T-Test Exercises in R: 20 Real-World Practice Problems

Explanation: t.test(x, mu = 5.85) defaults to a two-sided test against the supplied null mean. The p-value 0.697 is far above 0.05, so the data do not contradict the lab's claim. The 95 percent CI [5.71, 5.98] contains 5.85, which is the same conclusion expressed as an interval. Always pull mu from the scientific claim, never from the data itself.

Explanation: Setting alternative = "greater" halves the p-value compared to the two-sided test only when the sample mean is on the predicted side. The CI becomes one-sided: [18.41, Inf), and because its finite endpoint exceeds 18, the test rejects at alpha 0.05. The common mistake is using "greater" when the sample mean is below the null; in that case R still reports a finite p but it will be near 1, not near 0.

Explanation: A p-value of 0.24 is nowhere near 0.01, so the line is statistically on-spec. Small samples (n = 12) have low power, so a non-rejection is not proof of compliance; it just means this evidence is insufficient to flag drift. For ongoing monitoring, an SPC chart with control limits is more useful than a single t-test because it visualizes trend, not just a single snapshot.

Explanation: t.test() returns an htest list with $conf.int, $estimate, $statistic, $p.value, and $parameter (df). Wrapping in as.numeric() drops the conf.level attribute that tags along, which matters when you later paste the values into a report or pass them to a function that errors on attributes. For one-sided tests one endpoint will be -Inf or Inf, which as.numeric preserves.

Explanation: Formula Petal.Length ~ Species is the cleanest two-sample syntax when data live in a data frame. t.test() defaults to var.equal = FALSE (Welch), which is the right default because equal variances are rarely true and Welch is robust when they happen to be equal. Dropping unused factor levels with droplevels() prevents R from silently trying a three-group comparison that errors on a two-sample test.

Explanation: When sample sizes are equal and variances similar, Welch and Student give nearly identical p-values, as here (0.061 vs 0.060). The Welch df is fractional because Satterthwaite approximation accounts for unequal variances even when they are unequal only slightly. Practical guidance: use Welch by default and only use Student if you have a strong prior reason to assume equal variances, since Student is anti-conservative when variances differ.

Explanation: p = 0.25 is well above 0.10, so trt1 is not detectably different from control. The CI [-0.29, 1.03] crosses zero, confirming the same conclusion. If you suspected trt1 reduces weight, a one-sided test (alternative = "greater" for ctrl) might be motivated by prior science, but the convention is to pre-register one-sided tests; otherwise reviewers will assume you fished for the smaller p-value.

Explanation: Welch df collapses toward the smaller sample size whenever that group also has the bigger variance, which is the exact scenario that makes Student's pooled-variance test wildly anti-conservative. Here df = 5.1, not the n1 + n2 - 2 = 54 you would get from Student. If you had run Student here you would have inflated Type I error roughly fivefold. Lesson: never pool variances without testing them, and Welch makes the whole question moot.

Explanation: A paired t-test is mathematically a one-sample t-test on the within-subject differences (before minus after) against mu = 0. The order of arguments only flips the sign of the t-statistic and CI, not the p-value. If you forgot paired = TRUE here, R would run an independent two-sample test and produce a far larger p-value because between-subject variance swamps the consistent 2-3 kg drop. The pairing is what gives the test its power.

Explanation: Real paired studies almost always lose some subjects to follow-up, so filter(!is.na(t0), !is.na(t21)) is mandatory before pairing. Reshaping with pivot_wider() turns a long-format panel into a one-row-per-chick wide table, which is the shape paired = TRUE expects. The huge effect (mean diff = 134 g) is unsurprising biologically and the p-value floors at the machine epsilon (< 2.2e-16); always report < 1e-3 or < 0.001 rather than the exact floor value.

Explanation: Both tests agree on direction, but the paired p-value is roughly seven orders of magnitude smaller. Why: most of the variance in BP comes from between-patient differences (some run high, some run low). The paired test eliminates that nuisance variance by analyzing within-patient changes. Forgetting to pair when your design IS paired is the single most common t-test error in practice; the diagnostic is "is the same unit (person, chick, plot) measured twice?" If yes, pair.

