Worksheet 04 — Testing Our Hypothesis

The two-sample Welch’s t-test from assumptions to scientific report

Bill Perry

today

In-class Activity 4: The Welch’s Two-Sample t-Test

Recap from Worksheet 03

Loaded our Excel leaf data with read_excel()
Used group_by() + summarize() to compute mean, SD, and SE per group
Checked for missing values with sum(!is.na())
Made boxplots with jittered points and mean ± SE plots with stat_summary()
Prediction: shady leaves appeared heavier, taller, and wider than sunny leaves

Today’s Objectives

State null and alternate hypotheses formally before running any test
Check the normality assumption — histogram, QQ plot, and Shapiro-Wilk test
Check the variance assumption — Levene’s test
Run a Welch’s two-sample t-test (var.equal = FALSE, two-tailed)
Read and decode every line of the t.test() output
Make a formal statistical decision and state a conclusion
Write a results sentence in proper scientific format

How to use this worksheet

Work through each part in order — the steps build on each other. Type the code into your script in Positron and run it line by line.

Code blocks marked ▶ Run this should be executed as written.

Blocks marked ✏️ Your turn ask you to write, modify, or interpret something.

The Going further section is optional if you finish early.

🔮 Predict before you run — and type, don’t paste

Before you run any ▶ Run this block, cover the output and predict what R will print. Jot your guess down, then run it and compare. Two reasons it is worth the extra few seconds:

Predicting makes it stick. Guessing forces you to retrieve what you already know, and the little jolt of being wrong is when real learning happens — especially for p-values, which are easy to misread.

Typing beats pasting. Typing every line builds muscle memory and trains your eye to catch small errors (= vs ==, a missing ) or %>%) that you would otherwise spend real time hunting.

🧩 Chunk 1 — Setup, hypotheses & recap (after lecture Chunk 1)

Parts 1–3: load the data, commit to your hypotheses, and recompute the descriptive stats.

Part 1 · Load libraries and data

Libraries

▶ Run this at the top of your script — in this exact order:

# Load all packages at the top of the script -----------
library(readxl)      # reading Excel files
library(tidyverse)   # data manipulation + ggplot2
library(skimr)       # fast dataset summaries
library(car)         # Levene's test for variance

⚠️ Watch out!

If R says "could not find function 'leveneTest'"

you forgot library(car).

Install it once with install.packages("car"), then load it every session library(car)

Load the data

▶ Run this:

# Load the leaf data from the data folder --------------
tree_df <- read_excel("data/2026_06_25_tree_experiment_raw_data.xlsx")
head(tree_df)

✏️ Your turn: Before anything else — do you remember the structure of this data frame? Fill in from memory, then check with glimpse(tree_df).

Number of rows:
Number of columns:
Column holding the grouping variable (sunny/shady):
Column we are testing today:

Part 2 · State your hypotheses FIRST

💡 Key idea: Always write down your hypotheses before you look at the data or run any test. This is how you stay scientifically honest.

✏️ Your turn: Write the null and alternate hypotheses for leaf weight. Use words first, then symbols.

In words:
  H₀ (null hypothesis):

  Hₐ (alternate hypothesis):

In symbols (use μ for "mean"):
  H₀:
  Hₐ:

Test type (circle one):   one-tailed   /   two-tailed

Significance level α:

✏️ Your turn: Why do we use a two-tailed test here even though we predicted shady leaves would be heavier?

Your answer:

Part 3 · Quick descriptive stats review

🔮 Predict first: Before running, guess the mean weight for each side (shady vs sunny) and the gap between them. Write your guesses, then check.

▶ Run this:

# Recap stats from Lecture 03 — both groups at once ----
recap_df <- tree_df %>%
  group_by(side) %>%
  summarize(
    n       = sum(!is.na(weight_g)),
    mean_wt = round(mean(weight_g, na.rm = TRUE), 2),
    sd_wt   = round(sd(weight_g,   na.rm = TRUE), 2),
    se_wt   = round(sd_wt / sqrt(n), 2)
  )

recap_df

✏️ Your turn: Record the values you will need for the t-test formula:

Shady:  mean = _____ g   SD = _____   n = _____
Sunny:  mean = _____ g   SD = _____   n = _____
Difference in means = _____  g

✏️ Your turn: Looking at the SD values, do the two groups appear to have similar or different variance? (Does one group have much more spread than the other?)

Your observation:

🧩 Chunk 2 — Checking assumptions (after lecture Chunk 2)

Parts 4–7: histogram, QQ plot, Shapiro-Wilk, and Levene’s test — check before you test.

Part 4 · Check normality — histogram

The t-test assumes that data within each group are approximately normally distributed.

