Statistics & Probability Statistical Inference Syllabus

Statistical Inference syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Statistical Inference in Statistics & Probability, not a summary of it.

1. 1

   Point and Interval Estimation

   4 topics

   • Point Estimators and Their Properties
   • Confidence Intervals for a Mean
   • Confidence Intervals for a Proportion
   • Margin of Error and Sample Size
2. 2

   Hypothesis Testing Fundamentals

   5 topics

   • Null and Alternative Hypotheses
   • Type I and Type II Errors
   • Significance Level and P-Values
   • Statistical Power
   • One-Tailed vs Two-Tailed Tests
3. 3

   Tests for Means

   3 topics

   • One-Sample z-Test and t-Test
   • Two-Sample t-Tests
   • Assumptions and Diagnostics
4. 4

   Tests for Proportions and Variance

   4 topics

   • One and Two Proportion z-Tests
   • Chi-Square Goodness-of-Fit
   • Chi-Square Test of Independence
   • Tests for Variance
5. 5

   Analysis of Variance (ANOVA)

   4 topics

   • One-Way ANOVA
   • Two-Way ANOVA
   • Post-Hoc Tests
   • ANOVA Assumptions
6. 6

   Nonparametric Methods

   4 topics

   • Mann-Whitney U Test
   • Wilcoxon Signed-Rank Test
   • Kruskal-Wallis Test
   • Spearman Rank Correlation
7. 7

   Resampling Methods

   3 topics

   • Bootstrap
   • Permutation Tests
   • Cross-Validation

Statistical Inference flashcards for Statistics & Probability

20 of 50 cards from the Statistical Inference deck — real questions with worked answers.

1. What is a point estimator, and how does it differ from a point estimate?

   A point estimator is a statistic (a rule/function of the sample data) used to infer an unknown population parameter, e.g. $\bar{X}$ for $\mu$. A point estimate is the single numerical value the estimator takes for a specific sample. The estimator is the random variable; the estimate is one realized number.

2. Define an unbiased estimator.

   An estimator $\hat{\theta}$ is unbiased for $\theta$ if its expected value equals the parameter: $E(\hat{\theta}) = \theta$. The bias is $\text{Bias}(\hat{\theta}) = E(\hat{\theta}) - \theta$, so unbiasedness means zero bias for all sample sizes.

3. Why is the sample variance defined with $n-1$ in the denominator rather than $n$?

   Dividing by $n-1$ makes the sample variance unbiased for the population variance: $E\left(\frac{1}{n-1}\sum (X_i-\bar{X})^2\right) = \sigma^{2}$. Using $n$ underestimates $\sigma^{2}$ because deviations are taken about $\bar{X}$ (which minimizes them), costing one degree of freedom.

4. What does it mean for an estimator to be consistent?

   An estimator $\hat{\theta}_n$ is consistent if it converges in probability to the parameter as the sample grows: for every $\varepsilon>0$, $P(|\hat{\theta}_n - \theta| > \varepsilon) \to 0$ as $n \to \infty$. Roughly, the estimate gets arbitrarily close to $\theta$ with more data.

5. How is the efficiency of an unbiased estimator judged, and what is the MSE decomposition?

   Among unbiased estimators, the more efficient one has the smaller variance. Overall accuracy is measured by mean squared error: $$\text{MSE}(\hat{\theta}) = \text{Var}(\hat{\theta}) + \left[\text{Bias}(\hat{\theta})\right]^{2}.$$ A minimum-variance unbiased estimator is the most efficient unbiased choice.

6. What is the confidence interval for a population mean $\mu$ when the population standard deviation $\sigma$ is known?

   $$\bar{x} \pm z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}}$$ where $z_{\alpha/2}$ is the standard-normal critical value (e.g. $1.96$ for 95%). Valid when the population is normal or $n$ is large (CLT).

