Statistical Officer / Government Statistical Service (GSS) Assessment Probability and Statistical Inference Syllabus

State the density, mean, and variance of the continuous uniform distribution on $[a, b]$.

$f(x) = \dfrac{1}{b - a}$ for $a \leq x \leq b$; mean $E[X] = \dfrac{a + b}{2}$; variance $\operatorname{Var}(X) = \dfrac{(b - a)^{2}}{12}$.

State the density, mean, and variance of the exponential distribution with rate $\lambda$.

$f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$; mean $E[X] = \dfrac{1}{\lambda}$; variance $\operatorname{Var}(X) = \dfrac{1}{\lambda^{2}}$.

What is the memoryless property, and which common distributions have it?

$P(X > s + t \mid X > s) = P(X > t)$: the future is independent of elapsed time. The exponential distribution (continuous) and the geometric distribution (discrete) are the only distributions with this property.

Write the probability density function of the normal distribution $N(\mu, \sigma^{2})$.

$f(x) = \dfrac{1}{\sigma \sqrt{2\pi}} \exp\!\left(-\dfrac{(x - \mu)^{2}}{2\sigma^{2}}\right)$ for $-\infty < x < \infty$.

How do you standardize a normal random variable $X \sim N(\mu, \sigma^{2})$?

$Z = \dfrac{X - \mu}{\sigma}$, which follows the standard normal distribution $N(0, 1)$.

For a normal distribution, what proportion of values lie within 1, 2, and 3 standard deviations of the mean (the empirical rule)?

Approximately $68\%$ within $\mu \pm \sigma$, $95\%$ within $\mu \pm 2\sigma$, and $99.7\%$ within $\mu \pm 3\sigma$.

State the Central Limit Theorem.

For i.i.d. random variables with mean $\mu$ and finite variance $\sigma^{2}$, the sample mean $\bar{X}$ has a distribution that approaches $N\!\left(\mu, \dfrac{\sigma^{2}}{n}\right)$ as $n \to \infty$, regardless of the population's shape.

What is the sampling distribution of the sample mean $\bar{X}$, and what is its standard error?

$\bar{X}$ has mean $\mu$ and variance $\dfrac{\sigma^{2}}{n}$. Its standard error is $\text{SE}(\bar{X}) = \dfrac{\sigma}{\sqrt{n}}$ (estimated by $\dfrac{s}{\sqrt{n}}$ when $\sigma$ is unknown).

Define the standard error of a statistic.

The standard error is the standard deviation of the sampling distribution of a statistic — it quantifies how much the estimate would vary from sample to sample. For the mean it is $\dfrac{\sigma}{\sqrt{n}}$.

What distribution does the sample variance lead to under normality, and with how many degrees of freedom?

For a normal population, $\dfrac{(n-1)s^{2}}{\sigma^{2}} \sim \chi^{2}_{n-1}$, a chi-squared distribution with $n - 1$ degrees of freedom.

When should the $t$-distribution be used instead of the normal for inference about a mean?

When the population standard deviation $\sigma$ is unknown and estimated by $s$ (especially with small $n$). Use $t = \dfrac{\bar{X} - \mu}{s / \sqrt{n}}$ with $n - 1$ degrees of freedom. As $n \to \infty$ the $t$-distribution approaches $N(0,1)$.

Define what makes an estimator unbiased.

An estimator $\hat{\theta}$ is unbiased if $E[\hat{\theta}] = \theta$ for all $\theta$; that is, its expected value equals the true parameter. The bias is $E[\hat{\theta}] - \theta$.

Why is the sample variance defined with divisor $n - 1$ rather than $n$?

Dividing by $n - 1$ (Bessel's correction) makes $s^{2} = \dfrac{1}{n-1}\sum (x_i - \bar{x})^{2}$ an unbiased estimator of $\sigma^{2}$; dividing by $n$ underestimates the variance on average.

What is the mean squared error (MSE) of an estimator, and how does it decompose?

$\text{MSE}(\hat{\theta}) = E[(\hat{\theta} - \theta)^{2}] = \operatorname{Var}(\hat{\theta}) + [\text{Bias}(\hat{\theta})]^{2}$. It captures the bias–variance trade-off.

Define a consistent estimator.

An estimator $\hat{\theta}_n$ is consistent if it converges in probability to $\theta$ as $n \to \infty$: for any $\varepsilon > 0$, $P(|\hat{\theta}_n - \theta| > \varepsilon) \to 0$. Loosely, more data makes it arbitrarily accurate.

Describe the principle of maximum likelihood estimation (MLE).

Choose the parameter value $\hat{\theta}$ that maximizes the likelihood $L(\theta) = \prod_i f(x_i; \theta)$ (or, equivalently, the log-likelihood $\ell(\theta) = \sum_i \log f(x_i; \theta)$) given the observed data.

What is an efficient estimator, and what does the Cramér–Rao lower bound give?

