Statistics Interview Questions in R: 25 Solved Problems

Explanation: Standard deviation measures variability in the data itself and does not shrink with more observations. Standard error measures the precision of a statistic (here the mean) and equals $\sigma/\sqrt{n}$, so it shrinks as $n$ grows. The classic interview mistake is quoting the standard deviation when asked how confident you are in an average. Any interval you build around a mean uses the standard error, not the standard deviation.

Explanation: The exponential is strongly right-skewed, yet the distribution of its sample means is nearly Normal and centered at the population mean, with spread matching $\sigma/\sqrt{n}$. This is the Central Limit Theorem in action, and it is why t-tests and z-intervals work on non-Normal data once $n$ is moderate. The interviewer is checking that you know the CLT is about the distribution of the mean, not about the raw data becoming Normal.

Explanation: Because the standard error is $\sigma/\sqrt{n}$, precision improves with the square root of the sample size, not linearly. Going from 25 to 100 (a 4x increase) halves the error from 3.0 to 1.5; another 4x to 400 halves it again. The practical takeaway interviewers want is that diminishing returns are built in: each additional decimal place of precision costs roughly 100 times more data.

Explanation: The bootstrap treats the sample as a stand-in for the population, so resampling it with replacement mimics drawing fresh samples. The standard deviation of the resampled statistic estimates its standard error, which is invaluable for statistics like the median, trimmed mean, or a ratio where no clean formula exists. A common mistake is resampling without replacement, which just reshuffles the same values and gives zero variability.

Explanation: t.test() returns a list, and $conf.int carries the interval plus a conf.level attribute. The correct interpretation, and the one interviewers test, is procedural: if you repeated this sampling many times, about 95% of the intervals so constructed would contain the true mean. It does not mean there is a 95% probability the true mean lies in this particular interval, which is a fixed but unknown quantity. Question 2.3 demonstrates the coverage claim by simulation.

Explanation: The t interval is slightly wider because the t distribution has heavier tails, which is the correct penalty for estimating the standard deviation from the same small sample. As $n$ grows the t and z critical values converge, so the two intervals become indistinguishable. Interviewers use this to check that you know when the Normal approximation is safe (large $n$) and when the t correction matters (small $n$).

Explanation: About 95% of the intervals capture the true mean, which is exactly what the confidence level promises: it is a property of the interval-building procedure across many samples, not of any single interval. This simulation is the cleanest way to answer the interview trap "is there a 95% chance the mean is in this interval?" No: once computed, a given interval either contains the mean or it does not. The 95% describes how often the method succeeds.

Explanation: The Wald interval is symmetric around the point estimate and assumes Normality, which breaks down for small $n$ or extreme proportions, so it sits noticeably lower and undercovers here. The Wilson and exact intervals respect the boundary at 0 and give more honest coverage. The interview point is that "confidence interval for a proportion" is not one formula: reach for Wilson or exact when events are rare, and never let a Wald interval slip below 0 or above 1.

Explanation: The p-value of 0.001374 is the probability of a t statistic at least this extreme if the two group means were truly equal, so it is evidence against the null of no difference, not the probability the null is true. The fractional degrees of freedom flag the Welch correction for unequal variances. The confidence interval excludes 0, which agrees with the small p-value, and the group means show automatic transmissions average about 7 mpg lower.

Explanation: The one-sided p-value is half the two-sided value because it places all the significance in a single tail. The catch interviewers probe: you must commit to the direction before seeing the data. Choosing "less" only after noticing group 0 was lower is p-hacking, and it doubles your true Type I error rate. One-sided tests are defensible only when the opposite direction is genuinely irrelevant to the decision.

Explanation: With no true difference, the test still rejects about 5% of the time, which is precisely the definition of the significance level $\alpha$: the probability of a Type I error when the null holds. This reframes the interview answer to "what is a p-value threshold?" It is not a measure of truth; it is the false-alarm rate you are willing to tolerate. The next question shows the deeper reason this works.

Explanation: Under the null, the p-value is uniformly distributed on the unit interval, so its quantiles trace the identity line and about 5% of them fall below 0.05 (here 4.94%). This single fact explains Question 3.3: the 5% false positive rate is a direct consequence of uniformity. It also explains why running many tests is dangerous, since uniform p-values mean small values appear purely by chance, which is exactly the setup for the next problem.

Explanation: Every null hypothesis here is true, yet one raw p-value dips below 0.05, exactly the false positive uniformity predicts. Bonferroni controls the family-wise error rate by tightening the threshold, and Benjamini-Hochberg controls the false discovery rate; both correctly reject the lone "finding" (its smallest raw p-value is 0.0388). The interview lesson is that a single significant result among many tests is usually noise, and you must state your multiple-comparison strategy before you look.