Explanation: Neither group rejects normality (p = 0.75 and 0.45), so the t-test in 2.3 is safe to interpret. Shapiro-Wilk is sensitive at moderate n (10 to 50) and underpowered below n = 8, so absent rejection is weak evidence of normality at very small sizes. A pragmatic alternative is to look at a Q-Q plot directly with qqnorm() and rely on the Central Limit Theorem for n >= 30 since the t-statistic is robust to mild non-normality.

Explanation: Levene's test does not reject equal variances (p = 0.28). Pre-test then pick test is a known statistical pitfall called the conditional test problem: it inflates Type I error in the second-stage t-test. The cleaner modern advice is to skip Levene entirely and always use Welch, which costs essentially nothing in power when variances are equal and recovers the right alpha when they are not. Center = "median" (Brown-Forsythe) is more robust than the classic mean-centred Levene.

Explanation: Both tests reject at alpha 0.01, but Wilcoxon is testing a location shift in ranks rather than a difference of means. Use the rank-sum when (1) the data are clearly skewed and your sample is too small for the CLT to bail you out, or (2) the response is ordinal. Note that wilcox.test() reports W, not U; W = U + n1(n1+1)/2. R's continuity correction can be turned off with correct = FALSE for tiny samples where it over-shrinks the p-value.

Explanation: Cohen's d expresses the mean difference in pooled-SD units, so d = -10.5 means the two species centroids are over ten standard deviations apart on petal length, which is enormous by Cohen's rules of thumb (d = 0.2 small, 0.5 medium, 0.8 large). The pooled SD denominator is the conventional choice; using the SD of a single group or an averaged SD gives slightly different effect sizes (Glass's delta or Hedges' g). With unequal n the Hedges correction (1 - 3/(4(n1+n2)-9)) reduces small-sample bias.

Explanation: The result n = 63.77 means round UP to 64 per group, or 128 total. Always round up: rounding down would leave you short of 80 percent power. Note type = "two.sample" is critical; the default is "two.sample" but spelling it out prevents bugs when colleagues read your code. For unequal group sizes use pwr.t2n.test() and supply n1 and n2 directly. To bracket sensitivity, recompute at d = 0.4 and 0.6 to show how n explodes for smaller anticipated effects.

Explanation: Observed power = 0.48 explains exactly why the p-value (0.06) hovered just above 0.05: the design had less than a coin-flip chance of detecting the very effect it observed. Many journals now ask authors NOT to report post-hoc power because it is mathematically equivalent to the p-value and offers no extra information. The legitimate use is exactly what we did here: to justify a larger replication study, not to retroactively explain a non-significant result.

Explanation: A clinical workflow is rarely "run one test." You sequence a normality screen on the DIFFERENCES (not the raw before/after, which can each be non-normal even when their pairwise differences are normal), then the paired test, then a CI. Wrapping all three in a list keeps everything together for downstream kable() or gtsummary rendering. The CI for the mean change is the natural way to communicate effect magnitude to clinicians; p-values alone are not actionable in a clinical setting.

Explanation: Two practical points for A/B tests. First, gamma-distributed revenue violates normality; the CLT rescues a t-test at n = 750 and 250, but for n < ~50 per arm you should use a Mann-Whitney or a permutation test instead. Second, the one-sided test halves the two-sided p-value only when the observed sign matches the hypothesis. Pre-register one-sidedness; post-hoc switching is p-hacking. Many companies report both two-sided p-values and a separate predicted direction in dashboards.

Explanation: Reporters that wrap repeated stats output pay for themselves after the third paste. APA style asks for italic t and df; in markdown you would wrap the t and df in underscores. For p-values below 0.001 the convention is to print "p < 0.001" rather than the exact tiny value, which a robust function would handle with if (h$p.value < 0.001) "p < .001" else sprintf("p = %.3f", h$p.value). Pairing this with broom::tidy() gives you a tibble row per test, ideal for batch reporting.