Data Science & Statistics - MCQ Practice Questions

The arithmetic mean is calculated by summing all values and dividing by the count, so a single very large or very small outlier directly shifts the result. The median, being a positional measure, and the mode, being the most frequent value, are largely unaffected by extreme values. Hence, the mean is the most sensitive to outliers.

The empirical rule (68-95-99.7 rule) states:

Therefore, approximately $\boxed{95\%}$ of data lies within two standard deviations of the mean.

Mutually exclusive events cannot occur simultaneously, so their intersection is empty:

Option A defines independence, not mutual exclusivity. Option B is the general addition rule valid for all events. Option D describes statistical independence. Hence the correct answer is $\boxed{P(A\cap B)=0}$.

The coefficient of variation expresses standard deviation as a percentage of the mean, allowing comparison of variability across datasets with different units or scales:

It is a dimensionless measure of relative dispersion.

The ordinary least squares (OLS) estimator minimizes $\sum (y_i-{\hat{y}}_i{)}^2$. Solving the normal equations gives:

which is the ratio of the sample covariance of $x$ and $y$ to the sample variance of $x$.

The $p$-value is formally defined as:

It does NOT give the probability that $H_0$ is true (that would require a Bayesian posterior). A low $p$-value suggests the data is unlikely under $H_0$, but does not prove $H_1$. Failing to reject $H_0$ does not prove $H_0$.

For the ordered dataset $\{2,4,7,10,13,16,19\}$ with $n=7$:

By definition: Type I error (false positive) occurs when we reject $H_0$ even though it is actually true. Its probability equals the significance level $\alpha$. Type II error (false negative) occurs when we fail to reject $H_0$ even though it is actually false. Its probability is denoted $\beta$, and $1-\beta$ is the power of the test.

Using the variance rule for linear combinations of independent variables:

Note: the cross term vanishes because $X$ and $Y$ are independent, so $\text{Cov}(X,Y)=0$.

Bayes' theorem is derived from the definition of conditional probability:

Substituting $P(A\cap B)=P(B|A)\cdot P(A)$ gives:

This theorem is fundamental to Bayesian inference and many classification algorithms like Naive Bayes.