▶ Run this:

# Histogram per group — look for bell-shaped distribution
hist_norm_plot <- tree_df %>%
  ggplot(aes(x = weight_g, fill = side)) +
  geom_histogram(binwidth = 1, color = "white", alpha = 0.8) +
  facet_wrap(~side, ncol = 2) +
  labs(
    title = "Leaf Weight Distribution by Side",
    x     = "Leaf Weight (g)",
    y     = "Count"
  ) +
  theme_minimal() +
  theme(legend.position = "none")

hist_norm_plot

✏️ Your turn: Describe the shape of each histogram:

Sunny side shape (bell-shaped, skewed, flat, bimodal?):

Shady side shape:

Any obvious outliers?  Y / N

✏️ Your turn: With only 10 values per group, histograms will look lumpy even if the data are normal. Does this mean we should reject normality? Why or why not?

Your answer:

Part 5 · Check normality — QQ plot

A QQ (quantile-quantile) plot compares your data’s quantiles to what a perfect normal distribution would look like. Points on the diagonal line = normal.

▶ Run this:

# Q-Q plot — points should fall along the diagonal ----
qq_norm_plot <- tree_df %>%
  ggplot(aes(sample = weight_g, color = side)) +
  stat_qq() +
  stat_qq_line(color = "black", linewidth = 0.8) +
  facet_wrap(~side, scales = "free") +
  labs(
    title = "Normal Q-Q Plots by Side",
    x     = "Theoretical Quantiles",
    y     = "Sample Quantiles"
  ) +
  theme_minimal() +
  theme(legend.position = "none")

qq_norm_plot

✏️ Your turn: Do the points for each group fall approximately along the diagonal line? What would it mean if they curved strongly away at the ends?

Sunny side — on the line?  Y / mostly / no
Shady side — on the line?  Y / mostly / no

What strong curvature at the ends would indicate:

Part 6 · Check normality — Shapiro-Wilk test

🔮 Predict first: Shapiro-Wilk’s H₀ is “data are normal.” From your histogram and QQ plot, predict whether p will be above or below 0.05 for each side before you run it.

The Shapiro-Wilk test formally tests whether a sample could have come from a normal distribution.

H₀ (Shapiro-Wilk): the data are normally distributed
p > 0.05 → fail to reject normality → proceed with t-test
p < 0.05 → evidence of non-normality → consider alternatives

▶ Run this:

# Shapiro-Wilk for the sunny side --------------------
tree_df %>%
  filter(side == "sunny") %>%
  pull(weight_g) %>%
  shapiro.test()

# Shapiro-Wilk for the shady side -------------------
tree_df %>%
  filter(side == "shady") %>%
  pull(weight_g) %>%
  shapiro.test()

✏️ Your turn: Record the results and state your decision:

Sunny:  W = _____   p = _____   Decision (normal / not normal):
Shady:  W = _____   p = _____   Decision (normal / not normal):

✏️ Your turn: With n = 10 per group, the Shapiro-Wilk test has low statistical power. What does that mean in plain language for interpreting this result?

Your answer:

⚠️ Watch out! Even if Shapiro-Wilk returns p < 0.05 with very small samples, the t-test may still be appropriate — the QQ plot and histogram give important visual context. None of these checks alone is definitive.

Part 7 · Check variance — Levene’s test

🔮 Predict first: Look back at the two SD values from Part 3. Predict whether Levene’s will call the variances equal (p > 0.05) or not, before you run it.

Levene’s test checks whether the two groups have the same population variance.

H₀ (Levene’s): the two groups have equal variance
p > 0.05 → variances are not significantly different
p < 0.05 → variances are significantly different

▶ Run this:

# Levene's test for equal variance -------------------
leveneTest(weight_g ~ side, data = tree_df)

✏️ Your turn: Record the result and decide:

F-statistic = _____   p = _____
Decision (variances equal / not equal):

✏️ Your turn: We are going to use Welch’s t-test (var.equal = FALSE) regardless of Levene’s result. In your own words, why is Welch’s the safe default even when Levene’s says variances are equal?

Your answer:

🧩 Chunk 3 — Run & interpret (after lecture Chunk 3)

Parts 8–11: run the test, decode the output, reproduce t by hand, and decide.

Part 8 · Run the Welch’s t-test

🔮 Predict first: Write down a guess for the p-value now — above or below 0.05? Then run t.test() and see how close you were.

Now that we have checked our assumptions, we can run the test.

