Lesson 80 — Consolidation Term 8 — Applied Statistics and Probability

Sample: 4, 6, 8, 8, 9, 10, 10, 11, 12, 22. Calculate median, $Q_1$, $Q_3$, IQR, and identify outliers by the Tukey fence.

Same sample as exercise 80.1. Calculate the mean and sample standard deviation. Compare with the median: which is more representative of the central position? Why?

$X \sim \text{Bin}(20,\, 0{,}3)$. Calculate $E[X]$, $\text{Var}(X)$, and $\sigma$.

For $X \sim \text{Bin}(20,\, 0{,}3)$: verify the normal approximation criterion and, even if borderline, use the approximation with continuity correction to estimate $P(X \geq 10)$.

$X \sim \mathcal{N}(50,\, 100)$. Calculate $P(X > 65)$.

Sample $n = 100$ from population with $\mu = 10$, $\sigma = 3$. Calculate $P(\bar X > 10{,}3)$.

Pairs $(x, y)$: (1, 2), (2, 4), (3, 5), (4, 4), (5, 7). Calculate Pearson correlation coefficient $r$ and regression line $\hat y = b_0 + b_1 x$.

Disease with 2% prevalence, test with 90% sensitivity and 95% specificity. Calculate Positive Predictive Value (PPV).

$Y = 2X + 5$ where $X \sim \mathcal{N}(10,\, 4)$. Determine the distribution of $Y$.

A fair die is rolled 50 times. Calculate the expectation and standard deviation of the sum $S = X_1 + \cdots + X_{50}$.

Which statement about dispersion measures is correct?

Which relationship is true for the point probabilities of the binomials at the mode?

Explain in your own words: does the CLT state that, for large $n$, individual data points follow a normal distribution? If not, what exactly converges to normal?

Which statement about Pearson correlation is correct?

Pizza delivery time: $T \sim \mathcal{N}(32,\, 5^2)$ min. What is the maximum deadline covering 95% of deliveries?

With the model from exercise 80.15 (SLA of 40.2 min, 5% violation rate), calculate the expectation of the number of violations in 100 deliveries.

Real estate market: correlation between area and price is $r = 0{,}8$. Means $\bar x = 80\,\text{m}^2$, $\bar y = 450,000$ (R$); deviations$s_x = 20$,$s_y = 80,000$. Find the regression line and predict the price for an average-area property.

Financial portfolio: 100 independent stocks, each with daily return $\sim \mathcal{N}(0{,}001,\; 0{,}02^2)$. Determine the distribution of the daily return of the equal-weighted portfolio.

Six Sigma: parts with dimension $X \sim \mathcal{N}(10,\; 0{,}02^2)$ mm. Tolerance $10 \pm 0{,}1$ mm. Calculate the proportion of defects and estimate defects per million.

Election poll: $n = 2500$, $\hat p = 0{,}48$. Construct a 95% confidence interval for the true proportion $p$.

Spam filter: 80% of spam contains "FREE", 5% of ham contains it. $P(\text{spam}) = 0{,}3$. An email contains "FREE" — apply Bayes and classify.

Production line: 2% defect rate, batch of 200 parts. Estimate $P(X \geq 5\text{ defects})$ via Poisson approximation and normal approximation. Compare results.

Financial portfolio: assets A ($\sigma_A = 1\%$) and B ($\sigma_B = 2\%$) with correlation $\rho_{AB} = 0{,}3$. Calculate the standard deviation of a 50%/50% portfolio.

Two independent diagnostic tests, both positive: test 1 (sens 90%, spec 95%), test 2 (sens 85%, spec 90%). Prevalence 1%. Apply Bayes sequentially and calculate final PPV.

Vaccine clinical trial: 100 vaccinated, 5 sick; 100 placebo, 25 sick. Calculate vaccine efficacy and evaluate (informally) if the difference is statistically significant.

Call center: in each minute, each of 120 agents receives a call with 2% probability. Model the number of simultaneous calls in 1 minute and calculate $P(\text{at least 1 call})$.

Heights of adult men in Brazil: $\mu = 173$ cm, $\sigma = 7$ cm. What percentage does not pass through a 180 cm door? What door height covers 99% of the male population?

Explain, with a numerical example, why in highly right-skewed distributions the median is more informative than the mean, and IQR more informative than standard deviation.

Describe intuitively and mathematically how Bayesian inference converges to frequentist inference (MLE) as $n \to \infty$. Which theorem formalizes this convergence?

Construct a data example where $r > 0{,}8$ but the relationship between $X$ and $Y$ is entirely explained by a confounder $C$. Explicit the mathematical mechanism.

Generate (theoretically) 100 independent random variables $X_i \sim \text{Uniform}[0,1]$. Use CLT to approximate $P(X_1 + \cdots + X_{100} > 55)$.

ENEM: public school has $\mu_1 = 520$, $\sigma_1 = 90$ (Math); private school has $\mu_2 = 610$, $\sigma_2 = 80$. Samples of $n=100$ from each. What is the probability that the private sample mean exceeds the public one by more than 80 points?

Prove that $\text{Var}(X) = E[X^2] - (E[X])^2$ from the definition $\text{Var}(X) = E[(X-\mu)^2]$.

Show that if $X_i \sim \text{Bernoulli}(p)$ are iid, then $S = \sum_{i=1}^n X_i \sim \text{Bin}(n,p)$. Conclude that $\bar X = S/n \xrightarrow{P} p$ by the Law of Large Numbers.

Prove that $E[aX + b] = aE[X] + b$ and $\text{Var}(aX + b) = a^2 \text{Var}(X)$ for any $a, b \in \mathbb{R}$.

Derive Bayes' rule $P(H \mid E) = P(E \mid H)\,P(H)/P(E)$ from the definition of conditional probability and the law of total probability.

State the CLT formally. Sketch the proof via characteristic function (indicate the steps, justifying Lévy's continuity theorem is not required).