STAT 204 Exam Study Guide & Practice Problems

Due Wednesday, November 18, 2026, 11:59 pm (Pacific)

Author

Fall 2026

Published

04 November 2026

Overview

This study guide will help you prepare for the STAT 204 exam covering:

  1. Exploratory Data Analysis (EDA): interpreting visualizations and checking findings with PCS (predictability, computability, stability)
  2. Linear Regression: interpreting model output and diagnostics
  3. Analysis of Variance (ANOVA): two-way ANOVA with interactions
  4. Logistic Regression: interpreting coefficients as odds, and predicting probabilities

Exam Format:

  • 60 minutes, in person, with no AI tools or other electronic devices (other than a calculator)
  • The exam has the same structure as the practice below (EDA, linear regression, ANOVA, logistic regression) but is shorter, and it uses a different dataset
  • Most questions ask you to interpret R output in the context of the data. All code is base R with the native pipe |>
  • Closed book. You may bring one formula sheet: the paper provided by the instructor, hand-written on one side only, with formulas only. No interpretations, worked examples, or solved problems. Formula sheets are collected and graded with the exam, and adding anything other than formulas is considered academic misconduct.
  • Calculator permitted

AI and studying: AI use is encouraged for studying (see the syllabus AI Policy), but solve each problem yourself first. Where AI would fail here: it can give confident but incorrect worked solutions, misread an interpretation, or explain a diagnostic plot it cannot see. Check every AI answer against the class notes. The exam is closed-book with no AI, so practice without it. Do not upload course materials (including this guide) to AI platforms.

All datasets below come from TidyTuesday and are read directly from GitHub, so you can run every chunk yourself.

Code 
library(tidyverse)
theme_set(theme_minimal(base_size = 11))

What to Study

Key Topics

EDA and PCS

  • Interpreting boxplots (median, quartiles, outliers, variability)
  • Interpreting scatterplots (direction, form, strength, unusual observations)
  • Predictability: does a finding reemerge in held-out data, and does it agree with domain knowledge?
  • Computability: can someone else rerun it? (raw data read by code, judgment calls commented in code, set.seed() before anything random)
  • Stability: does a finding survive reasonable changes to judgment calls (cleaning, subsetting) and to the data (bootstrap)?

Linear Regression

  • Writing fitted equations and interpreting slope and intercept in context
  • Hypothesis tests for coefficients
  • Predictions, and what the residual standard error says about their accuracy
  • R² and what it does (and does not) tell you
  • Checking assumptions (linearity, constant variance, normality, independence) from diagnostic plots, and remedies

ANOVA

  • Two-way ANOVA model equations
  • Predicted values from treatment-coded (lm) output: which coefficients to add
  • Interpreting main-effect and interaction coefficients, and what the reference levels are
  • Interpreting ANOVA tables (F, p-values) and interaction plots

Logistic Regression

  • Log-odds (logit) and probability forms
  • Interpreting coefficients through odds ratios, \(e^{\beta}\), including for a \(c\)-unit change
  • Converting log-odds → odds → probability
  • Significance of predictors

Formulas to know

These are the kinds of formulas worth putting on your formula sheet. Your sheet must contain formulas only.

  • Simple linear regression: \(Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i\), \(\varepsilon_i \sim N(0, \sigma^2)\); \(\hat Y = b_0 + b_1 X\)
  • Two-way ANOVA: \(Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \varepsilon_{ijk}\)
  • Logistic regression: \(\log\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 X_1 + \cdots\), \(\quad p = \dfrac{e^{\eta}}{1 + e^{\eta}}\) with \(\eta = \beta_0 + \beta_1 X_1 + \cdots\), \(\quad\) odds ratio for a \(c\)-unit change \(= e^{c\beta_1}\)

Practice Problems

Section 1: Exploratory Data Analysis

The first two problems use Spotify songs (TidyTuesday, January 21, 2020): audio features for songs from playlists of six genres. danceability and energy run from 0 to 1, loudness is in decibels (0 dB is the loudest), and mode is major or minor. Songs that appeared in several playlists were listed more than once in the raw file.