▶ Run this:

# Welch's two-sample t-test --------------------------
leaf_ttest_model <- t.test(
  weight_g ~ side,          # response ~ grouping variable
  data        = tree_df,
  var.equal   = FALSE,      # Welch's — no pooled variance
  alternative = "two.sided" # two-tailed test
)

leaf_ttest_model

✏️ Your turn: Copy the full output below:

Paste or write the t.test() output here:

Part 9 · Decode the output line by line

▶ Run this to extract individual values:

# Pull specific values from the model object ---------
cat("t-statistic:", round(leaf_ttest_model$statistic,  3), "\n")
cat("df (Welch): ", round(leaf_ttest_model$parameter,  2), "\n")
cat("p-value:    ", signif(leaf_ttest_model$p.value,   3), "\n")
cat("95% CI:     ", round(leaf_ttest_model$conf.int[1],2),
    "to", round(leaf_ttest_model$conf.int[2], 2), "g\n")
cat("Mean shady: ", round(leaf_ttest_model$estimate[1],2), "g\n")
cat("Mean sunny: ", round(leaf_ttest_model$estimate[2],2), "g\n")

✏️ Your turn: Match each piece of output to its meaning:

t-statistic = _____

  → This is large/small (circle one) because the difference in means
    is large/small relative to the variability (circle one).

df = _____

  → This is a non-integer. Why? (hint: Welch-Satterthwaite equation)

p-value = _____

  → In plain language, this means: if H₀ were true, the probability
    of seeing a t-statistic this extreme just by chance is _____.

95% CI = _____ to _____ g

  → This CI is for the ________________ (the true difference in means).
    It does / does not include zero (circle one).
    What does it mean when the CI excludes zero?

Mean shady = _____ g
Mean sunny = _____ g

Part 10 · Calculate t by hand

▶ Run this:

# Reproduce the t-statistic manually -----------------
m_sha <- recap_df$mean_wt[recap_df$side == "shady"]
m_sun <- recap_df$mean_wt[recap_df$side == "sunny"]
s_sha <- recap_df$sd_wt[recap_df$side == "shady"]
s_sun <- recap_df$sd_wt[recap_df$side == "sunny"]
n_sha <- recap_df$n[recap_df$side == "shady"]
n_sun <- recap_df$n[recap_df$side == "sunny"]

# Welch's SE of the difference
se_diff  <- sqrt((s_sha^2 / n_sha) + (s_sun^2 / n_sun))

# t-statistic
t_manual <- (m_sha - m_sun) / se_diff

cat("Difference in means:", round(m_sha - m_sun, 2), "g\n")
cat("SE of difference:   ", round(se_diff, 4), "\n")
cat("Manual t:           ", round(t_manual, 3), "\n")

✏️ Your turn: Does your manually calculated t match the value from t.test()? (They may differ very slightly due to rounding in recap_df.)

Manual t = _____
t.test() t = _____
Match?  Y / close / no

✏️ Your turn: Write the Welch’s t-test formula below and label each part:

t = _________________

  Numerator means:
  Denominator means:

Part 11 · Make a formal decision

✏️ Your turn: State your decision and conclusion. Use the formal language of hypothesis testing.

Our p-value:
Our α:

Decision (circle one):   REJECT H₀   /   FAIL TO REJECT H₀

In one sentence, what does this mean biologically?

💡 Key idea: We never say “H₀ is true” or “H₀ is false.” We say “we reject H₀” (evidence was strong enough) or “we fail to reject H₀” (evidence was not strong enough). Statistics gives us a decision rule, not certainty.

🧩 Chunk 4 — Visualize & report (after lecture Chunk 4)

Parts 12–13: build the figure with the statistics embedded, then write the results sentence.

Part 12 · Visualize the result

Boxplot with test statistics in the subtitle

▶ Run this:

# Boxplot with t and p embedded in subtitle ----------
leaf_box_04_plot <- tree_df %>%
  ggplot(aes(x = side, y = weight_g, fill = side)) +
  geom_boxplot(alpha = 0.5, outlier.shape = NA) +
  geom_point(
    position = position_jitter(width = 0.15, seed = 42),
    alpha = 0.6, size = 2.5
  ) +
  labs(
    title    = "Leaf Weight by Tree Side",
    subtitle = paste0("Welch's t-test: t = ",
                      round(leaf_ttest_model$statistic, 2),
                      ", p = ",
                      signif(leaf_ttest_model$p.value, 2)),
    x = "Side of Tree",
    y = "Leaf Weight (g)"
  ) +
  theme_minimal() +
  theme(legend.position = "none")

leaf_box_04_plot

✏️ Your turn: The subtitle line uses paste0() to build the label from the model object. What happens to the subtitle if you re-run this with different data? Why is this better than typing the numbers yourself?

