Activity: Probability & Inference

From the normal curve to the confidence interval

Hands-on activity: the normal distribution, z-scores and areas under the curve, checking normality, standard error, confidence intervals, and one- and two-sample t-tests on the Arctic grayling length dataset.

Author

Bill Perry

Worksheet: Probability & Inference

📥 Download the data 📄 Download the script skeleton

Part 1 · Setup and Data

▶ Run this:

# ── Part 1.1 · Setup and Data ───────────────────────────────────────────
library(janitor)
library(tidyverse)

g_df <- read_csv("data/gray_I3_I8.csv")

i3_df <- g_df %>% filter(lake == "I3")

head(g_df)

▶ Run this — the three numbers everything today depends on:

# ── Part 1.2 · Setup and Data ───────────────────────────────────────────
stats_df <- g_df %>%
  group_by(lake) %>%
  summarize(
    mean_length = mean(length_mm, na.rm = TRUE),
    sd_length   = sd(length_mm, na.rm = TRUE),
    n           = sum(!is.na(length_mm)),
    se_length   = sd_length / sqrt(n),
    .groups = "drop"
  )
stats_df

✏️ Your turn: Which lake has more fish measured? Which has the larger mean length? Which has the larger SD?

________________________

A tool you will reuse all semester

You just typed out mean, sd, n and se by hand. You will want those same four numbers in every analysis from here on — so let’s package them into a function and never type that block again.

Note

This one is a tool, not a lesson — it is already filled in for you in the script skeleton. You do not have to type it. Read it, run it, and then use it.

{ variable } (the “curly-curly” braces) is what lets you hand the function a bare column name like length_mm instead of a quoted string. It is the one piece of syntax here worth remembering.

▶ Run this — define it once:

# ── Part 1.3 · A reusable summary function ──────────────────────────────
summary_stats <- function(data, variable) {
  data %>%
    summarize(
      n        = sum(!is.na({{ variable }})),
      mean     = mean({{ variable }}, na.rm = TRUE),
      sd       = sd({{ variable }}, na.rm = TRUE),
      se       = sd / sqrt(n),
      ci_lower = mean - qt(0.975, df = n - 1) * se,
      ci_upper = mean + qt(0.975, df = n - 1) * se,
      .groups  = "drop"
    ) %>%
    mutate(across(c(mean, sd, se, ci_lower, ci_upper),
                  ~ round_half_up(.x, 2)))
}

▶ Run this — now use it. Three ways, no retyping:

# ── Part 1.4 · A reusable summary function ──────────────────────────────
g_df %>% group_by(lake) %>% summary_stats(length_mm)   # per lake

g_df %>% summary_stats(length_mm)                      # both lakes pooled

g_df %>% group_by(lake) %>% summary_stats(mass_g)      # a different column

✏️ Your turn: Compare the first result to stats_df from Part 1.2. Do the mean, sd and se columns match?

________________________

✏️ Your turn: The function also returns ci_lower and ci_upper — a 95% confidence interval. We have not explained those yet. That is Part 6. For now, just notice they are there.

Part 2 · The Normal Distribution

📖 Read more: Whitlock & Schluter Ch. 10 (The Normal Distribution); Gotelli & Ellison Ch. 2.

A normal distribution is fully described by just two numbers: its mean (μ) and its standard deviation (σ). Let’s draw the normal curve that matches our fish and see how well it fits.

▶ Run this:

# ── Part 2.1 · The Normal Distribution ──────────────────────────────────
i3_mean <- mean(i3_df$length_mm, na.rm = TRUE)
i3_sd   <- sd(i3_df$length_mm, na.rm = TRUE)

i3_df %>%
  ggplot(aes(x = length_mm)) +
  geom_histogram(aes(y = after_stat(density)), binwidth = 10,
                 fill = "lightblue", colour = "white") +
  stat_function(fun = dnorm, args = list(mean = i3_mean, sd = i3_sd),
                colour = "red", linewidth = 1) +
  labs(x = "Length (mm)", y = "Density")

✏️ Your turn: Where does the red curve sit above the blue bars, and where does it sit below them? Look especially at the right-hand side.