An efficient estimator achieves the smallest possible variance among unbiased estimators. The Cramér–Rao lower bound states that for an unbiased $\hat{\theta}$, $\operatorname{Var}(\hat{\theta}) \geq \dfrac{1}{I(\theta)}$, where $I(\theta)$ is the Fisher information.

Give the general form of a confidence interval for a population mean when $\sigma$ is known.

$\bar{x} \pm z_{\alpha/2}\,\dfrac{\sigma}{\sqrt{n}}$, where $z_{\alpha/2}$ is the standard normal critical value (e.g. $1.96$ for $95\%$ confidence).

Give the confidence interval for a mean when $\sigma$ is unknown (small sample, normal population).

$\bar{x} \pm t_{\alpha/2,\, n-1}\,\dfrac{s}{\sqrt{n}}$, using the $t$-distribution with $n - 1$ degrees of freedom.

What is the correct frequentist interpretation of a 95% confidence interval?

If the sampling procedure were repeated many times, about $95\%$ of the constructed intervals would contain the true parameter. It does NOT mean there is a $95\%$ probability the parameter lies in this particular interval (the parameter is fixed).

Give the confidence interval for a population proportion $p$ (large-sample, Wald interval).

$\hat{p} \pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1 - \hat{p})}{n}}$, where $\hat{p}$ is the sample proportion.

What is the margin of error in a confidence interval, and how does it scale with sample size?

The margin of error is the half-width, e.g. $z_{\alpha/2}\,\dfrac{\sigma}{\sqrt{n}}$. It is proportional to $\dfrac{1}{\sqrt{n}}$, so quadrupling the sample size halves the margin of error.

Derive the sample size needed to estimate a mean with margin of error $E$ (known $\sigma$).

From $E = z_{\alpha/2}\,\dfrac{\sigma}{\sqrt{n}}$, solve $n = \left(\dfrac{z_{\alpha/2}\,\sigma}{E}\right)^{2}$, rounding up to the next integer.

What sample size is needed to estimate a proportion with margin $E$, and what value of $p$ is most conservative?

$n = \dfrac{z_{\alpha/2}^{2}\,p(1 - p)}{E^{2}}$. The worst case is $p = 0.5$ (maximizing $p(1-p) = 0.25$), giving the largest, most conservative sample size.

How do precision and confidence level trade off for a fixed sample size?

Increasing the confidence level (e.g. $95\% \to 99\%$) widens the interval, reducing precision; narrowing the interval lowers confidence. To improve both simultaneously you must increase $n$.

List the steps in the framework of a significance (hypothesis) test.

(1) State $H_0$ and $H_1$; (2) choose significance level $\alpha$; (3) select and compute a test statistic; (4) determine the rejection region or $p$-value; (5) decide to reject or fail to reject $H_0$; (6) interpret in context.

Define a Type I error and a Type II error.

Type I error: rejecting $H_0$ when it is true (probability $\alpha$, the significance level). Type II error: failing to reject $H_0$ when it is false (probability $\beta$).

Define the power of a statistical test and how it relates to $\beta$.

Power $= 1 - \beta = P(\text{reject } H_0 \mid H_0 \text{ false})$ — the probability of correctly detecting a true effect. Power increases with larger sample size, larger effect size, and larger $\alpha$.

Define the $p$-value precisely.

The $p$-value is the probability, assuming $H_0$ is true, of obtaining a test statistic at least as extreme as the one observed. A small $p$-value indicates the data are unlikely under $H_0$. Reject $H_0$ if $p \leq \alpha$.

Which significance test is appropriate for comparing two independent sample means, and what is its statistic?

The two-sample (independent) $t$-test: $t = \dfrac{\bar{x}_1 - \bar{x}_2}{\sqrt{\dfrac{s_1^{2}}{n_1} + \dfrac{s_2^{2}}{n_2}}}$ (Welch's form). Use a paired $t$-test instead when observations are matched.

What does a chi-squared ($\chi^{2}$) test assess, and what is its statistic?

It tests goodness-of-fit or independence in categorical data using $\chi^{2} = \sum \dfrac{(O - E)^{2}}{E}$, where $O$ are observed and $E$ are expected frequencies. For an $r \times c$ table of independence, degrees of freedom $= (r - 1)(c - 1)$.

Explain the distinction between statistical significance and practical (substantive) significance.

Statistical significance means an effect is unlikely due to chance ($p \leq \alpha$); practical significance concerns whether the effect is large enough to matter. With very large $n$, trivially small effects can be statistically significant yet practically meaningless.

Describe the multiple comparisons problem and a common correction.

Performing many tests inflates the family-wise Type I error rate: at $\alpha = 0.05$, about 1 in 20 true nulls is wrongly rejected by chance. The Bonferroni correction tests each of $m$ hypotheses at $\dfrac{\alpha}{m}$ to control the overall error rate.

What is the difference between correlation and causation, and why does it matter in inference?

Correlation measures association between variables; it does not establish that one causes the other. Confounding variables or reverse causation can produce association without causation — causal claims require controlled experiments or careful design, not just observed correlation.