Your answer:

Save the plot

▶ Run this:

ggsave("figures/leaf_weight_boxplot_ttest.png",
       plot   = leaf_box_04_plot,
       width  = 5,
       height = 5,
       units  = "in",
       dpi    = 300)

Part 13 · Write a scientific results sentence

✏️ Your turn: Write a complete results sentence as you would in a lab report or paper. Include:

Which group was larger
The word “significantly” (if appropriate)
Test name, t-statistic, df, and p-value in parentheses
Mean ± SE for both groups with units

Use this template:

"[Group] leaves were significantly [heavier / lighter] than [group] leaves
(Welch's two-sample t-test: t(___) = ___, p = ___).
Mean ± SE: [side 1] = ___ ± ___ g, [side 2] = ___ ± ___ g."

Write your sentence here:

Your results sentence:

💡 Key idea: Never write p = 0.000. Write p < 0.001 instead. A p-value is never exactly zero — it is just too small to display at three decimal places.

Part 14 · Review and checkpoint

At this point you should be able to:

✏️ Your turn: Run your entire script from top to bottom with Ctrl/Cmd + Shift + Enter. Does it run without errors?

Ran cleanly?  Y / N
If not, what error appeared:

Part 15 · Going further

This section is optional — work through it if you finish early or want to push deeper.

Test a second variable: `height_cm`

▶ Try this:

# Run the same t-test pipeline for height_cm ---------

# 1. Stats recap
tree_df %>%
  group_by(side) %>%
  summarize(
    n        = sum(!is.na(height_cm)),
    mean_ht  = round(mean(height_cm, na.rm = TRUE), 2),
    sd_ht    = round(sd(height_cm,   na.rm = TRUE), 2),
    se_ht    = round(sd_ht / sqrt(n), 2)
  )

# 2. Shapiro-Wilk for height_cm
tree_df %>% filter(side == "sunny") %>% pull(height_cm) %>% shapiro.test()
tree_df %>% filter(side == "shady") %>% pull(height_cm) %>% shapiro.test()

# 3. Levene's test for height_cm
leveneTest(height_cm ~ side, data = tree_df)

# 4. Welch's t-test for height_cm
height_ttest_model <- t.test(
  height_cm ~ side,
  data        = tree_df,
  var.equal   = FALSE,
  alternative = "two.sided"
)
height_ttest_model

✏️ Your turn: Is leaf height also significantly different between sides? Fill in the results:

t = _____   df = _____   p = _____
Decision:
One-sentence conclusion:

Change the confidence level

By default t.test() reports a 95% CI (conf.level = 0.95). Try 99%:

▶ Try this:

# Run the weight t-test with 99% confidence interval --
t.test(
  weight_g ~ side,
  data       = tree_df,
  var.equal  = FALSE,
  alternative = "two.sided",
  conf.level  = 0.99        # 99% CI instead of 95%
)

✏️ Your turn: How did the 99% CI compare to the 95% CI? Why is the 99% CI wider?

95% CI was: _____ to _____
99% CI was: _____ to _____
Why wider:

What if the groups were not significantly different?

✏️ Your turn: Think about what would happen if both sides had leaves with weight_g around 5.5 ± 2.0 g. Would the t-test be significant? What would change in the output?

Your prediction:

What your finished `figures/` folder should contain

project/
├── data/
│   └── 2026_06_25_tree_experiment_raw_data.xlsx
├── figures/
│   ├── leaf_weight_boxplot.png            <- from Worksheet 03
│   ├── leaf_weight_mean_se.png            <- from Worksheet 03
│   └── leaf_weight_boxplot_ttest.png      <- from this worksheet
└── scripts/
    └── 04_leaf_ttest.R                    <- your script

Getting unstuck

Read the error message out loud. R usually names the line and the problem.
Check the usual suspects:
- Loaded library(car), library(readxl), library(tidyverse)?
- Spelling of column names? Check with names(tree_df)
- Missing ) or %>%?
leveneTest not found? You need library(car) — not just car::leveneTest.
Shapiro-Wilk needs > 3 values. If you filtered to a tiny subset, you may get an error.
t.test() formula: the grouping variable goes on the right of ~ and must have exactly two levels. Check with unique(tree_df$side).
Cheat sheets — https://posit.co/resources/cheatsheets/
Bring the exact error (copy-paste it) to class, Canvas, or office hours.

💡 Key idea: The five-step framework (hypotheses → normality → variance → test → report) is the same for nearly every parametric test you will ever run. Master it here and it transfers directly to ANOVA and regression.

