Lecture 09 — T-Tests I: Setting Up the Test

Hypotheses, the t-statistic, and checking assumptions before we test

Why comparing means isn’t enough, the t-statistic, why Welch’s is the safe default, and checking normality and variance before we ever run a test.

Where we left off (Lectures 05–06)

• Wrangling — filter(), select(), mutate(), arrange() as the core tidyverse verbs
• Descriptive stats — mean, median, SD, SE with group_by() + summarize()
• NA handling — sum(!is.na()) is the safe way to count n
• Boxplot + jittered points; mean ± SE with stat_summary() (Lecture 03, GGPlot I)
• Prediction: shady-side leaves would be heavier than sunny-side leaves

Note

✅ Key idea from Lecture 06

We described the pattern in our data. Over the next two lectures we formally test whether the sides really differ — today we set the test up, next time we run it.

Goals for Today

• Understand why comparing means is not enough — we need a test
• Meet the two-sample t-test and why we default to Welch’s
• State null and alternate hypotheses formally
• Check assumptions before running any test
  • normality: histogram + Shapiro-Wilk
  • variance: Levene’s test

Tip

🖐 Try it yourself

By the end of today your data will be fully checked and ready to test. Next lecture, we run it.

Tools added today:

• car — Levene’s variance test

References:

• 📖 Whitlock & Schluter, Ch. 4 — Estimating with Uncertainty
• 📖 R4DS §3

Both are in the course readings/ folder.

Today’s naming:

• test objects → _model
• plots → _plot

Our Hypotheses — A Reminder

Biological prediction:

Shady-side leaves will be larger and heavier — they need more surface area to capture the limited light filtering through the canopy.

	Formal statement
H₀ — null	\(\mu_{shady} = \mu_{sunny}\) — no difference in mean leaf mass
Hₐ — alternate	\(\mu_{shady} \neq \mu_{sunny}\) — mean leaf mass differs by side

We use a two-tailed test — we will detect a difference in either direction.

Why two-tailed?

• We predicted shady > sunny
• A one-tailed test would only reject H₀ in that direction
• Two-tailed is more conservative and more broadly accepted
• If the result is significant two-tailed, it is also significant one-tailed

α = 0.05 — our threshold for rejecting H₀

How to Use These Slides — Predict · Type · Run

This lecture runs in two short chunks. After each chunk you switch to the activity and type the code yourself into your R script.

For every code block, do three things:

1. Predict — before it runs, say what you think the output will be
2. Type it out by hand — do not copy-paste
3. Run it and compare to your prediction

Note

✅ Why bother? (the evidence)

• A p-value is easy to memorize and hard to actually understand. Guessing “above or below 0.05” before you see a test result forces you to reason about the test instead of reading a number off the screen.
• Typing var.equal = FALSE yourself — rather than pasting it — is what makes you notice it’s there at all, instead of skating past the one argument that separates Welch’s from the standard t-test.

Load Libraries and Data

# Load all packages at the top ----------------------------
library(readxl) # reading Excel files
library(tidyverse) # data manipulation + ggplot2
library(janitor) # clean_names()
library(car) # Levene's test for variance equality

# Load the leaf data from the data folder -----------------
leaf_df <- read_excel("data/2026_09_03_data_sci_leaf_area.xlsx") %>%
  clean_names()

Note

car is the Companion to Applied Regression package. It contains leveneTest(), which we use to check whether our two groups have similar variance.

Install once: install.packages("car")

🧩 Chunk 1 of 2 · Why We Test & the t-Statistic

We will cover: why comparing means isn’t enough, the five-step framework we’ll use for every test this semester, what a two-sample t-test is, and why we default to Welch’s.

🛑 After this chunk: Activity Parts 1–2 (load data, recap the descriptive stats).

Why Comparing Means Is Not Enough

# Remind ourselves what the data look like ---------------
recap_plot <- leaf_df %>%
  ggplot(aes(x = shade, y = mass_g, color = shade)) +
  geom_point(
    position = position_jitter(width = 0.15, seed = 42),
    alpha = 0.5,
    size = 2
  ) +
  stat_summary(fun = mean, geom = "point", size = 4) +
  stat_summary(
    fun.data = mean_se,
    geom = "errorbar",
    width = 0.15,
    linewidth = 0.9
  ) +
  labs(x = "Shade", y = "Leaf Mass (g)") +
  theme_minimal() +
  theme(legend.position = "none")
recap_plot

The two means look close — and with this much scatter, they could easily be the same:

• About 25 leaves per group
• Lots of natural variability within each group
• The groups’ ranges overlap heavily

A p-value tells us:

“If H₀ were true, how often would we see a gap at least this large just by chance?”

