Homework 03 — Cisco Body Size in Trout Lake, Wisconsin

Descriptive statistics and a two-sample t-test on long-term fish monitoring data

tidyverse

descriptive-stats

t-test

Apply wrangling, descriptive statistics, and a two-sample Welch’s t-test to a real long-term fish dataset (weeks 2 lectures/worksheets 03–04).

Author

Name: _________________________________ | Date: _____________

A Cisco (*Coregonus artedi*), also called lake herring or tullibee — a cold-water species native to the Great Lakes region and an important prey fish for lake trout, walleye, and northern pike. (Public domain / Wikimedia Commons)

Overview

The North Temperate Lakes Long-Term Ecological Research Program (NTL-LTER) in Wisconsin has monitored fish populations since 1981 — one of the longest continuous fish datasets in North America. Cisco (Coregonus artedi) are especially important because they depend on cold, oxygen-rich water. As lakes warm with climate change, cisco are predicted to decline. Understanding how body size varies among years is one way biologists track whether populations are changing.

In Lectures 03–04 and Worksheets 03–04 you learned to wrangle data, calculate descriptive statistics, and run a two-sample Welch’s t-test. This homework asks you to apply all those skills to a new, real dataset and answer: Did cisco body weight differ between 1981 and 1982 in Trout Lake?

Submit to Canvas: (1) your completed .R script and (2) two saved PNG figures. Fill in all answer boxes in the script as # comments below the relevant code. The question numbers here match the # Q__ markers in the skeleton script.

The Data

Column	Type	What it holds
`lakeid`	chr	lake code (`TR` = Trout Lake)
`year4`	num	year of catch (1981–2006)
`sampledate`	chr	sampling date (MM/DD/YYYY)
`gearid`	chr	gill-net mesh size code
`spname`	chr	species (always `CISCO`)
`length`	num	total body length (mm)
`weight`	num	body weight (g)
`sex`	chr	M, F, I, or NA

Source: Ogle DH (2023). FSAdata R package / NTL-LTER, UW–Madison.

Part 1 · Load and inspect (→ see Lecture 02, Worksheet 02)

After loading the data what is the size of the dataframe

Q1 — Total rows in the raw dataset: ________

Q2 — Total columns: ________

How many fish were caught each year.

Q3 — Year with the most fish: ________ n = ________

Q4 — Year with the second most fish: ________ n = ________

Part 2 · Wrangle (→ see Lecture 03, Worksheet 03)

Complete the pipeline in the skeleton script. After filtering to 1981 and 1982 (removing any fish with missing length, missing weight, or weight = 0):

Q5 — Rows remaining after filtering: ________

Check with count(cisco_df, year):

Q6 — n for 1981: ________ Q7 — n for 1982: ________

Arrange the data by length descending and look at the five largest fish.

Q8 — Year of the heaviest fish: ________ Weight: ________ g

Part 3 · Descriptive statistics (→ see Lecture 03, Worksheet 03)

Run your group_by(year) %>% summarize(...) pipeline (see Q in the skeleton).

Fill in the table below from the cisco_stats_df output:

	1981	1982
n
Mean weight (g)
SD weight (g)
SE weight (g)
Mean length (mm)

Q9 — Is 1982 mean weight higher than 1981? Y / N

Q10 — Difference in mean weight (1982 − 1981): ________ g

Part 4 · Visualize (→ see Lecture 03, Worksheet 03)

Make and save a boxplot with jittered points and a mean ± SE plot (see skeleton).

Describe what you see in the boxplot:

Q11 — Degree of overlap between the two groups (none / some / substantial): ________________

Q12 — Approximate median weight for 1981 (from the box): ________ g

Q13 — Approximate median weight for 1982 (from the box): ________ g

Q14 — Do the SE error bars overlap? Y / N: ________ What does this suggest? ________________

Part 5 · Five-step Welch’s t-test (→ see Lecture 04, Worksheet 04)

Work through each step in order.

Step 1 — Hypotheses

Write your hypotheses before looking at the test result.

Q15 — H₀ (plain English):

Q16 — Hₐ (plain English):

Significance level: α = ________ Test type (one- or two-tailed): ________

Step 2 — Normality: histogram

Make the histogram of body weight, faceted by year (see skeleton).