End of Worksheet 04. Next: Worksheet 05 — one-way ANOVA for comparing more than two groups. ### Recap from Worksheet 03

Loaded our Excel leaf data with read_excel()
Used group_by() + summarize() to compute mean, SD, and SE per group
Checked for missing values with sum(!is.na())
Made boxplots with jittered points and mean ± SE plots with stat_summary()
Prediction: shady leaves appeared heavier, taller, and wider than sunny leaves

Today’s Objectives

State null and alternate hypotheses formally before running any test
Check the normality assumption — histogram, QQ plot, and Shapiro-Wilk test
Check the variance assumption — Levene’s test
Run a Welch’s two-sample t-test (var.equal = FALSE, two-tailed)
Read and decode every line of the t.test() output
Make a formal statistical decision and state a conclusion
Write a results sentence in proper scientific format

How to use this worksheet

Work through each part in order — the steps build on each other. Type the code into your script in Positron and run it line by line.

Code blocks marked ▶ Run this should be executed as written.

Blocks marked ✏️ Your turn ask you to write, modify, or interpret something.

The Going further section is optional if you finish early.

Part 1 · Load libraries and data

Libraries

▶ Run this at the top of your script — in this exact order:

# Load all packages at the top of the script -----------
library(readxl)      # reading Excel files
library(tidyverse)   # data manipulation + ggplot2
library(skimr)       # fast dataset summaries
library(car)         # Levene's test for variance

⚠️ Watch out!

If R says "could not find function 'leveneTest'"

you forgot library(car).

Install it once with install.packages("car"), then load it every session library(car)

Load the data

▶ Run this:

# Load the leaf data from the data folder --------------
tree_df <- read_excel("data/2026_06_25_tree_experiment_raw_data.xlsx")
head(tree_df)

✏️ Your turn: Before anything else — do you remember the structure of this data frame? Fill in from memory, then check with glimpse(tree_df).

Number of rows:
Number of columns:
Column holding the grouping variable (sunny/shady):
Column we are testing today:

Part 2 · State your hypotheses FIRST

💡 Key idea: Always write down your hypotheses before you look at the data or run any test. This is how you stay scientifically honest.

✏️ Your turn: Write the null and alternate hypotheses for leaf weight. Use words first, then symbols.

In words:
  H₀ (null hypothesis):

  Hₐ (alternate hypothesis):

In symbols (use μ for "mean"):
  H₀:
  Hₐ:

Test type (circle one):   one-tailed   /   two-tailed

Significance level α:

✏️ Your turn: Why do we use a two-tailed test here even though we predicted shady leaves would be heavier?

Your answer:

Part 3 · Quick descriptive stats review

▶ Run this:

# Recap stats from Lecture 03 — both groups at once ----
recap_df <- tree_df %>%
  group_by(side) %>%
  summarize(
    n       = sum(!is.na(weight_g)),
    mean_wt = round(mean(weight_g, na.rm = TRUE), 2),
    sd_wt   = round(sd(weight_g,   na.rm = TRUE), 2),
    se_wt   = round(sd_wt / sqrt(n), 2)
  )

recap_df

✏️ Your turn: Record the values you will need for the t-test formula:

Shady:  mean = _____ g   SD = _____   n = _____
Sunny:  mean = _____ g   SD = _____   n = _____
Difference in means = _____  g

✏️ Your turn: Looking at the SD values, do the two groups appear to have similar or different variance? (Does one group have much more spread than the other?)

Your observation:

Part 4 · Check normality — histogram

The t-test assumes that data within each group are approximately normally distributed.

▶ Run this:

# Histogram per group — look for bell-shaped distribution
hist_norm_plot <- tree_df %>%
  ggplot(aes(x = weight_g, fill = side)) +
  geom_histogram(binwidth = 1, color = "white", alpha = 0.8) +
  facet_wrap(~side, ncol = 2) +
  labs(
    title = "Leaf Weight Distribution by Side",
    x     = "Leaf Weight (g)",
    y     = "Count"
  ) +
  theme_minimal() +
  theme(legend.position = "none")

hist_norm_plot

✏️ Your turn: Describe the shape of each histogram:

Sunny side shape (bell-shaped, skewed, flat, bimodal?):

Shady side shape:

Any obvious outliers?  Y / N

✏️ Your turn: With only 10 values per group, histograms will look lumpy even if the data are normal. Does this mean we should reject normality? Why or why not?

Your answer:

Part 5 · Check normality — QQ plot

A QQ (quantile-quantile) plot compares your data’s quantiles to what a perfect normal distribution would look like. Points on the diagonal line = normal.

