Confidence Interval Exercises in R: 18 Practice Problems

Explanation: t.test() does a one-sample t-test against the default null mean of zero, but the $conf.int element is the same regardless of the null because it is built from the t critical value, the sample mean, and the standard error. Pulling $conf.int keeps the result tidy: a length-2 numeric vector with a conf.level attribute, easy to round(), diff(), or feed into a report. The full print() output mixes the interval with hypothesis-test machinery you do not need.

Explanation: Raising conf.level from 0.95 to 0.99 pushes the t critical value from roughly 2.04 to 2.74 (with df = 31), so the half-width multiplier grows by about 34%. The point estimate mean(mtcars$mpg) = 20.09 is unchanged, but the interval stretches further on both sides. This is the fundamental width-versus-confidence trade-off: you cannot be more sure without being less precise unless you collect more data.

Explanation: With n = 7, the t critical value (df = 6) is about 2.45 versus roughly 2.04 for the full sample, but the bigger driver is the smaller n in the standard error denominator. Subgroup intervals are almost always wider than full-sample intervals at the same confidence level. A tidyverse one-liner like mtcars |> filter(cyl == 6) |> pull(mpg) |> t.test() |> _$conf.int produces the same numbers in pipeline-friendly style.

Explanation: The t-interval assumes approximately symmetric (near-normal) data. Engine displacement is right-skewed, so the interval on raw disp would be inflated by the long tail. Logging straightens the distribution, the t-interval is valid on the log scale, and exp() maps it back. The back-transformed bounds are an interval for the geometric mean, not the arithmetic mean, that is the cost of the transformation, but it is the right summary for multiplicative quantities like displacement, income, or concentration.

Explanation: This is what t.test() does internally. qt(0.975, df = n - 1) is the upper 2.5% t critical value, so it covers the central 95% region. Note c(-1, 1) is a vectorisation trick: multiplying scalar tcr * se by the length-2 vector produces the lower-then-upper endpoints in one line. A common error is using qt(0.95, ...) (one-sided) instead of qt(0.975, ...) (two-sided), that gives a 90% interval, not 95%.

Explanation: When sigma is known you use qnorm(0.975) = 1.959964 instead of a t critical value, so degrees of freedom never enter. In practice you almost never have a truly known sigma, the t-interval is the safer default. The z-interval is narrower because it does not pay the "we estimated sigma from data" penalty that pushes the t critical value above 1.96 at small n. For n >= 50 and unknown sigma, the t and z intervals agree to two decimals.

Explanation: The half-width is the part of the interval that depends on data and confidence level, useful when you only care about precision and not the centre. diff(conf.int) gives the full width; dividing by 2 yields the half-width that appears in margin-of-error reporting. The two routes agree exactly because t.test() uses the same formula internally. unname() strips the leftover name from diff() so the final named vector has the names you assigned.

Explanation: t.test() with a formula does a Welch two-sample test (unequal variances by default), which is the safer choice when you have no evidence that group SDs match. The interval [-11.28, -3.21] is for mean(am=0) - mean(am=1), both endpoints negative, meaning automatics have lower mpg, and the difference is between 3.2 and 11.3 mpg with 95% confidence. The interval excludes zero, so the difference is also statistically significant at alpha = 0.05.

Explanation: Pairing assumes a meaningful 1-to-1 correspondence between observations in the two groups (here, by row order within the filtered slice). The paired t-interval is the t-interval of the differences oj - vc, so it has n - 1 = 9 degrees of freedom rather than the two-sample's roughly 18. Pairing removes between-subject variance, narrowing the interval whenever the two measurements are positively correlated within a subject. The interval here includes zero, so the OJ-vs-VC difference at dose 1.0 is not statistically significant.

Explanation: The pooled-variance interval is slightly narrower than Welch here (width about 7.21 versus 8.07) because pooling buys back degrees of freedom (30 vs Welch's roughly 18.3 for this split). The catch: if the equal-variance assumption is wrong, the pooled interval can be wildly off in coverage, Welch is robust either way. As a rule, use Welch unless you have strong prior reason to believe variances are equal (e.g., randomised assignment in a clean experiment).

