header = TRUE)
Note that currently R does not know what class the Date column is. We need to convert the Date
column into class “date” using the following line:
ice$Date <- as.Date(ice$Date, "%m/%d/%Y")
a. Produce a summary of a linear model of sea ice extent against time.
b. Plot the data and overlay the regression line. Does there seem to be a trend in this data?
c. Plot the residuals of the model over time and include a horizontal line. Comment what you
observe from the residual plot, and discuss about the goodness of fit of the model.
Sampling and simulating in R
We can use the sample() function to sample data from a vector, and the replicate() function to
simulate random events many times over. Note that the computer is not truly random, but it is
close enough for our purposes to consider it random. We also rely on the set.seed() function to
make our “random” results reproducible. Try the following examples in R and see the ##
comments for descriptions.
## Set seed for reproducibility
set.seed(1335)
## Create a names vector
names = c("Leslie", "Ron", "Andy", "April", "Tom", "Ben", "Jerry")
## Sample 3 of the names with replacement
sample(names, 3, replace = TRUE)
## Sample 3 of the names without replacement
sample(names, 3, replace = FALSE)
## Create a vector from 1 to 10
numbers = 1:10
## Simulate sampling 3 different numbers at random, 10 times
replicate(10, sample(numbers, 3, replace = FALSE))
## One more time, but save as an object
rand_draws = replicate(10, sample(numbers, 3, replace = FALSE))
## Perform analysis on the random draws
colMeans(rand_draws) ## Takes the mean of each sample
colSums(rand_draws) ## Takes the sum of each sample
Exercise 3
One of Adam’s favorite casino games is called “Craps”. In the first round of this game, two fair
6-sided dice are rolled. If the sum of the two dice equal 4 or 11, Adam doubles his money! If a
2, 5, or 12 are rolled, Adam loses all the money he bets. L
a. Based on your lecture notes, what is the chance Adam will double his money in the first round
of the game? What is the chance Adam will lose his money in the first round of the game?
b. Let’s now approximate the results in (a) by simulation. First, set the seed to 123. Then, create
an object that contains 10,000 sample first round Craps outcomes (simulate the sum of 2 dice,
10,000 times). Use the appropriate function to visualize the distribution of these outcomes (hint:
are the outcomes discrete or continuous?).
c. Imagine these sample results happened in real life for Adam. Using R functions of your
choice, calculate the percentage of time Adam doubled his money. Calculate the percentage of
time Adam lost his money.
d. Adam winning money and Adam losing money can both be considered events. Are these two
events independent, disjoint, or both? Explain why.
e. Quickly mathematically verify by calculator if those events are independent using part (a) and
what you learned in lecture. Show work.