7. What is the confidence interval for a mean $\mu$ when $\sigma$ is unknown?

   $$\bar{x} \pm t_{\alpha/2,\,n-1}\,\frac{s}{\sqrt{n}}$$ using the sample standard deviation $s$ and the $t$-distribution with $n-1$ degrees of freedom. The $t$ critical value exceeds the corresponding $z$ value to account for extra uncertainty in estimating $\sigma$.

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

   If we repeatedly drew samples and built an interval each time by the same method, about 95% of those intervals would contain the true parameter. It does NOT mean there is a 95% probability that this particular interval contains $\mu$ — the parameter is fixed, and a given interval either contains it or not.

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

   $$\hat{p} \pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$ where $\hat{p} = x/n$ is the sample proportion and $\sqrt{\hat{p}(1-\hat{p})/n}$ is the standard error of $\hat{p}$.

10. What conditions justify the normal-approximation confidence interval for a proportion?

    Independent observations (e.g. random sample, and $n \leq 10\%$ of the population) and a large enough sample so the sampling distribution is approximately normal: typically $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$ (some texts use $\geq 5$).

11. Define the margin of error in a confidence interval.

    The margin of error is the half-width of the interval: $E = (\text{critical value}) \times (\text{standard error})$. For a mean with known $\sigma$, $E = z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}}$; for a proportion, $E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.

12. How does the required sample size for estimating a mean depend on the desired margin of error $E$?

    Solving $E = z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}}$ gives $$n = \left(\frac{z_{\alpha/2}\,\sigma}{E}\right)^{2}.$$ Halving $E$ quadruples $n$, and $n$ grows with the square of the confidence critical value; always round up.

13. What sample size is needed to estimate a proportion within margin $E$, and what value maximizes it?

    $$n = \frac{z_{\alpha/2}^{2}\,\hat{p}(1-\hat{p})}{E^{2}}.$$ With no prior estimate, use the conservative $\hat{p} = 0.5$, which maximizes $\hat{p}(1-\hat{p}) = 0.25$ and thus the required $n$. Round up.

14. How do the null hypothesis $H_0$ and alternative hypothesis $H_a$ differ?

    $H_0$ is the default claim of no effect/no difference, always stated with equality (e.g. $\mu = \mu_0$). $H_a$ is the researcher's claim of an effect, stated with $\neq$, $<$, or $>$. Testing seeks evidence to reject $H_0$ in favor of $H_a$; we never 'prove' $H_0$.

15. What is a Type I error and how is its probability denoted?

    A Type I error is rejecting a true null hypothesis (a 'false positive'). Its probability equals the significance level: $P(\text{Type I}) = \alpha$.

16. What is a Type II error and how is its probability denoted?

    A Type II error is failing to reject a false null hypothesis (a 'false negative'). Its probability is denoted $\beta$, and $1-\beta$ is the power of the test.

17. How are the Type I and Type II error rates related as you change the decision threshold?

    For a fixed sample size, lowering $\alpha$ (stricter rejection rule) raises $\beta$, and vice versa — there is a trade-off. Both can be reduced simultaneously only by increasing the sample size $n$.

18. What is the significance level $\alpha$ of a hypothesis test?

    $\alpha$ is the pre-chosen threshold for the maximum tolerable probability of a Type I error — the probability of rejecting $H_0$ when it is true. Common choices are $0.05$, $0.01$, and $0.10$; it defines the size of the rejection region.

19. Define the p-value of a hypothesis test.

    The p-value is the probability, assuming $H_0$ is true, of obtaining a test statistic at least as extreme as the one observed (in the direction of $H_a$). A small p-value means the observed data would be unlikely under $H_0$.

20. State the p-value decision rule for a significance level $\alpha$.

    If $p \leq \alpha$, reject $H_0$ (result is statistically significant). If $p > \alpha$, fail to reject $H_0$. Note: a p-value is NOT the probability that $H_0$ is true, nor the probability the result occurred by chance.

See more Statistical Inference flashcards →

Planning Statistical Inference for Statistics & Probability

Statistical Inference (Statistics & Probability) FAQ