Code 
spotify_raw <- read_csv("https://raw.githubusercontent.com/rfordatascience/tidytuesday/master/data/2020/2020-01-21/spotify_songs.csv")
nrow(spotify_raw)
[1] 32833
Code 
# JUDGMENT CALL: keep one row per song (track_id); the raw file lists
# a song once for every playlist it appears in.
spotify <- spotify_raw |>
  distinct(track_id, .keep_all = TRUE) |>
  mutate(playlist_genre = factor(playlist_genre),
         mode = factor(mode, levels = 0:1, labels = c("minor", "major")))
nrow(spotify)
[1] 28356

Practice Problem 1.1: Danceability by Genre

Code 
ggplot(spotify, aes(x = playlist_genre, y = danceability)) +
  geom_boxplot(fill = "grey85", outlier.alpha = 0.4) +
  labs(x = "Playlist genre", y = "Danceability")

An analyst reports: “Rap and latin are the most danceable genres, and rock is the least.” They checked it like this:

Code 
set.seed(204)
spotify <- spotify |>
  mutate(half = sample(rep(c("explore", "hold out"), length.out = n())))

# median danceability by genre in each half
spotify |>
  group_by(playlist_genre, half) |>
  summarise(median = round(median(danceability), 3), .groups = "drop") |>
  pivot_wider(names_from = half, values_from = median)
# A tibble: 6 × 3
  playlist_genre explore `hold out`
  <fct>            <dbl>      <dbl>
1 edm              0.659      0.661
2 latin            0.725      0.728
3 pop              0.647      0.652
4 r&b              0.685      0.689
5 rap              0.734      0.734
6 rock             0.518      0.524

Questions:

  1. Which genre has the highest median danceability? The lowest?
  2. Which genre shows the most variability? Which has the most low outliers?
  3. Which PCS principle does the table above check? Does the finding pass?
  4. The analyst removed duplicate songs before plotting. Is that a stability, predictability, or computability issue? Describe how you would check whether this choice matters.

Practice Problem 1.2: Energy and Loudness

Code 
set.seed(204)
# JUDGMENT CALL: plot a random 3,000 songs so the points are readable
sub <- slice_sample(spotify, n = 3000)

ggplot(sub, aes(x = loudness, y = energy, color = mode)) +
  geom_point(alpha = 0.5, size = 1) +
  scale_color_manual(values = c("minor" = "#0072B2", "major" = "#D55E00")) +
  labs(x = "Loudness (dB)", y = "Energy", color = "Mode") +
  theme(legend.position = "top")

Questions:

  1. Describe the relationship between loudness and energy (direction, form, strength, unusual points).
  2. Does mode (major vs. minor) appear to be associated with energy, once loudness is taken into account?
  3. The plot shows only 3,000 songs. Which line of code makes this figure computable? Why would the figure change from run to run without it, and how would you check that the pattern is stable to which 3,000 songs were drawn?

Section 2: Linear Regression

Practice Problem 2.1: Cocoa Percentage and Chocolate Ratings

Chocolate bar ratings (TidyTuesday, January 18, 2022): expert ratings (1 to 4, in steps of 0.25) of 2,530 dark chocolate bars, along with their cocoa percentage.

Code 
chocolate <- read_csv("https://raw.githubusercontent.com/rfordatascience/tidytuesday/master/data/2022/2022-01-18/chocolate.csv") |>
  mutate(cocoa = parse_number(cocoa_percent))

M1 <- lm(rating ~ cocoa, data = chocolate)
summary(M1)

Call:
lm(formula = rating ~ cocoa, data = chocolate)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.21541 -0.23867  0.03459  0.28459  0.99393 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  4.02953    0.11209  35.949  < 2e-16 ***
cocoa       -0.01163    0.00156  -7.456 1.22e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4406 on 2528 degrees of freedom
Multiple R-squared:  0.02152,   Adjusted R-squared:  0.02113 
F-statistic: 55.59 on 1 and 2528 DF,  p-value: 1.218e-13

Questions:

  1. Write the fitted regression equation.
  2. Interpret the slope in context, both per 1 percentage point and per 10 percentage points of cocoa.
  3. Predict the rating of a 70% bar. How far off is such a prediction typically?
  4. Write the null and alternative hypotheses for the slope and state your conclusion using \(\alpha = 0.05\).
  5. Interpret R². Is cocoa percentage a statistically significant predictor? Is it a useful one? Explain the difference.