Q17 — Shape of 1981 distribution (bell-shaped / right-skewed / other): ________________

Q18 — Shape of 1982 distribution: ________________

Step 3 — Normality: QQ plot + Shapiro-Wilk

Run shapiro.test() on each year’s weights separately (see skeleton).

Q19 — Shapiro-Wilk 1981: W = ________ p = ________

Q20 — Shapiro-Wilk 1982: W = ________ p = ________

Note

Both groups may fail Shapiro-Wilk (p < 0.05). With n > 80 the t-test is robust to non-normality because of the Central Limit Theorem — the sampling distribution of the mean is approximately normal even when raw data are not. Check the QQ plot: if points roughly follow the line without extreme curvature, proceed. (See Lecture 04, Step 3)

Q21 — Decision: proceed with the t-test? Y / N Reason:

Step 4 — Variance: Levene’s test

Run leveneTest(weight ~ year, data = cisco_df) (see skeleton).

Q22 — Levene’s F: ________ p: ________

Q23 — Are the variances equal? Y / N

Q24 — Does this change which t-test we run? Y / N Why:

Step 5 — Run the test and interpret

Run t.test(y ~ x, data = df, var.equal = FALSE, alternative = "two.sided").

Q25 — t-statistic: ________

Q26 — df (Welch-Satterthwaite — probably not a whole number): ________

Q27 — p-value (write the full number, not just “< 0.05”): ________

Q28 — 95% CI for the difference in means: ________ to ________ g

Q29 — Mean weight 1981: ________ g Mean weight 1982: ________ g

Q30 — Is p < α? Y / N Decision (reject / fail to reject H₀): ________________

Part 6 · Results sentence (→ see Lecture 04)

Complete the sentence below by filling in every blank:

Q31: “Cisco body weight was significantly [higher / lower] in ______ than in ______ (Welch’s t-test: t(______) = ______, p ______). Mean ± SE body weight was ______ ± ______ g in 1981 and ______ ± ______ g in 1982, a difference of ______ g.”

Then write 1–2 sentences suggesting a biological explanation for why body size differed between these two consecutive years:

Q32 — Biological explanation:

Submission checklist

Before uploading to Canvas, confirm:

cisco_skeleton.R runs top to bottom without errors
All # Q__ answers are filled in the script
figures/cisco_weight_boxplot.png exists (5 × 5 in, dpi = 300)
figures/cisco_weight_mean_se.png exists (5 × 5 in, dpi = 300)
Results sentence (Q31) is complete with all blanks filled
Biological explanation (Q32) is written

Going further (optional — no points)

Run the code below in your script to visualize the full 25-year time series of mean Cisco body weight. Does body size show a trend over the monitoring period?

cisco_raw_df %>%
  filter(!is.na(weight), weight > 0) %>%
  group_by(year4) %>%
  summarize(n = n(), mean_wt = mean(weight, na.rm = TRUE),
            se_wt = sd(weight, na.rm = TRUE) / sqrt(n)) %>%
  ggplot(aes(x = year4, y = mean_wt)) +
  geom_point(aes(size = n), color = "steelblue", alpha = 0.8) +
  geom_line(color = "steelblue", linewidth = 0.6) +
  geom_errorbar(aes(ymin = mean_wt - se_wt, ymax = mean_wt + se_wt),
                width = 0.3, alpha = 0.5) +
  geom_smooth(method = "lm", se = FALSE, linetype = "dashed", color = "tomato") +
  labs(title = "25-Year Cisco Body Weight Trend — Trout Lake, WI",
       subtitle = "Annual mean ± SE; point size = sample size",
       x = "Year", y = "Mean Body Weight (g)",
       caption = "Dashed line = linear trend. NTL-LTER / FSAdata") +
  theme_minimal()

Notice: the long-term trend in mean body weight is less dramatic than the year-to-year variation you found in your t-test. This is why both single-year comparisons and long-term trends matter in ecology.

Data: Ogle DH (2023). FSAdata / NTL-LTER, University of Wisconsin–Madison, 1981–2006.