________________________

Part 3 · Z-Scores — Putting Everything on One Ruler

📖 Read more: Whitlock & Schluter Ch. 10 — standardizing a value.

There is a different normal curve for every μ and σ. Standardizing rescales any of them into the same units: standard deviations from the mean.

▶ Run this:

# ── Part 3.1 · Z-Scores ─────────────────────────────────────────────────
i3_df <- i3_df %>%
  mutate(z_score = (length_mm - i3_mean) / i3_sd)

i3_df %>%
  select(lake, length_mm, z_score) %>%
  head()

i3_df %>%
  ggplot(aes(x = z_score)) +
  geom_histogram(binwidth = 0.25, fill = "blue", alpha = 0.7) +
  labs(x = "z (SDs from the mean)", y = "Count")

✏️ Your turn: Compare this histogram to the one in Part 2. What changed — the shape, or just the axis numbers?

________________________

▶ Run this — what fraction of our fish really are within 1 SD?

# ── Part 3.2 · Z-Scores ─────────────────────────────────────────────────
within_1sd <- mean(abs(i3_df$z_score) <= 1, na.rm = TRUE)
round_half_up(100 * within_1sd, 1)

✏️ Your turn: A perfect normal distribution gives 68%. What did you get, and is the gap big or small?

________________________

Part 4 · Area Under the Curve = Probability

📖 Read more: Whitlock & Schluter Ch. 5 (Probability) and Ch. 10.

The whole point of a density curve is that area under it is probability. pnorm() gives us that area without any integration.

Function	You give it	You get back
`pnorm(z)`	a z	the area to the left
`qnorm(p)`	an area	the z that cuts it off
`dnorm(z)`	a z	the height of the curve

▶ Run this:

# ── Part 4.1 · Area Under the Curve ─────────────────────────────────────
pnorm(1.96)          # area to the LEFT of 1.96
1 - pnorm(1.96)      # area to the RIGHT
qnorm(0.975)         # the z with 97.5% to its left - the inverse of pnorm

▶ Run this — the worked example from lecture: how rare is a 300 mm grayling?

# ── Part 4.2 · Area Under the Curve ─────────────────────────────────────
z_300 <- (300 - i3_mean) / i3_sd
round_half_up(z_300, 2)                          # should be about 1.22

round_half_up(100 * (1 - pnorm(z_300)), 1)       # % of fish LONGER than 300 mm

▶ Run this — and now run it backwards, from a probability to a length:

# ── Part 4.3 · Area Under the Curve ─────────────────────────────────────
z_top5 <- qnorm(0.95)
length_top5 <- i3_mean + z_top5 * i3_sd
round_half_up(length_top5, 1)

✏️ Your turn: In one sentence, what does that last number mean to someone fishing lake I3?

________________________

✏️ Your turn: Use pnorm() to find the percentage of I3 fish shorter than 240 mm. Convert 240 to a z-score first.

# ── Part 4.4 · Area Under the Curve ─────────────────────────────────────
# Write your code in the R script:

Tip

🚀 If you finish early: Shade the tail you just calculated.

# ── Part 4.5 · Area Under the Curve ─────────────────────────────────────
i3_df %>%
  ggplot(aes(x = length_mm)) +
  geom_density(fill = "lightblue", alpha = 0.5) +
  geom_vline(xintercept = 300, colour = "red", linetype = "dashed")

Part 5 · Checking the Assumption

📖 Read more: Whitlock & Schluter Ch. 13 (Handling Violations of Assumptions).

Every number in Part 4 assumed the fish are normally distributed. Now we test that.

Watch the hypotheses — they run backwards from what you expect:

H₀: the data ARE normal
Hₐ: the data are NOT normal
so p < 0.05 means NOT normal — a small p-value here is bad news

▶ Run this:

# ── Part 5.1 · Checking the Assumption ──────────────────────────────────
shapiro.test(i3_df$length_mm)

i3_df %>%
  ggplot(aes(sample = length_mm)) +
  stat_qq() +
  stat_qq_line(colour = "red") +
  labs(x = "Theoretical quantiles", y = "Sample quantiles")

✏️ Your turn: What is the p-value, and what does it say about normality? On the Q-Q plot, where do the points leave the red line?