Large p → a gap this small is nothing unusual.

Five Steps to a Hypothesis Test

Every hypothesis test this semester — t-test, ANOVA, regression — follows the same sequence:

Step	Action	When
1	State H₀ and Hₐ	Today
2	Check normality	Today
3	Check variance	Today
4	Run the test	Next lecture
5	Interpret and report	Next lecture

Important

Check assumptions BEFORE running the test.

If your data seriously violate assumptions, the p-value from the t-test is unreliable.

With ~25 per group, our main concern is gross non-normality. The t-test is quite robust to mild departures.

What Is a Two-Sample t-Test?

A two-sample t-test compares the means of two independent groups.

It asks: Could both groups plausibly be drawn from populations with the same mean?

The t-statistic is:

\[t = \frac{\bar{x}_1 - \bar{x}_2}{\text{SE}_{\text{diff}}}\]

• Numerator: the observed difference in means
• Denominator: the uncertainty in that difference (SE of the difference)

Large |t| → the difference is large relative to the noise → small p-value

Component	Our context
Group 1	Shady leaves
Group 2	Sunny leaves
Measurement	mass_g
H₀	\(\mu_{shady} = \mu_{sunny}\)
Hₐ	\(\mu_{shady} \neq \mu_{sunny}\)

Why Welch’s? — Don’t Assume Equal Variance

Standard t-test — pools the two variances into one estimate. Requires both groups to have the same population variance.

Welch’s t-test — uses each group’s variance separately. Does not require equal variance.

\[t_W = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}}\]

The denominator is the true SE of the difference — no pooling.

Why always use Welch’s?

• When variances are equal: almost identical result — minimal loss of power
• When variances are not equal: standard t-test gives inflated Type I error (false positives)
• Welch’s protects you either way

Tip

Best practice: use Welch’s as your default for all two-group comparisons.

In R: var.equal = FALSE

Seeing the Formula in Action

Note

🔮 Predict first: t = difference ÷ SE-of-the-difference, and the two means differ by only about 0.02 g. Do you expect |t| to land nearer 0.5, 3, or 10? Predict before the numbers appear.

# Watch the instructor plug numbers into the formula ----
stats_df <- leaf_df %>%
  group_by(shade) %>%
  summarize(
    n = sum(!is.na(mass_g)),
    m = mean(mass_g, na.rm = TRUE),
    s = sd(mass_g, na.rm = TRUE)
  )

m_sha <- stats_df$m[stats_df$shade == "shady"]
m_sun <- stats_df$m[stats_df$shade == "sunny"]
s_sha <- stats_df$s[stats_df$shade == "shady"]
s_sun <- stats_df$s[stats_df$shade == "sunny"]
n_sha <- stats_df$n[stats_df$shade == "shady"]
n_sun <- stats_df$n[stats_df$shade == "sunny"]

se_diff <- sqrt((s_sha^2 / n_sha) + (s_sun^2 / n_sun))
t_manual <- (m_sha - m_sun) / se_diff

cat("Difference in means:", round(m_sha - m_sun, 3), "g\n")

Difference in means: 0.018 g

cat("SE of difference:   ", round(se_diff, 4), "\n")

SE of difference:    0.0505 

cat("t-statistic:        ", round(t_manual, 3), "\n")

t-statistic:         0.356 

Step by step:

1. Difference in means: shady − sunny ≈ 0.02 g
2. SE of the difference: \(\sqrt{\frac{s^2_{shady}}{n_{shady}} + \frac{s^2_{sunny}}{n_{sunny}}}\)
3. t = difference / SE

The observed difference is well under one standard error from zero — right in the range you’d expect if H₀ were true. You’ll try this yourself with a calculator in the activity’s optional section — today the goal is just to see where the number comes from.

→ ACTIVITY 9 Parts 1–2 now

🛑 Do Activity Parts 1–2 now

Load the data and recompute the group means, SD, and SE. Predict each output, type it, run it.

🧩 Chunk 2 of 2 · Steps 1–3: Hypotheses & Assumptions

We will cover: stating H₀ / Hₐ formally, checking normality (histogram + Shapiro-Wilk), and checking equal variance (Levene’s).

🛑 After this chunk: Activity Parts 3–6 (hypotheses, histogram, Shapiro-Wilk, Levene’s).

Step 1 — Hypotheses (Formal)

Null hypothesis:

\[H_0: \mu_{shady} = \mu_{sunny}\]

Shady and sunny leaves have the same mean mass in the population.

