The variance of B(n, p) is: — np(1−p).Var(X) = np(1−p). Standard deviation = √(np(1−p)).
A Poisson distribution is used when: — Only the average rate λ is known.Poisson: use when you know the average rate λ of rare events per interval, but not n and p separately.
For Poisson, P(X=k) = λᵏe^(−λ)/k!. The mean and variance both equal: — λ.For a Poisson distribution, μ = σ² = λ. Mean equals variance.
10 coin flips with p=0.5. Using binomial, P(exactly 6 heads) = ? — All three are equivalent.All three expressions equal C(10,6)·(0.5)¹⁰ = 210/1024 ≈ 0.205. They are the same formula.
The normal approximation to the binomial is valid when: — np ≥ 10 and n(1−p) ≥ 10.The rule of thumb: np ≥ 10 AND n(1−p) ≥ 10 ensures the binomial is approximately normal.
Poisson approximates binomial when n is large and p is: — Small (close to 0).Use Poisson when n is large and p is small, with λ = np. Classic examples: rare defects, accidents.
The 68-95-99.7 rule for the normal distribution states: — 68% of data is within σ, 95% within 2σ, 99.7% within 3σ.Empirical rule: 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ of the mean.
For Binomial, which additional condition must be satisfied? — Trials must be independent.Binomial requires: (1) fixed n, (2) each trial independent, (3) exactly 2 outcomes, (4) constant p.
If X ~ B(10, 0.3): P(X = 0) = ? — (0.7)¹⁰.P(X=0) = C(10,0)·(0.3)⁰·(0.7)¹⁰ = (0.7)¹⁰ ≈ 0.028.
Which distribution models the number of email spam per day if average is 5 per day? — Poisson with λ=5.Rare events with a known average rate → Poisson. Spam emails per day with λ = 5.
For X ~ N(100, 225) (μ=100, σ=15): What z-score corresponds to X = 130? — z = 2.z = (x − μ)/σ = (130 − 100)/15 = 30/15 = 2.
The significance level α represents: — The probability of rejecting H₀ when it is true (Type I error rate).α = P(Type I error) = P(rejecting H₀ | H₀ is true). Common values: 0.05 or 0.01.
A p-value is defined as: — P(observing data this extreme or more extreme, given H₀ is true).p-value = P(data as extreme as observed | H₀). Small p-value → data unlikely under H₀ → evidence against H₀.
Type I error is: — Rejecting H₀ when H₀ is true (false positive).Type I error = false positive. Probability = α. We incorrectly reject a true null hypothesis.
Type II error is: — Failing to reject H₀ when H₀ is false (false negative).Type II error = false negative. Probability = β. We fail to detect a real effect.
The power of a test is: — 1 − β.Power = P(reject H₀ | Hₐ is true) = 1 − β. Higher power → better at detecting true effects.
The z-test statistic z = (x̄ − μ₀)/(σ/√n). What does σ/√n represent? — Standard error of the mean.σ/√n is the standard error of the mean (SEM) — how much x̄ varies from sample to sample.
A t-test is used instead of z-test when: — The population σ is unknown (we use s).Use t-test when σ is unknown and you must estimate it with the sample standard deviation s.
Degrees of freedom for a one-sample t-test with n observations: — n − 1.df = n − 1 for a one-sample t-test. We lose one degree of freedom for estimating μ.
If p-value = 0.12 and α = 0.05, we: — Fail to reject H₀ (but do not prove H₀).0.12 > 0.05, so we fail to reject H₀. We never "accept" H₀ — we just don't have enough evidence against it.
"Statistical significance" at α = 0.05 means: — There is less than a 5% chance the results are due to random variation if H₀ is true.Statistical significance says only that the observed result is unlikely under H₀ (p < α). It says nothing about practical importance.
A two-tailed test is used when Hₐ states: — μ ≠ μ₀.A two-tailed test (Hₐ: μ ≠ μ₀) rejects H₀ for deviations in either direction.
A one-tailed test is preferred when: — The researcher predicts a specific direction (greater or less).One-tailed: Hₐ: μ > μ₀ or Hₐ: μ < μ₀. Use when theory/prior evidence specifies the direction.
The chi-square test for independence tests whether: — Two categorical variables are related in a contingency table.Chi-square test of independence: tests if two categorical variables in a contingency table are associated or independent.
A z-test for population mean: sample n=100, x̄=52, σ=10, H₀: μ=50. z = ? — z = 2.z = (52−50)/(10/√100) = 2/1 = 2.
We "never accept H₀" because: — Failing to reject doesn't prove H₀ is true — the test just lacks evidence against it.Absence of evidence ≠ evidence of absence. Failing to reject H₀ means insufficient evidence — not proof that H₀ is true.
A 95% confidence interval means: — 95% of the time, intervals constructed this way will contain the true parameter.Correct interpretation: if we repeated the study many times, 95% of all intervals would contain the true parameter. For THIS interval, the parameter is either in it or not.
The critical value z* for a 90% CI is approximately: — 1.645.z* ≈ 1.645 for a 90% CI.
The critical value z* for a 99% CI is approximately: — 2.576.z* ≈ 2.576 for a 99% CI.
The margin of error E = z*·(σ/√n). To halve E, multiply n by: — 4.E ∝ 1/√n. To halve E: 1/√(4n) = (1/2)·(1/√n). So multiply n by 4.
The formula to find required sample size for margin of error E is: — n = (z*σ/E)².E = z*·σ/√n → √n = z*σ/E → n = (z*σ/E)². Always round UP.
Convenience sampling (surveying whoever is nearby) leads to: — Convenience bias / non-random sample.Convenience sampling is non-random, leading to biased estimates that may not represent the population.
A voluntary response sample (people who choose to respond) is biased because: — It overrepresents people with strong opinions.Voluntary response attracts people with strong opinions (often negative), biasing the results.
A confounding variable is one that: — Is related to both the explanatory and response variable, creating spurious associations.A confound is associated with both the treatment and outcome, making it hard to isolate the true effect.
If σ doubles and n stays the same, the margin of error: — Doubles.ME = z*σ/√n. If σ → 2σ, then ME → 2ME. Margin of error doubles.
If n quadruples and σ stays the same, the margin of error: — Halves.ME = z*σ/√n. √(4n) = 2√n. So ME is halved.
Which study design best establishes causation? — Randomized controlled experiment.Randomized experiments (with random assignment to treatment/control) best establish causation by controlling confounders.