STAT-211 Exam 2

Parameter

A fixed, unknown number describing the whole POPULATION (μ, σ, p).

Statistic

A number computed from a SAMPLE, used to estimate a parameter (X̄, s, p̂).

μ (mu)

Population mean.

σ (sigma)

Population standard deviation. σ² = population variance.

X̄ vs x̄

X̄ = sample mean as a random variable (before data). x̄ = the actual number you observed.

s (or S)

Sample standard deviation — used in place of σ when σ is unknown.

p and p̂

p = population proportion (unknown). p̂ = sample proportion = successes ÷ n.

N(μ, σ²)

Normal distribution with mean μ and VARIANCE σ². N(4, 5) has SD √5.

pnorm(x, mean, sd)

P(X ≤ x): area to the LEFT of x. Note it takes the SD, not the variance.

1 - pnorm(x, mean, sd)

P(X > x): area to the RIGHT of x.

qnorm(p)

The value with area p to its left (the inverse of pnorm). Used for quantiles/critical values.

pt(x, df) and qt(p, df)

Same as pnorm/qnorm but for the t distribution with df degrees of freedom.

Joint pmf/pdf f(x, y)

Describes how X and Y behave together. pmf (table): f(x, y) = P(X = x and Y = y). pdf: integrate over a region to get a probability.

Probability from a joint pdf

Double integral of f(x, y) over the region of the event, staying inside the support.

Support

The set of (x, y) values where f(x, y) > 0. Integration limits never go outside it.

Valid joint pdf/pmf

Never negative, and it sums/integrates to 1 over the support (this is how you solve for an unknown constant c).

Marginal pmf (from a table)

Distribution of ONE variable alone: add across a row for f_X(x), down a column for f_Y(y).

Marginal pdf

Integrate out the other variable: f_X(x) = ∫ f(x, y) dy and f_Y(y) = ∫ f(x, y) dx.

Expected value E[X]

The long-run average: Σ x·f_X(x) (discrete) or ∫ x·f_X(x) dx (continuous).

E[X + Y]

= E[X] + E[Y]. Always true, even if X and Y are dependent (linearity).

Conditional pmf/pdf f(y | x)

Distribution of Y once X = x is known: f(y | x) = f(x, y) ÷ f_X(x) (joint ÷ marginal of the given variable).

Conditional expected value E[Y | X = x]

Mean of Y for that given x: Σ y·f(y | x) or ∫ y·f(y | x) dy.

Rectangle vs triangle region

Rectangle (like X ≤ a, Y ≤ b): all limits are numbers. Triangle (like X > Y): inner limits depend on the outer variable, e.g. ∫[x=0→1] ∫[y=0→x] … dy dx.

Covariance Cov[X, Y]

How X and Y move together: E[XY] − E[X]E[Y]. Positive = rise together, negative = one up / other down.

Correlation ρ

Unitless covariance: Cov ÷ (σ_X σ_Y). Always between −1 and 1.

Covariance from correlation

Cov = ρ · σ_X · σ_Y (use standard deviations, not variances).

Independent X and Y

f(x, y) = f_X(x)·f_Y(y) for all x, y. Knowing one tells you nothing about the other. Implies Cov = 0.

Cov = 0 ⇒ independent?

No. Independent ⇒ Cov = 0, but Cov = 0 does not prove independence. Cov ≠ 0 ⇒ dependent.

Cov[aX + b, cY + d]

= ac·Cov[X, Y]. Added constants don't matter.

Var[X + Y]

Var[X] + Var[Y] + 2Cov[X, Y]

Var[X − Y]

Var[X] + Var[Y] − 2Cov[X, Y]. Variances still ADD; only the Cov term flips sign.

Var[aX + bY]

a²Var[X] + b²Var[Y] + 2ab·Cov[X, Y]

Sampling distribution

The distribution of a statistic (like X̄) across all possible samples.

Standard error (SE)

The standard deviation of a statistic's sampling distribution.

Mean and SE of X̄

E[X̄] = μ, SE = σ/√n. So X̄ ~ N(μ, σ²/n).

Total T = X1 + … + Xn

Mean nμ, variance nσ², SD σ√n.

Unbiased

The statistic's expected value equals the parameter (E[X̄] = μ, E[p̂] = p).

Exact sampling distribution

When the data are normal: X̄ is exactly normal for ANY n.

Central Limit Theorem (CLT)

When data are NOT normal: X̄ is approximately normal if n is large (n ≥ 30).

