Name: ____________________________
Answer Questions 1–4 in words. For Questions 5–12, download the CSV linked in each question and use R to complete the analysis. Submit the requested numerical answers, figures, and written explanations. Label each answer by question and part.
| Questions | Topic | Points |
|---|---|---|
| 1–4 | P-values, alpha, and interpretation | 16 |
| 5–8 | t-tests | 36 |
| 9 | Binomial test | 10 |
| 10 | Chi-square test | 14 |
| 11 | Pearson correlation | 10 |
| 12 | Simple linear regression | 14 |
| Total | 100 |
Use the individual CSV download link beneath each study description, or download all eight datasets together.
Download all eight exam datasets (ZIP)Explain its role in hypothesis testing and how it differs from a p-value.
Why is 0.05 often chosen as alpha? Is it a universal scientific rule? Give one reason a researcher might choose a different alpha value.
a. (2 points) A study reports p = 0.03 using alpha = 0.05. What statistical decision should the researcher make? Does this result alone establish that the effect is large or biologically important? Explain.
b. (2 points) Another study reports p = 0.20 using alpha = 0.05. What statistical decision should the researcher make? Does this result establish that there is no effect? Explain.
Biologists independently sample 45 bluegill from Willow Lake. Each fish is measured once. They want to know whether the population mean total length differs from 20 cm.
File: 01_fish_lengths.csv. Columns:
fish_id, length_cm.
a. (2 points) Choose the appropriate test and explain your choice.
b. (2 points) State the null and alternative hypotheses in terms of the population parameter.
c. (4 points) Run the test. Report the sample mean, 95% confidence interval for the population mean, t statistic, degrees of freedom, and p-value.
d. (2 points) Give a conclusion at alpha = 0.05. Does your result establish that the population mean is exactly 20 cm? Explain.
Biologists measure 35 independently sampled bluegill from Willow Lake and 55 from Cedar Lake. These are different fish. There is no justification for assuming equal population variances.
File: 06_lake_comparison.csv. Columns:
fish_id, lake, length_cm.
a. (2 points) Choose the appropriate t-test and explain how both the sampling design and the variance assumption support your choice.
b. (2 points) State the null and alternative hypotheses. Define the difference as Willow minus Cedar.
c. (4 points) Report both sample means, the estimated mean difference, its 95% confidence interval, the t statistic, degrees of freedom, and p-value.
d. (2 points) Interpret the direction of the difference and state your conclusion at alpha = 0.05.
Systolic blood pressure is recorded for 30 people before and four weeks after starting amlodipine. Different people are independent, and each row contains the two measurements from one person. There is no untreated comparison group.
File: 08_blood_pressure.csv. Columns:
patient_id, before_mmhg,
after_mmhg.
a. (2 points) Choose the appropriate t-test and explain why an independent two-sample test would not match this design.
b. (2 points) State the null and alternative hypotheses for before minus after.
c. (4 points) Report the estimated mean difference, its 95% confidence interval, the t statistic, degrees of freedom, and p-value.
d. (2 points) Explain what the sign of the difference means and state your statistical conclusion. Does this study alone establish that amlodipine caused the change? Why or why not?
A separate study samples 40 independent bluegill from each lake. For this question, assume normal populations with equal population variances, justified independently of these observations. Use the pooled, equal-variance version of the two-sample t-test.
File: 04_lake_comparison.csv. Columns:
fish_id, lake, length_cm.
a. (2 points) State the additional population assumption required by the pooled test. Explain why equal sample sizes alone would not justify that assumption.
b. (2 points) Test whether the population means differ. Use Willow minus Cedar and report the estimated difference, t statistic, degrees of freedom, and p-value.
c. (2 points) State your conclusion at alpha = 0.05. Explain why failing to reject the null would not prove that the population means are equal.
A fictional registry contains 320 independent singleton births. Test whether the probability of a birth recorded as male differs from the specified teaching benchmark of 0.50. Assume a common probability across births. The benchmark is not a claim about the true sex ratio of every human population.
File: 10_birth_records.csv. Columns:
birth_id, recorded_sex (Male or
Female).
a. (2 points) Report the number of male births, total births, and observed male proportion.
b. (2 points) State the null and alternative hypotheses.
c. (4 points) Run an exact binomial test. Report its p-value and the 95% confidence interval for the male-birth probability.
d. (2 points) State a conclusion in context at alpha = 0.05.
A survey records the species of 25 independent fish from each lake. Each fish appears once, the same sampling protocol is used in both lakes, and the three species categories are mutually exclusive. First test whether lake and species are independent. Then investigate which cells depart from their expected counts.
File: 18_lake_species.csv. Columns:
fish_id, lake, species
(Bluegill, Bass, or Perch).
a. (1 point) Construct and display the lake-by-species count table.
b. (1 point) State the null and alternative hypotheses for the overall test in words.
c. (2 points) Report the expected counts under independence. Explain whether these counts support using the usual chi-square approximation.
d. (3 points) Use a Monte Carlo test with at least 100,000 randomizations to report the observed chi-square statistic and overall p-value. Describe what your null simulation randomizes and what it holds fixed. State your conclusion at alpha = 0.05.
e. (5 points) Report the raw Monte Carlo p-value for every cell. Apply a Bonferroni correction for all six cells, capping adjusted values at 1. Identify the cells that remain significant at alpha = 0.05, and describe whether each has more or fewer fish than expected under independence.
f. (2 points) Why do the two lakes have identical two-sided p-values for a given species? If we had defined the comparisons in advance as three species comparisons, what multiplier would we use instead?
Researchers measure total length and body depth on each of 36 independently sampled fish. They want to assess the strength and direction of a linear association between these measurements. Neither variable was experimentally manipulated.
File: 15_fish_measurements.csv. Columns:
fish_id, length_cm,
body_depth_cm.
a. (1 point) Make a scatterplot with length on the horizontal axis and body depth on the vertical axis. Label the axes and units.
b. (2 points) State the null and alternative hypotheses for a Pearson correlation test.
c. (3 points) Report Pearson’s correlation coefficient, its 95% confidence interval, and the p-value.
d. (2 points) State two relevant assumptions or features you would check before interpreting the test.
e. (2 points) Interpret the direction and strength of the association. Explain why correlation does not, by itself, establish causation.
Researchers record water temperature and opercular beats per minute for 36 fish, each measured once in a separate tank. Treat fish as independent. Use temperature as the explanatory variable and ventilation rate as the response. The observed temperatures span 16–28 degrees C.
File: 16_fish_ventilation.csv. Columns:
fish_id, water_temp_c,
opercular_beats_min.
a. (3 points) Fit a simple linear regression and write its fitted equation, including the estimated intercept and slope.
b. (2 points) Interpret the slope in biological terms, including units.
c. (2 points) State the null and alternative hypotheses for the slope. Report the slope’s p-value and your conclusion at alpha = 0.05.
d. (1 point) Report and interpret R-squared.
e. (2 points) Predict the mean ventilation rate at 22 degrees C. Is this interpolation or extrapolation?
f. (4 points) Plot residuals against fitted values. Explain what curvature or a funnel shape would suggest, and describe whether you see either pattern in this dataset. Can this plot establish independence of fish?