Inferential Statistics — Part II: Confidence Intervals

Before you begin, take a few minutes to check what you already know and how confident you feel. You will see the same questions again at the end of the module — this helps both you and us measure what you have learned. Click an option for every question and confidence rating, then click Next to continue.

In Part I you saw the Central Limit Theorem make the sample mean bell-shaped, and you learned that the standard error measures how much that mean would wobble if you ran the study again. This part cashes that in: it turns a single estimate into a confidence interval — a stated range of plausible values for the population truth you could not measure directly.

This part of the module covers five skills — building each interval in R, plus reading what it means and reporting it cleanly:

• A confidence interval for a mean — build it once by hand as the estimate give or take 1.96 standard errors, then let t.test() do it properly, and read the range off with $conf.int.
• What the 95% really means — a property of the procedure across many repeated studies, not a probability for the one interval on your screen.
• Intervals for a proportion — a large-sample interval with prop.test() and the exact binom.test() you reach for when the event count is small.
• A skewed median by bootstrap — resample your own data with sample(x, replace = TRUE), recompute the median many times, and read the middle 95% off with quantile() — no formula required.
• Reporting the result — in one consistent shape, "estimate, 95% CI [low, high]", pulling the numbers cleanly with broom::tidy() rather than copying them by hand.

By the end of this part you will be able to build a 95% confidence interval for a mean, a proportion, and a skewed median in R, state correctly what the 95% refers to, and report any estimate as "estimate, 95% CI [low, high]".

Try every snippet in the R Scratchpad on the right. The dataset diabetes_clinic.csv is already loaded, and the small vectors and bootstrap simulations build their own data with c(), set.seed(), and replicate(), so you can run everything as you read.

The CLT lets us go from a point estimate to a range. A confidence interval is a range of plausible values for the population parameter, built from your sample so that the procedure traps the true value 95% of the time. For a mean, R hands you the whole interval through t.test().

You already have the pieces. A rough 95% interval is the estimate give or take about 1.96 standard errors — that 1.96 is the qnorm(0.975) cut-off from the Normal curve in Part I. Build it by hand once, then let t.test() do it properly (it uses the slightly wider t-distribution for small samples).

hba1c <- c(7.2, 8.1, 6.5, 9.4, 7.8, 6.9, 8.4, 7.1)
se <- sd(hba1c) / sqrt(length(hba1c))
mean(hba1c) + c(-1.96, 1.96) * se

You do not need the t-distribution algebra. Call t.test on your vector and read the confidence interval off the result, or pull it out cleanly with $conf.int.

hba1c <- c(7.2, 8.1, 6.5, 9.4, 7.8, 6.9, 8.4, 7.1)
t.test(hba1c)
t.test(hba1c)$conf.int

Now the interpretation that examiners and reviewers care about most. The 95% describes the long-run behaviour of the procedure, not a probability for this one interval. If you repeated the whole study many times and built an interval each time, about 95% of those intervals would contain the true mean.

Say it the wrong way and you have made the most famous error in statistics. It is wrong to say "there is a 95% probability the true mean lies in this interval." The true mean is a fixed number; your specific interval either caught it or did not. The 95% is a property of the method across many studies, not of the single interval on your screen.

A wider interval means more uncertainty. Intervals shrink as the sample grows, for the same square-root reason the SE shrinks. And never report a bare point estimate with no interval — a mean of 7.8 alone hides whether the truth could plausibly be 7.2 or 8.4, which changes the clinical story entirely.

Means are not the only estimates that need an interval. Proportions and medians do too, and R has a clean function for each. This section finishes your toolkit.

Suppose 38 of 200 patients reached their HbA1c target. The sample proportion is 0.19, but you owe the reader an interval. prop.test gives a large-sample interval; binom.test gives an exact one, which you prefer when counts are small.

prop.test(38, 200)
binom.test(38, 200)

Read the interval the same procedural way as the mean: the method, repeated across many studies, would trap the true proportion 95% of the time. Reach for the exact binom.test() when you have only a handful of events, where the large-sample approximation gets shaky.

What if you want an interval for a median from skewed data, where no neat formula applies? You bootstrap it: resample your own data with replacement many times, recompute the median each time, and read the middle 95% of those medians as your interval.

IMAGE GIF_bootstrap_median

A median's confidence interval can be read straight off your own data by resampling it with replacement many times and keeping the middle 95% of the resulting medians, no formula required.

The building blocks are familiar. sample(x, replace = TRUE) draws a resample the same size as x; replicate() repeats it; quantile with c(0.025, 0.975) cuts off the lowest and highest 2.5% to leave the central 95%.

set.seed(2025)
los <- rexp(120, rate = 1 / 5)
boot_meds <- replicate(2000, median(sample(los, replace = TRUE)))
quantile(boot_meds, c(0.025, 0.975))

The two numbers you get back are the lower and upper limits of a 95% bootstrap confidence interval for the median. The same set.seed() habit applies — fix it so the interval is reproducible.

However you got the interval, report it in one consistent shape: the point estimate, then "95% CI", then the low and high in square brackets — for example, "mean HbA1c 7.8, 95% CI [7.2, 8.4]". The estimate tells the reader your best guess; the interval tells them how sure to be.

Pull the numbers programmatically rather than copying by hand. tidy from the broom package turns any t.test() or prop.test() result into a tidy one-row table with estimate, conf.low, and conf.high columns you can read or drop straight into a report.

library(broom)
hba1c <- c(7.2, 8.1, 6.5, 9.4, 7.8, 6.9, 8.4, 7.1)
tidy(t.test(hba1c))

The worked example below fades its support: first study a full solution, then fill the gap, then solve a fresh one on your own. It estimates a mean with a t.test interval and recomputes the standard error by hand.

sbp <- c(132, 145, 128, 151, 139, 136)
t.test(sbp)$conf.int
sd(sbp) / sqrt(length(sbp))

You have reached the end of the module. Try the same questions again — your answers here, paired with your pre-test answers, are how we measure what the module taught you. Answer every question and confidence rating, then click Submit and see results to view your score.

Here is how your post-test answers compare with your pre-test answers. The pre/post pairing is the most reliable way to see what this module actually taught you.