Lab 03: Coffee grades

Inference for simple linear regression using mathematical models

Important

Due:

  • Monday, September 26 , 11:59pm (Thursday labs)
  • Tuesday, September 27, 11:59pm (Friday labs)

Introduction

In today’s lab you will analyze data from over 1,000 different coffees to explore the relationship between a coffee’s sweetness and it’s flavor grade.

Learning goals

By the end of the lab you will…

  • be able to use mathematical models to conduct inference for the slope
  • be able to assess conditions for simple linear regression

Packages

The follow packages are used in the lab.

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

The Data

The dataset for this lab comes from the Coffee Quality Database and was obtained from the #TidyTuesday GitHub repo. It includes information about the origin, producer, measures of various characteristics, and the quality measure for over 1,000 coffees. The coffees can be reasonably be treated as a random sample.

This lab will focus on the following variables:

  • sweetness: Sweetness grade, 0 (least sweet) - 10 (most sweet)
  • flavor: Flavor grade, 0 (worst flavor) - 10 (best flavor)

Click here for the definitions of all variables in the data set. Click here for more details about how these measures are obtained.

coffee <- read_csv("data/coffee-grades.csv")

Exercises

Important

Make sure to do the following as you complete the assignment:

  • Write all code and narrative in your Quarto file. We should be able to read all your code in the rendered PDF.
  • Write all narrative in complete sentences.
  • Use informative axis titles and labels on all graphs.
  • Implement version control in your reproducible workflow. This means throughout the assignment, you should periodically render your Quarto document to produce the updated PDF, commit the changes in the Git pane, and push the updated files to GitHub. You should aim to push changes to GitHub at least three times as you work on the assignment.

Goal: The goal of this analysis is to use linear regression to understand variability in coffee flavor grades based on the sweetness grade.

Exercise 1

Visualize the relationship between the sweetness and flavor grades. Write two observations from the plot.

Exercise 2

Fit the linear model using sweetness grade to understand variability in the flavor grade. Neatly display the model using three digits, and include the 98% confidence interval for the model coefficients in the output.

Exercise 3

  • Interpret the slope in the context of the data.

  • Assume you are a coffee drinker. Would you drink a coffee represented by the intercept? Why or why not?

Exercise 4

Fill in the name of your model object from Exercise 2 to obtain the estimated regression standard error, \(\hat{\sigma}_\epsilon\).

glance(_____$fit)$sigma

Describe what this value means in the context of the data.

Exercise 5

Do the data provide evidence of a statistically significant linear relationship between sweetness and flavor grades? Conduct a hypothesis test using mathematical models to answer this question. In your response

  • State the null and alternative hypotheses in words and in mathematical notation.
  • What is the test statistic? State with the test statistic means in the context of this problem.
  • What distribution was used to calculate the p-value? Be specific.
  • State the conclusion in the context of the data using a threshold of \(\alpha = 0.02\) to make your decision.

Exercise 6

  • What is critical value used to calculate the 98% confidence interval displayed in Exercise 2? Show the code and output used to get your response.

  • Is the confidence interval consistent with the conclusions from the hypothesis test? Briefly explain why or why not.

Exercise 7

We’d like to check the model conditions to assess the reliability of the results in Exercises 4 - 6.

  • Make a scatterplot of the residuals vs. fitted values. Add a horizontal dotted line at \(residuals = 0\).

    Tip

    You need to get the augmented data. See the lecture notes for example code.

  • Make a histogram of the residuals.

Exercise 8

  • Is the linearity condition satisfied? Briefly explain your response using the plot of residuals vs. fitted to support your assessment.

  • Is the constant variance condition satisfied? Briefly explain your response using the plot of residuals vs. fitted to support your assessment.

Exercise 9

Is the normality condition satisfied? Briefly explain your response.

Exercise 10

Is the independence condition satisfied? Briefly explain your response.

Submission

In this class, the final PDF documents will be submitted to Gradescope.

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.

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

To submit your assignment:

  • Go to http://www.gradescope.comand click Log in in the top right corner.
  • Click School Credentials ➡️ Duke NetID and log in using your NetID credentials.
  • Click on your STA 210 course.
  • Click on the assignment, and you’ll be prompted to submit it.
  • Mark the pages associated with each exercise. All of the pages of your lab should be associated with at least one question (i.e., should be “checked”).
  • Select the first page of your .PDF submission to be associated with the “Workflow & formatting” & “Reproducible report” sections.

Grading (50 pts)

Component Points
Ex 1 - 10 45
Workflow & formatting 31
Reproducible report 22

Footnotes

  1. The “Workflow & formatting” grade is to assess the reproducible workflow. This includes implementing version control with at least at least 3 informative commit messages and having a neatly formatted PDF document that is easily readable with an updated name and date.↩︎

  2. This means we will be able to render the .qmd file in your GitHub repo and exactly reproduce the .pdf file in the Github repo and on Gradescope.↩︎