________________________

▶ Run this — so how wrong was our 300 mm answer? Compare theory to the actual fish:

# ── Part 5.2 · Checking the Assumption ──────────────────────────────────
round_half_up(100 * (1 - pnorm(300, i3_mean, i3_sd)), 1)   # normal theory
round_half_up(100 * mean(i3_df$length_mm > 300, na.rm = TRUE), 1)  # actual fish

✏️ Your turn: How far apart are those two numbers? Why does it matter that the disagreement is out in the tail rather than in the middle?

________________________

Tip

🚀 If you finish early: Run shapiro.test() on lake I8 as well. Is it normal? Does the Q-Q plot agree with the p-value?

# ── Part 5.3 · Checking the Assumption ──────────────────────────────────
# Write your code in the R script:

Part 6 · SD, SE, and Confidence Intervals

📖 Read more: Whitlock & Schluter Ch. 4 (Estimating with Uncertainty) — this is the chapter that separates SD from SE.

SD describes how much individual fish vary. SE describes how much a sample mean would vary from study to study. They answer different questions and must not be swapped.

▶ Run this:

# ── Part 6.1 · SD, SE, and Confidence Intervals ─────────────────────────
i3_n  <- sum(!is.na(i3_df$length_mm))
i3_se <- i3_sd / sqrt(i3_n)

tibble(
  n  = i3_n,
  SD = round_half_up(i3_sd, 1),
  SE = round_half_up(i3_se, 1)
)

✏️ Your turn: If we measured 4× as many fish, which of those two numbers would shrink, and by how much?

________________________

▶ Run this — build a 95% CI by hand, the way the formula reads: \(\bar{y} \pm t \cdot SE\)

# ── Part 6.2 · SD, SE, and Confidence Intervals ─────────────────────────
t_crit <- qt(0.975, df = i3_n - 1)   # two-tailed, 95%
t_crit

ci_lower <- i3_mean - t_crit * i3_se
ci_upper <- i3_mean + t_crit * i3_se

round_half_up(c(ci_lower, ci_upper), 1)

▶ Run this — now check yourself two ways: against R’s t.test(), and against the function you wrote back in Part 1.3.

# ── Part 6.3 · SD, SE, and Confidence Intervals ─────────────────────────
t.test(i3_df$length_mm)$conf.int          # R's answer

i3_df %>% summary_stats(length_mm)        # your function's answer

✏️ Your turn: Do all three intervals — yours by hand, t.test(), and summary_stats() — agree?

________________________

💡 That is the payoff. The ci_lower / ci_upper columns were in your function’s output back in Part 1.4, before you knew what they meant. Now you do — and you get them on any grouped dataset for free:
g_df %>% group_by(lake) %>% summary_stats(length_mm)
One line, and every lake gets a mean, an SE, and a confidence interval.

✏️ Your turn: Write the I3 interval as a sentence you could put in a paper.

________________________

▶ Run this — why t and not 1.96? Take a small sample and watch the multiplier change:

# ── Part 6.4 · SD, SE, and Confidence Intervals ─────────────────────────
set.seed(42)
small_sample <- i3_df %>% slice_sample(n = 10)

s_mean <- mean(small_sample$length_mm)
s_se   <- sd(small_sample$length_mm) / sqrt(10)

tibble(
  method = c("z (wrong here)", "t (correct)"),
  crit   = round_half_up(c(1.96, qt(0.975, df = 9)), 3),
  lower  = round_half_up(s_mean - crit * s_se, 1),
  upper  = round_half_up(s_mean + crit * s_se, 1),
  width  = round_half_up(upper - lower, 1)
)

✏️ Your turn: Which interval is wider, and why is the wider one the honest one?

________________________

Note

The interpretation to get right

❌ “There is a 95% chance μ is in my interval.” ✅ “95% of intervals built this way capture μ.”

μ is a fixed number. The interval is what is random.

