Lab 10 — Bootstrap intervals

Resampling your own data to build a percentile confidence interval

Purpose. This lab is the hands-on companion to Week 10 — Bootstrap inference. The note develops the bootstrap principle — resample the sample with replacement to estimate sampling variability; here you build a bootstrap distribution for the mean, read a standard error and a percentile interval off it, and confirm it matches the theory-based interval from Week 7.

The idea

The bootstrap treats your one sample as a stand-in for the population: resample it with replacement many times, recompute the estimate each time, and the spread of those estimates approximates the sampling distribution — no formula required. This lab builds that distribution for the mean reading-gain, reads the bootstrap standard error and a percentile confidence interval, and checks both against the \(t\)-interval from Week 7. The two agreeing is the point: simulation and theory measuring the same variability.

Goal

From a sample of \(n = 36\) gains with mean \(8.0\) and SD \(6.0\), generate a bootstrap distribution of the sample mean, report the bootstrap SE (about \(1.0\)) and the 95% percentile interval (about \((6.0, 10.0)\)), and compare to the Week-7 \(t\)-interval \((5.97, 10.03)\).

Setup

Open R and a fresh Quarto document; fix the seed. We build a synthetic sample whose mean and SD match the locked study values, then bootstrap that sample — the bootstrap only ever uses the data you have.

set.seed(35103)
# a synthetic sample of 36 gains with mean ~ 8 and SD ~ 6 (stands in for the observed cohort)
gains <- rnorm(36, mean = 8, sd = 6)
mean(gains); sd(gains)    # close to 8.0 and 6.0
B <- 10000                # number of bootstrap resamples

Steps

Step 1 — one bootstrap resample

A single bootstrap resample draws 36 gains with replacement from the 36 observed gains; some appear twice, some not at all. Its mean is one bootstrap statistic.

set.seed(35103)
one <- sample(gains, size = 36, replace = TRUE)   # a resample
mean(one)                                          # one bootstrap mean

Step 2 — many resamples, the bootstrap distribution

Repeat B times to get the whole bootstrap distribution of the mean, and look at it.

boot_means <- replicate(B, mean(sample(gains, replace = TRUE)))
hist(boot_means, breaks = 30,
     main = "Bootstrap distribution of the mean gain", xlab = "bootstrap mean")

Step 3 — read the SE and the percentile interval

The bootstrap SE is the SD of the bootstrap means; the percentile interval is their middle 95%.

sd(boot_means)                          # bootstrap SE   ~ 1.0
quantile(boot_means, c(0.025, 0.975))   # percentile 95% CI  ~ (6.0, 10.0)

# compare to the Week-7 theory interval
mean(gains) + c(-1, 1) * qt(0.975, df = 35) * (sd(gains) / sqrt(36))   # ~ (5.97, 10.03)

Verify

Center. The bootstrap distribution is centered near the sample mean (\(\approx 8.0\)) — the bootstrap describes variability around the estimate, not a shift away from it.
SE matches the formula. sd(boot_means) is about \(1.0\), matching \(s/\sqrt n = 6/6\). The simulation reproduced the standard error.
Interval matches theory. The percentile interval \((6.0, 10.0)\) is essentially the \(t\)-interval \((5.97, 10.03)\). When they agree, both are trustworthy; if they disagreed sharply, you would suspect skew or a too-small sample and investigate.
Replacement matters. If you drop replace = TRUE, every resample is just the original sample, the bootstrap SE collapses to \(0\), and the interval vanishes — a fast way to confirm replacement is doing the work.

AI use note

Field	What to record
Tool	which assistant you used, with approximate date or version
Purpose	what you used it for (e.g. explaining `sample(..., replace = TRUE)`, debugging `quantile`)
Verification	how you checked it: compared the bootstrap SE to \(s/\sqrt n\), compared the percentile interval to the \(t\)-interval, or re-ran with the fixed seed

Verification is the load-bearing line: an AI can write the resampling loop, but you confirm the bootstrap SE matches \(s/\sqrt n\) and the interval matches Week 7 yourself.