Alternate hypothesis (two-tailed):

\[H_A: \mu_{shady} \neq \mu_{sunny}\]

The mean leaf masses differ between sides — in either direction.

Significance level: α = 0.05

We will reject H₀ if p < 0.05.

Why state hypotheses first?

• Keeps the analysis honest
• Prevents “HARKing” (Hypothesizing After Results are Known)
• Forces you to commit to a direction (one-tailed) or not (two-tailed) before seeing the data

Our commitment: two-tailed, α = 0.05, before running any test.

Step 2 — Check Normality: Histogram

# Histogram per group to visually check normality -------
hist_norm_plot <- leaf_df %>%
  ggplot(aes(x = mass_g, fill = shade)) +
  geom_histogram(binwidth = 0.05, color = "white", alpha = 0.8) +
  facet_wrap(~shade, ncol = 2) +
  labs(
    title = "Leaf Mass Distribution by Shade",
    x = "Leaf Mass (g)",
    y = "Count"
  ) +
  theme_minimal() +
  theme(legend.position = "none")
hist_norm_plot

What to look for:

• Roughly bell-shaped — not heavily skewed
• No obvious bimodal peaks
• No extreme outliers

With ~25 per group these are readable, but still lumpy — that is normal for real ecological data.

Note

A QQ plot is another common visual check — same idea, different picture. You’ll find one in the activity’s optional section if you want to compare.

Step 2 — Check Normality: Shapiro-Wilk Test

Note

🔮 Predict first: Shapiro-Wilk’s H₀ is “the data are normal.” From the histogram you just saw, predict for each side: will p be above or below 0.05?

# Shapiro-Wilk test for the sunny side -----------------
leaf_df %>%
  filter(shade == "sunny") %>%
  pull(mass_g) %>%
  shapiro.test()

    Shapiro-Wilk normality test

data:  .
W = 0.95266, p-value = 0.2876

# Shapiro-Wilk test for the shady side -----------------
leaf_df %>%
  filter(shade == "shady") %>%
  pull(mass_g) %>%
  shapiro.test()

    Shapiro-Wilk normality test

data:  .
W = 0.92364, p-value = 0.0427

Shapiro-Wilk H₀: the data are normally distributed.

• p > 0.05 → fail to reject normality → proceed with t-test
• p < 0.05 → evidence of non-normality → consider a non-parametric alternative

Important

One side may fail, the other pass.

A borderline Shapiro result (p just under 0.05) with a roughly bell-shaped histogram is not a reason to abandon the t-test — it is robust to mild non-normality. Read the histogram and the test together.

Step 3 — Check Variance: Levene’s Test

# Levene's test — are the variances similar? -----------
leveneTest(mass_g ~ shade, data = leaf_df)

Levene's Test for Homogeneity of Variance (center = median)
      Df F value  Pr(>F)  
group  1  2.9667 0.09105 .
      51                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Levene’s test H₀: the two groups have equal variance.

• p > 0.05 → variances are not significantly different
• p < 0.05 → variances are significantly different

Our response regardless of the result:

We will use Welch’s t-test (var.equal = FALSE).

Welch’s is valid whether or not the variances are equal. We run Levene’s to understand our data, not to choose our test.

Live Demo — Watch It Break (on purpose)

Suppose I forgot library(car) and run Levene’s:

leveneTest(mass_g ~ shade, data = leaf_df)

R stops with:

Error in leveneTest(mass_g ~ shade, data = leaf_df) :
  could not find function "leveneTest"

The fix — load the package that contains the function:

library(car)
leveneTest(mass_g ~ shade, data = leaf_df)

Tip

✅ Why show a broken run?

leveneTest() lives inside car, not base R — so R has no idea what the function is until the package is loaded. Seeing that exact error message once, on purpose, means you won’t waste ten minutes hunting for a typo the next time it happens to you.

→ ACTIVITY 9 Parts 3–6 now

🛑 Go to Activity 9 — Parts 3–6

Close the slides. State your hypotheses, then check normality (histogram + Shapiro-Wilk) and variance (Levene’s). Predict every p-value before you run it.

What We Learned Today

• Why we need a hypothesis test — means alone are not enough
• The five-step framework we’ll reuse all semester
• Welch’s t-test — use var.equal = FALSE as the safe default
• Two-tailed test — detects differences in either direction
• Checking assumptions before testing: normality (histogram + Shapiro-Wilk), variance (Levene’s)

References:

• 📖 Whitlock & Schluter, Ch. 4 — Estimating with Uncertainty
• 📖 R4DS §3

Up next — Lecture 10, T-Tests II:

• Run Welch’s t.test()
• Decode the output line by line
• Make a formal decision and write a results sentence