Lecture 04 — Testing Our Hypothesis
The two-sample Welch’s t-test from assumptions to results
2026-07-05
Why Comparing Means Is Not Enough
# Remind ourselves what the data look like ---------------
recap_plot <- tree_df %>%
ggplot(aes(x = side, y = weight_g, color = side)) +
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 = "Side", y = "Leaf Weight (g)") +
theme_minimal() +
theme(legend.position = "none")
recap_plot
The means look different — but:
- We only have 10 leaves per group
- There is natural variability within each group
- The groups’ ranges overlap somewhat
A p-value tells us:
“If H₀ were true, how often would we see a gap this large just by chance?”
Small p → the gap is unlikely to be chance.
Step 2 — Check Normality: Histogram
# Histogram per group to visually check normality -------
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
What to look for:
- Roughly bell-shaped — not heavily skewed
- No obvious bimodal peaks
- No extreme outliers
With n = 10 per group, histograms look lumpy — that is normal for small samples.
Histograms are a visual check only. We also use QQ plots and Shapiro-Wilk.
Step 3 — Check Normality: QQ Plot
# Q-Q plot — points should fall along the diagonal line --
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
How to read a QQ plot:
- Points on the line → data follow a normal distribution
- Points curving away at ends → heavy tails or skew
- With only n = 10, some deviation is expected and acceptable
If points follow the line roughly, we are comfortable proceeding with the t-test.
Visualize the Result — Boxplot
# Final boxplot with all points -----------------------
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
Including the test result in the figure:
subtitlepulls t and p directly from the model object- R fills in the actual numbers automatically
- If you re-run with new data, the subtitle updates
Tip
Embedding test statistics in the plot subtitle saves you from copying numbers by hand — and avoids typos in your report.
Visualize the Result — Mean ± SE
# Mean ± SE plot with individual points ---------------
leaf_mean_se_04_plot <- tree_df %>%
ggplot(aes(x = side, y = weight_g, color = side)) +
geom_point(
position = position_jitter(width = 0.15, seed = 42),
alpha = 0.35,
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(
title = "Mean ± SE Leaf Weight by Tree Side",
subtitle = paste0(
"Welch's t: t(",
round(leaf_ttest_model$parameter, 1),
") = ",
round(leaf_ttest_model$statistic, 2),
", p < 0.001"
),
x = "Side of Tree",
y = "Leaf Weight (g)"
) +
theme_minimal() +
theme(legend.position = "none")
leaf_mean_se_04_plot
What the error bars tell you:
- Bars show ± 1 SE — precision of the mean estimate
- Non-overlapping bars suggest a significant difference
Rule of thumb:
If error bars overlap by less than half the bar length, the difference is likely significant at α = 0.05. The p-value from the t-test is the definitive answer.