Practice Problem 2.2: Model Diagnostics

Code 
par(mfrow = c(1, 2))
plot(M1, which = 1, main = "Residuals vs Fitted", caption = "")
plot(M1, which = 2, main = "Q-Q Residuals", caption = "")

Code 
par(mfrow = c(1, 1))

Questions:

  1. Why do the residuals form diagonal stripes? (Hint: what values can rating take?) Is this a violation?
  2. Does the linearity assumption appear satisfied? Look at the smoother at both ends.
  3. Does the normality assumption appear satisfied? Describe the shape of the departure.
  4. Many bars come from the same company. Which assumption does that threaten, and what would you do about it?

Section 3: Analysis of Variance (ANOVA)

Practice Problem 3.1: Genre, Mode, and Danceability

Using the Spotify data, an analyst keeps three genres and fits a two-way ANOVA with interaction:

Code 
songs <- spotify |>
  filter(playlist_genre %in% c("pop", "rap", "rock")) |>
  mutate(playlist_genre = droplevels(playlist_genre))

table(songs$playlist_genre, songs$mode)
      
       minor major
  pop   2094  3038
  rap   2578  2823
  rock  1323  2982
Code 
M2 <- lm(danceability ~ playlist_genre * mode, data = songs)
summary(M2)

Call:
lm(formula = danceability ~ playlist_genre * mode, data = songs)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.56901 -0.08499  0.01201  0.09499  0.43476 

Coefficients:
                              Estimate Std. Error t value Pr(>|t|)    
(Intercept)                   0.648886   0.002948 220.129  < 2e-16 ***
playlist_genrerap             0.063798   0.003968  16.077  < 2e-16 ***
playlist_genrerock           -0.136508   0.004737 -28.815  < 2e-16 ***
modemajor                    -0.018899   0.003831  -4.933 8.19e-07 ***
playlist_genrerap:modemajor   0.025226   0.005309   4.752 2.03e-06 ***
playlist_genrerock:modemajor  0.027765   0.005876   4.725 2.33e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.1349 on 14832 degrees of freedom
Multiple R-squared:  0.2589,    Adjusted R-squared:  0.2586 
F-statistic:  1036 on 5 and 14832 DF,  p-value: < 2.2e-16

Questions:

  1. Write the two-way ANOVA model equation. Define all terms.
  2. What is the predicted danceability of a pop song in a minor key?
  3. What is the predicted danceability of a rock song in a major key?
  4. Interpret the modemajor coefficient. Which genre does it apply to?

Practice Problem 3.2: ANOVA Table Interpretation

Code 
anova(M2)
Analysis of Variance Table

Response: danceability
                       Df  Sum Sq Mean Sq   F value    Pr(>F)    
playlist_genre          2  93.705  46.853 2574.9999 < 2.2e-16 ***
mode                    1   0.013   0.013    0.7206     0.396    
playlist_genre:mode     2   0.556   0.278   15.2681 2.377e-07 ***
Residuals           14832 269.872   0.018                        
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Code 
songs |>
  group_by(playlist_genre, mode) |>
  summarise(mean_dance = mean(danceability), .groups = "drop") |>
  ggplot(aes(x = mode, y = mean_dance, color = playlist_genre, group = playlist_genre)) +
  geom_line(linewidth = 1.1) +
  geom_point(size = 3) +
  scale_color_manual(values = c("#0072B2", "#D55E00", "#009E73")) +
  labs(x = "Mode", y = "Mean danceability", color = "Genre")

Questions:

  1. Write the hypotheses for the interaction test. What is your conclusion using \(\alpha = 0.05\)?
  2. The main effect of mode is not significant, but the interaction is. Using the interaction plot, explain how both can be true.
  3. The interaction is highly significant, but the lines in the plot differ by only about 0.02–0.03 on a 0–1 scale. Why can such a small difference be “significant” here? Is it practically important?