▶ Run this:

# Q-Q plot — points should fall along the diagonal ----
qq_norm_plot <- tree_df %>%
  ggplot(aes(sample = weight_g, color = side)) +
  stat_qq() +
  stat_qq_line(color = "black", linewidth = 0.8) +
  facet_wrap(~side, scales = "free") +
  labs(
    title = "Normal Q-Q Plots by Side",
    x     = "Theoretical Quantiles",
    y     = "Sample Quantiles"
  ) +
  theme_minimal() +
  theme(legend.position = "none")

qq_norm_plot

✏️ Your turn: Do the points for each group fall approximately along the diagonal line? What would it mean if they curved strongly away at the ends?

Sunny side — on the line?  Y / mostly / no
Shady side — on the line?  Y / mostly / no

What strong curvature at the ends would indicate:

Part 6 · Check normality — Shapiro-Wilk test

The Shapiro-Wilk test formally tests whether a sample could have come from a normal distribution.

H₀ (Shapiro-Wilk): the data are normally distributed
p > 0.05 → fail to reject normality → proceed with t-test
p < 0.05 → evidence of non-normality → consider alternatives

▶ Run this:

# Shapiro-Wilk for the sunny side --------------------
tree_df %>%
  filter(side == "sunny") %>%
  pull(weight_g) %>%
  shapiro.test()

# Shapiro-Wilk for the shady side -------------------
tree_df %>%
  filter(side == "shady") %>%
  pull(weight_g) %>%
  shapiro.test()

✏️ Your turn: Record the results and state your decision:

Sunny:  W = _____   p = _____   Decision (normal / not normal):
Shady:  W = _____   p = _____   Decision (normal / not normal):

✏️ Your turn: With n = 10 per group, the Shapiro-Wilk test has low statistical power. What does that mean in plain language for interpreting this result?

Your answer:

⚠️ Watch out! Even if Shapiro-Wilk returns p < 0.05 with very small samples, the t-test may still be appropriate — the QQ plot and histogram give important visual context. None of these checks alone is definitive.

Part 7 · Check variance — Levene’s test

Levene’s test checks whether the two groups have the same population variance.

H₀ (Levene’s): the two groups have equal variance
p > 0.05 → variances are not significantly different
p < 0.05 → variances are significantly different

▶ Run this:

# Levene's test for equal variance -------------------
leveneTest(weight_g ~ side, data = tree_df)

✏️ Your turn: Record the result and decide:

F-statistic = _____   p = _____
Decision (variances equal / not equal):

Your answer:

Part 8 · Run the Welch’s t-test

Now that we have checked our assumptions, we can run the test.

▶ Run this:

# Welch's two-sample t-test --------------------------
leaf_ttest_model <- t.test(
  weight_g ~ side,          # response ~ grouping variable
  data        = tree_df,
  var.equal   = FALSE,      # Welch's — no pooled variance
  alternative = "two.sided" # two-tailed test
)

leaf_ttest_model

✏️ Your turn: Copy the full output below:

Paste or write the t.test() output here:

Part 9 · Decode the output line by line

▶ Run this to extract individual values:

# Pull specific values from the model object ---------
cat("t-statistic:", round(leaf_ttest_model$statistic,  3), "\n")
cat("df (Welch): ", round(leaf_ttest_model$parameter,  2), "\n")
cat("p-value:    ", signif(leaf_ttest_model$p.value,   3), "\n")
cat("95% CI:     ", round(leaf_ttest_model$conf.int[1],2),
    "to", round(leaf_ttest_model$conf.int[2], 2), "g\n")
cat("Mean shady: ", round(leaf_ttest_model$estimate[1],2), "g\n")
cat("Mean sunny: ", round(leaf_ttest_model$estimate[2],2), "g\n")

✏️ Your turn: Match each piece of output to its meaning:

t-statistic = _____

  → This is large/small (circle one) because the difference in means
    is large/small relative to the variability (circle one).

df = _____

  → This is a non-integer. Why? (hint: Welch-Satterthwaite equation)

p-value = _____

  → In plain language, this means: if H₀ were true, the probability
    of seeing a t-statistic this extreme just by chance is _____.

95% CI = _____ to _____ g

  → This CI is for the ________________ (the true difference in means).
    It does / does not include zero (circle one).
    What does it mean when the CI excludes zero?

