Learning outcomes
- Explain the key idea in your own words.
- Apply the concept to a realistic but fictional research scenario.
- Identify assumptions and limits before interpreting numbers.
Conceptual lesson
With random sampling, different samples from the same population yield different estimates. A sampling distribution describes how a statistic would vary across hypothetical repeated samples. Standard error measures the variability of an estimate across samples, not the spread of individual observations.
A 95% confidence-interval procedure has approximately 95% long-run coverage of the fixed population parameter under its assumptions. It does not mean that a fixed parameter randomly moves into the particular observed interval with 95% probability.
Statistical power is the probability of rejecting a false null hypothesis under a specified effect and test. Sample size, true effect, noise and the significance threshold influence power. Large samples do not remove selection bias.
Worked research example
Calculate a t-based confidence interval for the mean of 36 fictional quiz scores. But a mathematically computed interval from volunteers with unknown representativeness does not automatically justify inference about an actual university population.
Common reasoning trap
Saying the fixed population parameter has a literal 95% frequentist probability of being inside the observed interval is not the standard confidence-interval interpretation.
Practice tasks
- 1. Distinguish standard deviation from standard error in words.
- 2. Calculate a mean and t-based confidence interval for the 36 records; state a limitation to generalisation.
- 3. Predict how standard error changes as sample size increases while variability stays constant.
Explain your reasoning, not just the numerical result.
Optional R practice
Download the fictional CSV into your working folder, then run the script below or download the .R file. No additional packages are needed.
Download CSV · Download R lab · Module 0 refresher
# Statistics Lab / ferdo — synthetic data (no real student records)
# Save stats_synthetic.csv in your working directory.
d <- read.csv("stats_synthetic.csv", stringsAsFactors = TRUE)
stopifnot(nrow(d) == 36, all(c("score_pre","score_post","group") %in% names(d)))
# Groups are observational labels: differences cannot establish a treatment effect.
d <- read.csv("stats_synthetic.csv")
n <- sum(!is.na(d$score_post))
xbar <- mean(d$score_post)
se <- sd(d$score_post) / sqrt(n)
c(mean=xbar, standard_error=se)
t.test(d$score_post)$conf.int
# Inferential validity still depends on the data-generating process.Stretch: mathematical track
Compare SE = s/sqrt(n) for n = 25, 100 and 400. What happens when sample size quadruples?
Exit ticket
Write one evidence-bounded claim and one limitation about this week's example.
This week's student workbook
Four original written problems (20 marks) and a formative four-question self-check.
Optional mathematical derivations
Extend this topic with worked mathematics, written exercises and base R checks.