Lab 04: Exam 01 Review

ImportantDue date

Turn in what you have at the end of your lab section on February 16 for credit on the lab. To be considered on time, the following must be done by the due date:

• Final .qmd and .pdf files pushed to your team’s GitHub repo
• Final .pdf file submitted on Gradescope

Getting started

• Go to the sta210-sp26 organization on GitHub. Click on the repo with the prefix lab-04-. It contains the starter documents you need to complete the lab.

• Clone the repo and start a new project in RStudio. See the Lab 00 instructions for details on cloning a repo, starting a new project in R, and configuring git.

Packages

You will need the following packages for the lab:

library(tidyverse)
library(tidymodels) 
library(knitr)
library(patchwork)

# add other packages as needed

Restaurant tips

What factors are associated with the amount customers tip at a restaurant? To answer this question, we will use data collected in 2011 by a student at St. Olaf who worked at a local restaurant.¹

The variables we’ll focus on for this analysis are

• Tip: amount of the tip
• Party: number of people in the party
• Age: Age of the payer

View the data set to see the remaining variables.

tips <- read_csv("data/tip-data.csv")

Exploratory data analysis

p1 <- ggplot(data = tips, aes(x = Tip)) + 
  geom_histogram(color = "white", binwidth = 2) +
  labs(x = "Tips ($)",
       title = "Tips at local restaurant")

p2 <- ggplot(data = tips, aes(x = Party)) + 
  geom_histogram(color = "white") +
  labs(x = "Party",
       title = "Number of diners in party") +
  xlim(c(0, 7))

p3 <- ggplot(data = tips, aes(x = Age)) + 
  geom_bar(color = "white") +
  labs(x = "",
       title = "Age of Payer") 

p1 / (p2 + p3)

p4 <- ggplot(data = tips, aes(x = Party, y = Tip)) + 
  geom_jitter() + 
  labs(x = "Number of diners in party", 
       y = "Tips ($)",
       title = "Tips vs. Party")

p5 <- ggplot(data = tips, aes(x = Age, y = Tip)) + 
  geom_boxplot() + 
  labs(x = "Age of payer", 
       y = "Tips ($)",
       title = "Tips vs. Age")

p4 + p5

We will use the number of diners in the party and age of the payer to understand variability in the tips.

Exercise 1

We will start with the main effects model that includes Age and Party.

1. Fit the model and neatly display the results with 95% confidence intervals for the coefficients, using 2 decimal places.

2. Interpret the coefficient of Party in the context of the data.

3. Compute the regression standard error \(\hat{\sigma}_\epsilon\) and interpret this value in the context of the data.

Exercise 2

1. You wish to use the model in Exercise 1 to test whether there is a linear relationship between tips and the number of diners in the party, after adjusting for the age of the payer. Explain what the test statistic means in the context of the data.

2. Explain what the p-value means in the context of the data.

3. State the conclusion from the test in the context of the data.

4. An article claims that one can expect about $2 in tips for each additional person at a table. Does the model support this claim? Why or why not.

Exercise 3

Let’s check the model conditions for the model from Exercise 1. Below are plots of residuals versus fitted values and the distribution of residuals.

For each model condition (linearity, independence, normality, equal variance), comment on whether the condition is satisfied and briefly explain your response.

tip_model <- lm(Tip ~ Age + Party, data = tips)

tip_model_aug <- augment(tip_model)

resid_fitted <- ggplot(data = tip_model_aug, aes(x = .fitted, y = .resid)) +
  geom_point() + 
  geom_hline(yintercept = 0, linetype = 2, color = "red") +
  labs(x = "Fitted values", 
       y = "Residuals", 
       title = "Residuals versus fitted values")

dist_resid <- ggplot(data = tip_model_aug, aes( x= .resid)) + 
  geom_histogram() + 
  labs(x = "Residuals", 
       title = "Distribution of residuals")

resid_fitted + dist_resid

Exercise 4

One observation in the data set has a leverage of 0.190, standardized residual of -1.499, and Cook’s distance of 0.134. Use these values to comment on the following:

• whether the observation has large leverage.

• whether the observation has an outlying residual.

• whether the observation is an influential point.

Exercise 5

Now let’s fit a model using Age, Party and the interaction between the two variables. Display the 95% confidence interval for the coefficients.

1. State what the standard error for the coefficient of AgeSenCit:Party means in the context of the data.

2. Write code to show how the 95% confidence interval for AgeSenCit:Party was computed.

3. Based on the confidence interval, is there evidence that senior citizens pay less per additional person in the party compared to middle aged diners? Briefly explain.

Exercise 6

We want to consider a new variable other_diners = Party - 1 that is the number of people in the party aside from the payer. We will use this variable in a model that includes Age, other_diners, and the interaction between the two variables.

1. What is an advantage to using other_diners instead of Party in the model?
2. How will the responses to the previous exercise change, if at all, by using other_diners instead of Party in the model?

Submission

Warning

Before you wrap up the assignment, make sure all documents are updated on your GitHub repo. We will be checking these to make sure you have been practicing how to commit and push changes.

Reminder: you must turn in a PDF file to the Gradescope page before the submission deadline for full credit.

To submit the assignment:

• Access Gradescope through the STA 210 Canvas site.

• Click on the assignment, and you’ll be prompted to submit it.

• Mark the pages for the “Exercises” question.

Grading (10 points)

Component	Points
Submitted by the end of lab	10

Footnotes

1. Dahlquist, Samantha, and Jin Dong. 2011. “The Effects of Credit Cards on Tipping.” Project for Statistics 212-Statistics for the Sciences, St. Olaf College.