Pre-course exam

Conditional probability

A wildlife disease has prevalence 0.05 in a population. A field-portable test has sensitivity 0.92 (P(positive | infected) = 0.92) and specificity 0.95 (P(negative | not infected) = 0.95).

A randomly sampled animal tests positive. What is the probability the animal is actually infected? Show your reasoning.

Independence vs mutually exclusive

In one or two sentences, explain the difference between two events being independent and being mutually exclusive. Give one ecological example of each.

Joint probability

Two camera traps each independently have a 30% chance of detecting a fisher on a given night, given a fisher is present. What is the probability that at least one camera detects the fisher, given presence?

Expected value

A management decision pays \$10,000 if a population survives and costs \$2,000 if not. The probability of survival under the management plan is 0.7. What is the expected payoff?

Pick the distribution

For each scenario, name the discrete distribution you would use:

1. The number of nests that successfully fledge out of 12 monitored, each with the same independent success probability.
2. The number of whale sightings per 1-hour acoustic monitoring period.
3. The classification of 100 surveyed elk into bulls/cows/calves/yearlings.

Mean and variance

If $X \sim \text{Poisson}(\lambda = 4)$, what is its mean and variance? Give both numbers, and state in plain English what their relationship implies (one sentence).

Continuous distribution choice

You have a response variable that is strictly positive and right-skewed (e.g., fish biomass per haul). Name one continuous distribution that is appropriate and explain in one sentence why.

Normal probability

Heights of a population of male elk are approximately Normal with mean 152 cm and SD 8 cm. Roughly what fraction of the population is taller than 160 cm? (You may sketch a Normal curve, mark the threshold, and estimate; calculator not required.)

Reading a coefficient

You fit lm(weight ~ length) on fish data and get an intercept of 5 and a slope of 0.8 (units: g per mm). Write a one-sentence biological interpretation of the slope.

Choosing a GLM

For each response variable, name the GLM family and link you would use:

1. Whether each of 100 nests succeeded (yes/no).
2. Count of birds detected per 5-minute point count.
3. Continuous body mass in grams, right-skewed and strictly positive.

Log-link interpretation

You fit a Poisson regression and the coefficient on a continuous predictor $x$ is $\hat\beta = 0.30$. What is the biological effect of a 1-unit increase in $x$ on the expected response? (Hint: think exponentiated effect.)

Logit-link interpretation

You fit a logistic regression of survival on age. The age coefficient on the logit scale is $\hat\beta = -0.5$. Roughly how much does the odds of survival change per year of age? You may answer "I don't know how to interpret a logit coefficient" if appropriate.

When to use a random effect

You have count data from 12 sites, with 6 plots per site. Why might it be problematic to fit a single regression that ignores site identity?

Random vs fixed

For each predictor, would you treat it as a fixed effect, a random effect, or possibly either? Briefly justify.

1. Sex (male/female).
2. Year (12 years of monitoring, treated as repeats of the same population).
3. Treatment (control vs. exclosure).

Shrinkage intuition

In a mixed-effects model, group-level estimates "shrink toward the grand mean." In two or three sentences, describe what that means and why it might be desirable. You may answer "I'm not sure" if the term is unfamiliar.

What is a likelihood?

Define "likelihood" in plain English in 1–2 sentences. Distinguish it from a probability.

MLE for a proportion

You observe 30 successes in 50 trials of a binomial process with unknown success probability $p$. What is the maximum likelihood estimate of $p$? Show your reasoning (it's fine to argue by intuition or by algebra).

Prior, likelihood, posterior

In Bayesian inference, what are the prior, the likelihood, and the posterior? One sentence each is fine.

Posterior interpretation

You fit a Bayesian model and get a 95% credible interval for a parameter of $(0.2, 0.8)$. Write a one-sentence interpretation of this interval. How does it differ from a frequentist 95% confidence interval?

Why MCMC

Why do Bayesian analyses typically use MCMC (Markov chain Monte Carlo) instead of computing the posterior in closed form? You may answer "I don't know" — this is a topic we'll spend significant time on.

Tell me about your data

Briefly describe (3–5 sentences) the kind of data you work with — or expect to work with — and one statistical question you would like to be able to answer by the end of this course. There are no wrong answers; this helps the instructor tune examples for the workshop.