Explanation: The Wilson interval is what most modern stats texts now recommend over the textbook normal-approximation interval ($\hat{p} \pm 1.96\sqrt{\hat{p}(1-\hat{p})/n}$), which can produce nonsensical bounds below 0 or above 1 at extreme proportions. prop.test() returns the Wilson interval by default. With correct = TRUE (the default), R applies a Yates continuity correction that widens the interval slightly; setting correct = FALSE gives the cleaner uncorrected Wilson endpoints used in most teaching material.

Explanation: With only n = 10, a normal-approximation interval is unreliable, so the Clopper-Pearson exact interval is the conservative go-to. It inverts the exact binomial test, guaranteeing coverage at least 95% (often more, that is the conservatism cost). The width here, roughly 0.59, reflects the genuine uncertainty when 3 events out of 10 could plausibly come from a true rate anywhere from 7% to 65%. Compare against prop.test(3, 10)$conf.int to see how the Wilson interval (the other small-sample option) is slightly narrower.

Explanation: prop.test() accepts two-element vectors for x (successes) and n (totals) to compute a CI for the difference. The interval is entirely negative, so variant A converts at a lower rate than B by between 0.1 and 5.9 percentage points with 95% confidence. Because the upper bound is right at -0.0011, the result is barely significant at alpha = 0.05; if you had run with correct = TRUE, the slightly wider interval would just cross zero. This is exactly the kind of "borderline call" where reporting the interval, not just the p-value, lets stakeholders see how thin the evidence is.

Explanation: Each bootstrap iteration resamples the data with replacement and recomputes the statistic; with R = 2000 you get an empirical sampling distribution of the median, and the central 95% range of that distribution is the percentile CI. boot.ci() returns several CI types, perc is the simplest, bca is bias-corrected and usually better when the bootstrap distribution is skewed. Always set a seed before boot() so the interval is reproducible; the percentile endpoints can wobble by a few units of the last decimal place between runs.

Explanation: confint() uses the t critical value with df = n - p - 1 (here 29) and the coefficient standard error from summary(fit)$coefficients to build each interval. Both intervals are entirely negative, so each predictor has a negative partial effect on mpg holding the other constant. The interval width matters as much as significance: the hp interval is tight near zero (effect is small but precise), while the wt interval is wide (effect is large but the magnitude is uncertain). Reporting both endpoints in a paper is far more informative than reporting only the p-value.

Explanation: A confidence band is the interval for the conditional mean E(mpg | wt), not for a single new observation, that wider one is interval = "prediction". The band is narrowest near the centre of the predictor distribution (around mean(wt) = 3.22) and fans out at the extremes because extrapolation amplifies coefficient uncertainty. This is also the band ggplot2::geom_smooth(method = "lm", se = TRUE) renders by default with level = 0.95; building it manually is what you do when you need the numbers for a table or downstream code.

Explanation: This is the most important practical fact about CIs: to halve the half-width you need 4x the data, to shrink it 10x you need 100x the data. Diminishing returns are baked into the $\sqrt{n}$ denominator. When a stakeholder asks "how much more data do we need to get a tight estimate", invert the formula: $n = (z_{\alpha/2}\sigma / \text{target half-width})^2$. Sample-size planning lives entirely in this rearrangement.

Explanation: 94.8% is within Monte Carlo noise of the nominal 95%, confirming that the t-interval is calibrated when its assumptions (independent normal data) are met. The same simulation with rexp(n) minus 1 (heavily skewed data) would show under-coverage at small n, exactly why log transforms or bootstrap intervals exist. Coverage simulations are also how you would empirically check that your favourite shrinkage estimator, robust SE, or Bayesian credible interval delivers the coverage it advertises. Trust the formula at scale, verify it at small n.