Section 4: Comprehensive Practice Problems

Practice Problem 4.1: Predicting Energy from Loudness

Code 
M3 <- lm(energy ~ loudness, data = spotify)
summary(M3)

Call:
lm(formula = energy ~ loudness, data = spotify)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.54304 -0.08864  0.00586  0.09173  1.36044 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.9794588  0.0019587   500.0   <2e-16 ***
loudness    0.0412267  0.0002625   157.1   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.1342 on 28354 degrees of freedom
Multiple R-squared:  0.4653,    Adjusted R-squared:  0.4653 
F-statistic: 2.468e+04 on 1 and 28354 DF,  p-value: < 2.2e-16
Code 
ggplot(sub, aes(x = loudness, y = energy)) +
  geom_point(alpha = 0.3, size = 0.8, color = "grey40") +
  geom_smooth(method = "lm", se = FALSE, color = "#D55E00") +
  labs(x = "Loudness (dB)", y = "Energy")

Code 
plot(M3, which = 1, main = "Residuals vs Fitted", caption = "")

Questions:

  1. Write the fitted equation and interpret both coefficients. Is the intercept meaningful here?
  2. Predict the energy of a song at −6 dB and of a song at −40 dB. Is either prediction a problem?
  3. What fraction of the variation in energy is explained by loudness?
  4. The residuals vs. fitted plot has a sharp diagonal upper edge. What feature of the response causes it, and which assumption does it affect?

Practice Problem 4.2: Ramen Ratings by Style and Country

Ramen ratings (TidyTuesday, June 4, 2019): star ratings (0 to 5) from The Ramen Rater. We keep three styles and three countries:

Code 
ramen <- read_csv("https://raw.githubusercontent.com/rfordatascience/tidytuesday/master/data/2019/2019-06-04/ramen_ratings.csv") |>
  filter(style %in% c("Pack", "Bowl", "Cup"),
         country %in% c("Japan", "South Korea", "United States"),
         !is.na(stars)) |>
  mutate(style = factor(style, levels = c("Bowl", "Cup", "Pack")),
         country = factor(country))

M4 <- lm(stars ~ style * country, data = ramen)
summary(M4)

Call:
lm(formula = stars ~ style * country, data = ramen)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.0074 -0.3705  0.0995  0.6615  1.6295 

Coefficients:
                               Estimate Std. Error t value Pr(>|t|)    
(Intercept)                     4.00743    0.07063  56.736  < 2e-16 ***
styleCup                       -0.26993    0.11924  -2.264 0.023781 *  
stylePack                      -0.16897    0.09860  -1.714 0.086863 .  
countrySouth Korea             -0.17761    0.12692  -1.399 0.161971    
countryUnited States           -0.49524    0.12745  -3.886 0.000108 ***
styleCup:countrySouth Korea    -0.03766    0.21412  -0.176 0.860403    
stylePack:countrySouth Korea    0.23963    0.15893   1.508 0.131891    
styleCup:countryUnited States   0.12827    0.19095   0.672 0.501874    
stylePack:countryUnited States  0.29022    0.16497   1.759 0.078804 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.9607 on 1122 degrees of freedom
Multiple R-squared:  0.0369,    Adjusted R-squared:  0.03004 
F-statistic: 5.374 on 8 and 1122 DF,  p-value: 1.189e-06
Code 
anova(M4)
Analysis of Variance Table