Part 7 · One-Sample t-Test

📖 Read more: Whitlock & Schluter Ch. 11 (Inference for a Normal Population) and Ch. 6 (Hypothesis Testing).

A CI asks where is μ? A t-test asks is μ equal to some specific value?

▶ Run this — do I3 fish differ from a historical mean of 260 mm?

# ── Part 7.1 · One-Sample t-Test ────────────────────────────────────────
t.test(i3_df$length_mm, mu = 260)

✏️ Your turn: Fill these in from the output.

H0            = ______________________________________________
Ha            = ______________________________________________
t             = __________    df = __________
p-value       = __________
95% CI        = __________ to __________
Decision at α = 0.05: ________________________________________

✏️ Your turn: Does the confidence interval contain 260? Explain how that answer and the p-value are telling you the same thing.

________________________

Part 8 · Two-Sample t-Test

📖 Read more: Whitlock & Schluter Ch. 12 (Comparing Two Means) and Ch. 6 on one- vs. two-tailed tests.

Formulate the hypotheses first, for the question “Are fish in Lake I8 longer than fish in Lake I3?”

H0 = ________________________________________________________
Ha = ________________________________________________________

This question has a direction, so it is one-tailed. R orders lake alphabetically and tests I3 − I8, so “I8 is longer” means that difference is less than zero.

▶ Run this:

# ── Part 8.1 · Two-Sample t-Test ────────────────────────────────────────
t.test(length_mm ~ lake, data = g_df, alternative = "less")

✏️ Your turn: What do you conclude? Report the difference in mm, not just the p-value.

________________________

✏️ Your turn: The output says “Welch Two Sample t-test”. Look back at the SDs in Part 1.2. Why is Welch the right default here?

________________________

Tip

🚀 If you finish early: Rerun with alternative = "two.sided". Does the p-value change? Does the conclusion? Which should you have chosen before seeing the data?

# ── Part 8.2 · Two-Sample t-Test ────────────────────────────────────────
# Write your code in the R script:

Review and checkpoint

At this point you can:

Draw a normal curve over real data and see where it fits and where it doesn’t
Convert values to z-scores and explain what standardizing does and does not change
Use pnorm() for areas and qnorm() for cutoffs, in both directions
Run and read a Shapiro-Wilk test, remembering that p < 0.05 means not normal
Say when to use SD and when to use SE
Build a 95% CI by hand and state correctly what the 95% refers to
Explain why the multiplier is t rather than 1.96
Run and interpret one- and two-sample t-tests

Note

📤 What to turn in before next class

Your script — scripts/05_probability_inference.R
This worksheet, with your written answers

Readings

Before next class — chapter PDFs are on Canvas:

Whitlock & Schluter — Ch. 10, The Normal Distribution; Ch. 4, Estimating with Uncertainty; Ch. 6, Hypothesis Testing. If you read only one, read Ch. 4
Gotelli & Ellison — Ch. 2, Random Variables and Probability Distributions; Ch. 4, Framing and Testing Hypotheses

Worksheet part	Read
Parts 1-2 — summary stats, the normal curve	W&S Ch. 3, Ch. 10
Parts 3-4 — z-scores, area as probability	W&S Ch. 10; Gotelli Ch. 2
Part 5 — checking normality	W&S Ch. 13
Part 6 — SD, SE, confidence intervals	W&S Ch. 4; Ch. 11
Part 7 — one-sample t-test	W&S Ch. 11, Ch. 6
Part 8 — two-sample t-test	W&S Ch. 12; Gotelli Ch. 4

If you want more: R4DS §13 — Numbers for mean(), sd() and rounding, and §9 — Layers for the plots.

Other Formats

Worksheet: Probability & Inference

Part 1 · Setup and Data

A tool you will reuse all semester

Part 2 · The Normal Distribution

Part 3 · Z-Scores — Putting Everything on One Ruler

Part 4 · Area Under the Curve = Probability

Part 5 · Checking the Assumption

Part 6 · SD, SE, and Confidence Intervals

Part 7 · One-Sample t-Test

Part 8 · Two-Sample t-Test

Review and checkpoint

Readings