Explanation: Power is the probability of rejecting the null when the alternative is genuinely true, so here you would catch a real difference of 5 only about 70% of the time, below the conventional 80% target. Interviewers want you to see that a "non-significant" result from an underpowered study is uninformative: absence of evidence is not evidence of absence. The four levers are effect size, variability, sample size, and $\alpha$; fixing any three determines the fourth.

Explanation: Detecting a 2-point absolute lift needs about 3,841 users per arm, roughly 7,700 total, which surprises people who expect a few hundred. Round up to be safe, since a fractional sample size cannot be met by rounding down. The interview point is that small effects demand large samples, and quoting a sample size without stating the effect size, power, and baseline rate is meaningless. This is the single most common applied A/B question.

Explanation: With 5,000 users per arm you can reliably detect a lift only if the true conversion rate reaches about 11.7%, a 1.7-point absolute (17% relative) improvement. Anything smaller will usually go undetected, so promising to catch a "1% lift" with this sample is not credible. Framing the minimum detectable effect up front stops teams from running tests that were doomed to be inconclusive, which is exactly the planning maturity interviewers look for.

Explanation: Peeking five times and stopping at the first significant result inflates the false positive rate from 5% to about 14%, because each look is a fresh opportunity for noise to cross the line. This is why stopping an experiment as soon as it "wins" is a classic way to ship an illusory effect. The disciplined fixes interviewers expect are a fixed sample size decided in advance, or sequential methods (alpha spending, always-valid p-values) that budget the error across looks.

Explanation: The slope of -5.34 means each additional 1,000 lbs of weight is associated with about 5.3 fewer miles per gallon, holding nothing else in this one-predictor model. The p-values print as 0 only because rounding hides values far below 0.0001; the large t statistics confirm strong evidence. Interviewers want the word "associated": regression on observational data quantifies association, and calling the slope a causal effect without a design to support it is the classic overreach.

Explanation: R-squared is the share of variance in mpg explained by the model and can never decrease when you add a predictor, even a useless one, which makes it a bad tool for comparing models of different sizes. Adjusted R-squared subtracts a penalty for each parameter, so it rises only when a new predictor improves fit more than chance would. That is why interviewers ask you to prefer the adjusted version, or a held-out metric, for model selection.

Explanation: The p-value of 0.10 does not reject Normality, so the residuals are consistent with the assumption. The interview nuance is that linear regression assumes Normal, constant-variance, independent errors, and it is the residuals that carry those assumptions, not the outcome variable itself. Also remember that with large samples a Shapiro-Wilk test rejects trivial departures, so residual plots (a QQ plot and residuals-versus-fitted) usually matter more than the formal test.

Explanation: A variance inflation factor tells you how much a coefficient's variance is inflated because that predictor is explained by the others; disp at 7.3 is the most redundant here. Rules of thumb flag VIF above 5 or 10 for attention. This hand-rolled version reveals what the packaged vif function does under the hood, which is a favourite follow-up. The fix for high collinearity is to drop or combine predictors, not to trust the individual coefficients, whose signs can flip unpredictably.

Explanation: The horsepower coefficient more than halves once weight enters, because heavier cars tend to have more powerful engines, so the single-predictor model blamed horsepower for damage that weight was really doing. This is omitted-variable bias, and it is why interviewers distrust a coefficient from a model that ignores obvious confounders. The direction and size of the bias depend on how the omitted variable correlates with both the predictor and the response.

Explanation: Treatment B wins on the pooled numbers (0.826 versus 0.78), yet treatment A has the higher success rate for both small and large stones. The reversal happens because A was used mostly on hard (large-stone) cases while B handled easy ones, so the aggregate is confounded by case mix. Simpson's paradox is the interview reminder that you must condition on the lurking variable; blindly pooling groups of different composition can flip the conclusion.

Explanation: Despite a "99% accurate" test, a positive result means only about a 9% chance of actually being sick, because the disease is so rare that false positives from the healthy 99.9% swamp the true positives. This base-rate neglect is one of the most-asked interview traps: accuracy figures are meaningless without the prevalence. The same arithmetic underlies fraud detection and spam filtering, where rare positive classes make precision collapse even when sensitivity looks impressive.

Explanation: The correlation is exactly zero even though y is a deterministic function of x, because the symmetric parabola has no linear trend for Pearson correlation to pick up. The interview lesson is that a correlation of zero rules out only a linear relationship, never dependence in general; the two variables here are as related as they could possibly be. To catch nonlinear association you need scatterplots, rank correlations like Spearman, or mutual-information measures.