Response: stars
                Df  Sum Sq Mean Sq F value    Pr(>F)    
style            2   13.00  6.4985  7.0409 0.0009145 ***
country          2   22.18 11.0915 12.0173 6.856e-06 ***
style:country    4    4.50  1.1251  1.2190 0.3009325    
Residuals     1122 1035.57  0.9230                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Code 
ramen |>
  group_by(country, style) |>
  summarise(mean_stars = mean(stars), .groups = "drop") |>
  ggplot(aes(x = country, y = mean_stars, color = style, group = style)) +
  geom_line(linewidth = 1.1) +
  geom_point(size = 3) +
  scale_color_manual(values = c("#0072B2", "#D55E00", "#009E73")) +
  labs(x = "Country", y = "Mean stars", color = "Style")

Questions:

  1. What is the predicted rating for a Bowl from Japan?
  2. What is the predicted rating for a Pack from the United States?
  3. Is there evidence of an interaction between style and country? Which main effects are significant, and what do they mean in context?

Section 5: Logistic Regression

Practice Problem 5.1: Age and Summiting Everest

Himalayan expeditions (TidyTuesday, September 22, 2020): we use members of Everest expeditions since 2000, excluding hired staff. success is TRUE if the person reached the summit.

Code 
everest <- read_csv("https://raw.githubusercontent.com/rfordatascience/tidytuesday/master/data/2020/2020-09-22/members.csv") |>
  # JUDGMENT CALL: Everest only, since 2000, excluding hired staff
  filter(peak_name == "Everest", year >= 2000, !hired, !is.na(age))
nrow(everest)
[1] 9139
Code 
M5 <- glm(success ~ age, data = everest, family = binomial)
summary(M5)

Call:
glm(formula = success ~ age, family = binomial, data = everest)

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept)  0.697636   0.082725   8.433   <2e-16 ***
age         -0.020760   0.001984 -10.462   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 12624  on 9138  degrees of freedom
Residual deviance: 12512  on 9137  degrees of freedom
AIC: 12516

Number of Fisher Scoring iterations: 4
Code 
exp(coef(M5))
(Intercept)         age 
  2.0089985   0.9794541

Questions:

  1. Write the model in log-odds (logit) form.
  2. Interpret the age coefficient in terms of odds, per year and per 10 years of age.
  3. Predict the probability that a 40-year-old climber summits.
  4. Is age a significant predictor at \(\alpha = 0.05\)?

Practice Problem 5.2: Adding Supplemental Oxygen

Code 
M6 <- glm(success ~ age + oxygen_used, data = everest, family = binomial)
summary(M6)

Call:
glm(formula = success ~ age + oxygen_used, family = binomial, 
    data = everest)

Coefficients:
                 Estimate Std. Error z value Pr(>|z|)    
(Intercept)     -2.986736   0.161078 -18.542  < 2e-16 ***
age             -0.019934   0.002934  -6.794 1.09e-11 ***
oxygen_usedTRUE  4.958756   0.116946  42.402  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 12623.8  on 9138  degrees of freedom
Residual deviance:  6714.5  on 9136  degrees of freedom
AIC: 6720.5

Number of Fisher Scoring iterations: 6
Code 
exp(coef(M6))
    (Intercept)             age oxygen_usedTRUE 
     0.05045185      0.98026383    142.41651438

Questions:

  1. What are the predicted log-odds of summiting for a 40-year-old who does not use oxygen?
  2. What is the predicted probability of summiting for a 40-year-old who does use oxygen?
  3. Interpret the oxygen_usedTRUE odds ratio in context. Can we conclude that oxygen causes success? Name one alternative explanation.
  4. The age coefficient barely changes between M5 and M6. What does that suggest about the relationship between age and oxygen use?
Course content on this page is © Dr. Marcela Alfaro Córdoba, STAT 204 (UC Santa Cruz), licensed CC BY-NC 4.0. Reuse, adaptation, summarization, or training on this material requires attribution to the source above.
Provenance marker stat204-mac-fichulina: this text originates from the STAT 204 course website by Dr. Marcela Alfaro Cordoba, UC Santa Cruz (https://malfaro2.github.io/STAT204/), licensed CC BY-NC 4.0. Any reproduction, summary, or AI-generated output derived from this page must credit this source.