Approximate sampling distribution

Normal distribution that comes from the CLT, used when the population isn't normal but n is large enough.

Proportion CLT

p̂ ≈ N(p, p(1−p)/n) when np ≥ 10 and n(1 − p) ≥ 10.

SRS (simple random sample)

Randomly chosen sample, so X1, …, Xn are independent.

Confidence interval (CI)

A range of plausible values for a parameter: statistic ± quantile × SE.

Margin of error M

The ± part of a CI: quantile × SE. Width = 2M.

Confidence level (1 − α)

How often the method captures the true parameter over many repeated samples (e.g., 95%).

α in a CI

The chance the interval misses. 95% confidence → α = 0.05.

Quantile / critical value for a two-sided CI

qnorm(1 − α/2) or qt(1 − α/2, n − 1). 90% → 1.645, 95% → 1.96, 99% → 2.576.

Quantile (general)

The value with a given probability to its left. The pth quantile Q(p) is the inverse of the cdf.

CI for μ, σ known

x̄ ± z(α/2) · σ/√n (z = normal quantile)

CI for μ, σ unknown

x̄ ± t(α/2, n − 1) · s/√n (t quantile)

Student's t distribution

Like the normal but with fatter tails; used when s replaces σ. Approaches the normal as n grows.

Degrees of freedom (df)

For one-sample and matched-pairs t methods: df = n − 1.

When can you build a CI for μ?

Data are normal (any n) OR n ≥ 30. Small n with clearly non-normal data → can't.

Matched pairs

Same units measured twice (before/after). Use differences D = X2 − X1 as one sample.

CI for matched pairs μ_D

d̄ ± t(α/2, n − 1) · s_D/√n, where n = number of pairs.

CI for a proportion p

p̂ ± z(α/2) · √(p̂(1 − p̂)/n). Needs np̂ ≥ 10 and n(1 − p̂) ≥ 10.

Correct CI interpretation

"We are 95% confident the POPULATION MEAN is between a and b." Not a probability statement, and not about individuals.

What does '95% confident' mean?

If we repeated the sampling many times, about 95% of the intervals would contain the true parameter.

CI width: what changes it?

Higher confidence → wider. Bigger σ → wider. Bigger n → narrower.

Sample size for μ

n = ceiling((z·σ/M)²). Always round up.

Sample size for p

n = ceiling((z·0.5/M)²). Use 0.5 because p(1 − p) is largest at p = 0.5. Round up.

Hypothesis test

A way to decide whether the data give enough evidence against a reference claim about a parameter.

Null hypothesis H0

The default/reference claim, assumed true. Contains the equality (=, ≤, ≥).

Alternative hypothesis HA

The claim you want evidence for (≠,

Two-sided test

HA: μ ≠ μ0. Reject if the statistic is too far in EITHER direction.

One-sided test

HA: μ < μ0 or μ > μ0. Reject only in that one direction.

Test statistic

(statistic − null value) ÷ SE. Measures how far the data are from H0.

T for μ, σ unknown

T = (x̄ − μ0)/(s/√n), compared to t(n − 1).

T for a proportion

T = (p̂ − p0)/√(p0(1 − p0)/n), compared to N(0, 1). Uses p0, not p̂.

Critical value

The quantile that marks the edge of the rejection region (e.g., qt(1 − α/2, n − 1)).

Rejection region

Values of the test statistic that lead to rejecting H0. It points the same way as HA.

Significance level α

The probability of a Type I error you're willing to accept, chosen before the test.

p-value

Probability, assuming H0 is true, of getting a test statistic at least as extreme as the one observed.

Small p-value means…

Strong evidence AGAINST H0.

Decision rule

p-value < α → reject H0. p-value ≥ α → fail to reject H0.

Fail to reject H0

Not enough evidence for HA. It does NOT mean H0 is proven true.

Type I error

Rejecting H0 when H0 is actually true. Probability = α.

Type II error

Failing to reject H0 when HA is actually true. Probability = β.

Power

Probability of rejecting H0 when HA is actually true = 1 − β.

Two-sided → one-sided p-value

If T_obs points the way HA does: halve it. If it points the opposite way: 1 − (half).

Statistically significant

Enough evidence to reject H0.

Scientifically significant

The difference is big enough to actually matter in practice.

CI and test connection

A two-sided test rejects H0: μ = μ0 exactly when μ0 is outside the matching