Standard deviation is the measure of how spread out a random variable is around its mean. It is defined as the square root of the variance: σX = sqrt(Var(X)). The variance captures the average squared deviation from the mean, which ends up in squared units, making interpretation harder. Taking the square root brings it back to the same units as X, giving a intuitive sense of typical deviation. The variance itself is Var(X), the mean is the expected value, and skewness describes asymmetry in the distribution. So the square root of the variance is the standard deviation, denoted σX.

When the population standard deviation is unknown, confidence intervals for the mean are built using the t-distribution because the variability is estimated from the sample data. Instead of dividing by the true sigma, you use the sample standard deviation s, which adds extra uncertainty. The resulting t-statistic has heavier tails and uses n−1 degrees of freedom, so the interval width reflects that extra variability: X̄ ± t_{α/2, n−1} · (s/√n). As n grows large, the t-distribution looks more like the standard normal, but the standard practice remains to use the t distribution whenever sigma is unknown, especially for smaller samples. This is why the statement about using the t-distribution when sigma is unknown is true.

Power is the probability that a statistical test will correctly reject a false null hypothesis, i.e., it detects an effect when one truly exists. This is why the correct description is the probability of rejecting a false null. Power is equal to 1 minus the probability of a Type II error (failing to reject a false null). The other concepts refer to different error rates: rejecting a true null is a Type I error; not rejecting a true null is the complement of the significance level; and failing to reject a false null is the Type II error. Power increases with larger sample size, larger true effect size, less variability, and, depending on the context, may be affected by the chosen significance level (higher alpha can raise power but also increases the risk of a Type I error).

A value of R-squared near 1 can signal overfitting when the data set is small or there are many predictors. R-squared measures how much of the variation in the observed outcome is explained by the model, but it is an in-sample, training metric. Adding predictors tends to raise R-squared regardless of whether those predictors truly capture real, generalizable relationships, so with limited data you can end up fitting random noise rather than a true signal. This makes the model look almost perfectly predictive on the data you used, while its performance on new data may be poor. To guard against this, look at adjusted R-squared, which penalizes extra predictors and often lowers the value if those predictors don’t add real explanatory power. Cross-validation or a separate test set provides a better check of generalization. In short, an R-squared near 1 in small samples or with many predictors is a warning sign of overfitting rather than a guarantee of real predictive strength.