Mean shady = _____ g
Mean sunny = _____ g

Part 10 · Calculate t by hand

▶ Run this:

# Reproduce the t-statistic manually -----------------
m_sha <- recap_df$mean_wt[recap_df$side == "shady"]
m_sun <- recap_df$mean_wt[recap_df$side == "sunny"]
s_sha <- recap_df$sd_wt[recap_df$side == "shady"]
s_sun <- recap_df$sd_wt[recap_df$side == "sunny"]
n_sha <- recap_df$n[recap_df$side == "shady"]
n_sun <- recap_df$n[recap_df$side == "sunny"]

# Welch's SE of the difference
se_diff  <- sqrt((s_sha^2 / n_sha) + (s_sun^2 / n_sun))

# t-statistic
t_manual <- (m_sha - m_sun) / se_diff

cat("Difference in means:", round(m_sha - m_sun, 2), "g\n")
cat("SE of difference:   ", round(se_diff, 4), "\n")
cat("Manual t:           ", round(t_manual, 3), "\n")

✏️ Your turn: Does your manually calculated t match the value from t.test()? (They may differ very slightly due to rounding in recap_df.)

Manual t = _____
t.test() t = _____
Match?  Y / close / no

✏️ Your turn: Write the Welch’s t-test formula below and label each part:

t = _________________

  Numerator means:
  Denominator means:

Part 11 · Make a formal decision

✏️ Your turn: State your decision and conclusion. Use the formal language of hypothesis testing.

Our p-value:
Our α:

Decision (circle one):   REJECT H₀   /   FAIL TO REJECT H₀

In one sentence, what does this mean biologically?

💡 Key idea: We never say “H₀ is true” or “H₀ is false.” We say “we reject H₀” (evidence was strong enough) or “we fail to reject H₀” (evidence was not strong enough). Statistics gives us a decision rule, not certainty.

Part 12 · Visualize the result

Boxplot with test statistics in the subtitle

▶ Run this:

# Boxplot with t and p embedded in subtitle ----------
leaf_box_04_plot <- tree_df %>%
  ggplot(aes(x = side, y = weight_g, fill = side)) +
  geom_boxplot(alpha = 0.5, outlier.shape = NA) +
  geom_point(
    position = position_jitter(width = 0.15, seed = 42),
    alpha = 0.6, size = 2.5
  ) +
  labs(
    title    = "Leaf Weight by Tree Side",
    subtitle = paste0("Welch's t-test: t = ",
                      round(leaf_ttest_model$statistic, 2),
                      ", p = ",
                      signif(leaf_ttest_model$p.value, 2)),
    x = "Side of Tree",
    y = "Leaf Weight (g)"
  ) +
  theme_minimal() +
  theme(legend.position = "none")

leaf_box_04_plot

Your answer:

Save the plot

▶ Run this:

ggsave("figures/leaf_weight_boxplot_ttest.png",
       plot   = leaf_box_04_plot,
       width  = 5,
       height = 5,
       units  = "in",
       dpi    = 300)

Part 13 · Write a scientific results sentence

✏️ Your turn: Write a complete results sentence as you would in a lab report or paper. Include:

Which group was larger
The word “significantly” (if appropriate)
Test name, t-statistic, df, and p-value in parentheses
Mean ± SE for both groups with units

Use this template:

"[Group] leaves were significantly [heavier / lighter] than [group] leaves
(Welch's two-sample t-test: t(___) = ___, p = ___).
Mean ± SE: [side 1] = ___ ± ___ g, [side 2] = ___ ± ___ g."

Write your sentence here:

Your results sentence:

💡 Key idea: Never write p = 0.000. Write p < 0.001 instead. A p-value is never exactly zero — it is just too small to display at three decimal places.

Part 14 · Review and checkpoint

At this point you should be able to:

✏️ Your turn: Run your entire script from top to bottom with Ctrl/Cmd + Shift + Enter. Does it run without errors?

Ran cleanly?  Y / N
If not, what error appeared:

Part 15 · Going further

This section is optional — work through it if you finish early or want to push deeper.

Test a second variable: `height_cm`

▶ Try this:

# Run the same t-test pipeline for height_cm ---------

# 1. Stats recap
tree_df %>%
  group_by(side) %>%
  summarize(
    n        = sum(!is.na(height_cm)),
    mean_ht  = round(mean(height_cm, na.rm = TRUE), 2),
    sd_ht    = round(sd(height_cm,   na.rm = TRUE), 2),
    se_ht    = round(sd_ht / sqrt(n), 2)
  )

# 2. Shapiro-Wilk for height_cm
tree_df %>% filter(side == "sunny") %>% pull(height_cm) %>% shapiro.test()
tree_df %>% filter(side == "shady") %>% pull(height_cm) %>% shapiro.test()

# 3. Levene's test for height_cm
leveneTest(height_cm ~ side, data = tree_df)

# 4. Welch's t-test for height_cm
height_ttest_model <- t.test(
  height_cm ~ side,
  data        = tree_df,
  var.equal   = FALSE,
  alternative = "two.sided"
)
height_ttest_model

✏️ Your turn: Is leaf height also significantly different between sides? Fill in the results:

t = _____   df = _____   p = _____
Decision:
One-sentence conclusion:

Change the confidence level

By default t.test() reports a 95% CI (conf.level = 0.95). Try 99%:

▶ Try this:

# Run the weight t-test with 99% confidence interval --
t.test(
  weight_g ~ side,
  data       = tree_df,
  var.equal  = FALSE,
  alternative = "two.sided",
  conf.level  = 0.99        # 99% CI instead of 95%
)

✏️ Your turn: How did the 99% CI compare to the 95% CI? Why is the 99% CI wider?

95% CI was: _____ to _____
99% CI was: _____ to _____
Why wider:

What if the groups were not significantly different?

✏️ Your turn: Think about what would happen if both sides had leaves with weight_g around 5.5 ± 2.0 g. Would the t-test be significant? What would change in the output?

Your prediction:

What your finished `figures/` folder should contain

project/
├── data/
│   └── 2026_06_25_tree_experiment_raw_data.xlsx
├── figures/
│   ├── leaf_weight_boxplot.png            <- from Worksheet 03
│   ├── leaf_weight_mean_se.png            <- from Worksheet 03
│   └── leaf_weight_boxplot_ttest.png      <- from this worksheet
└── scripts/
    └── 04_leaf_ttest.R                    <- your script

Getting unstuck

Read the error message out loud. R usually names the line and the problem.
Check the usual suspects:
- Loaded library(car), library(readxl), library(tidyverse)?
- Spelling of column names? Check with names(tree_df)
- Missing ) or %>%?
leveneTest not found? You need library(car) — not just car::leveneTest.
Shapiro-Wilk needs > 3 values. If you filtered to a tiny subset, you may get an error.
t.test() formula: the grouping variable goes on the right of ~ and must have exactly two levels. Check with unique(tree_df$side).
Cheat sheets — https://posit.co/resources/cheatsheets/
Bring the exact error (copy-paste it) to class, Canvas, or office hours.

💡 Key idea: The five-step framework (hypotheses → normality → variance → test → report) is the same for nearly every parametric test you will ever run. Master it here and it transfers directly to ANOVA and regression.

End of Worksheet 04. Next: Worksheet 05 — one-way ANOVA for comparing more than two groups.

Worksheet 04 — Testing Our Hypothesis

In-class Activity 4: The Welch’s Two-Sample t-Test

In-class Activity 4: The Welch’s Two-Sample t-Test

Recap from Worksheet 03

Today’s Objectives

How to use this worksheet

🔮 Predict before you run — and type, don’t paste

🧩 Chunk 1 — Setup, hypotheses & recap (after lecture Chunk 1)

Part 1 · Load libraries and data

Libraries

Load the data

Part 2 · State your hypotheses FIRST

Part 3 · Quick descriptive stats review

🧩 Chunk 2 — Checking assumptions (after lecture Chunk 2)

Part 4 · Check normality — histogram

Part 5 · Check normality — QQ plot

Part 6 · Check normality — Shapiro-Wilk test

Part 7 · Check variance — Levene’s test

🧩 Chunk 3 — Run & interpret (after lecture Chunk 3)

Part 8 · Run the Welch’s t-test

Part 9 · Decode the output line by line

Part 10 · Calculate t by hand

Part 11 · Make a formal decision

🧩 Chunk 4 — Visualize & report (after lecture Chunk 4)

Part 12 · Visualize the result

Boxplot with test statistics in the subtitle

Save the plot

Part 13 · Write a scientific results sentence

Part 14 · Review and checkpoint

Part 15 · Going further

Test a second variable: height_cm

Change the confidence level

What if the groups were not significantly different?

What your finished figures/ folder should contain

Getting unstuck

Today’s Objectives

How to use this worksheet

Part 1 · Load libraries and data

Libraries

Load the data

Part 2 · State your hypotheses FIRST

Part 3 · Quick descriptive stats review

Part 4 · Check normality — histogram

Part 5 · Check normality — QQ plot

Part 6 · Check normality — Shapiro-Wilk test

Part 7 · Check variance — Levene’s test

Part 8 · Run the Welch’s t-test

Part 9 · Decode the output line by line

Part 10 · Calculate t by hand

Part 11 · Make a formal decision

Part 12 · Visualize the result

Boxplot with test statistics in the subtitle

Save the plot

Part 13 · Write a scientific results sentence

Part 14 · Review and checkpoint

Part 15 · Going further

Test a second variable: height_cm

Change the confidence level

What if the groups were not significantly different?

What your finished figures/ folder should contain

Getting unstuck

Test a second variable: `height_cm`

What your finished `figures/` folder should contain

Test a second variable: `height_cm`

What your finished `figures/` folder should contain