A random variable X is: — A variable that takes values determined by a random experiment.A random variable X takes numerical values determined by the outcome of a random process.
For a binomial distribution B(n, p), n represents: — The number of trials.In B(n, p): n = number of independent trials, p = probability of success on each trial.
P(X = k) in binomial = C(n,k)·pᵏ·(1−p)^(n−k). What does C(n,k) count? — The number of ways to choose k successes from n trials.C(n,k) counts the number of different sequences with exactly k successes in n trials.
The mean of B(n, p) is: — np.E(X) = np for a binomial distribution. (n trials each with probability p of success.)
The normal distribution is characterized by: — Two parameters: μ (mean) and σ² (variance).Normal distribution N(μ, σ²): completely defined by mean μ and variance σ².
The normal distribution's pdf is symmetric about: — μ (mean).The normal distribution is symmetric about its mean μ. The bell curve peaks at the mean.
For standard normal Z ~ N(0,1), P(Z < 0) = ? — 0.5.The standard normal is symmetric about 0. P(Z < 0) = 0.5.
The standardizing formula for normal distributions is: — z = (x − μ)/σ.z = (x − μ)/σ converts any normal X to the standard normal Z ~ N(0,1).
P(X > 0) for X ~ N(0, 1) equals: — 0.5.By symmetry of the standard normal, P(Z > 0) = 0.5.
What is the total area under any probability density function? — 1.All probability distributions must have total probability = 1. The total area under any pdf equals 1.
The normal distribution's bell curve is symmetric, meaning: — Mean = median = mode.For a symmetric bell curve: mean = median = mode. All three measures of center coincide.
The null hypothesis H₀ typically represents: — The status quo / no effect.H₀ = "no effect" or "status quo" — the hypothesis of no change that we try to disprove with data.
The alternative hypothesis Hₐ represents: — Both B and C.Hₐ is both what we seek evidence for and the logical negation of H₀. Options B and C are equivalent.
Decision rule: reject H₀ when: — p-value < α.Reject H₀ when p-value < α. The data is too unlikely under H₀ to be due to chance.
A chi-square goodness-of-fit test uses χ² = Σ (O−E)²/E. "O" stands for: — Observed frequency.O = Observed frequency (actual count from data). E = Expected frequency (from null hypothesis).
If p-value = 0.01 and α = 0.05, we: — Reject H₀.0.01 < 0.05 = α, so we reject H₀. The result is statistically significant.
The general formula for a confidence interval is: — x̄ ± z*·(σ/√n).CI = x̄ ± z* · (σ/√n), where z* is the critical value for the desired confidence level.
The critical value z* for a 95% CI is approximately: — 1.960.z* ≈ 1.960 for a 95% CI. (This cuts off 2.5% in each tail of the standard normal.)
A t-interval uses t* instead of z* when: — σ is unknown (use s).Use t* when σ is unknown and you estimate it with the sample standard deviation s.
Higher confidence level (90% → 99%) with same n gives: — Wider interval.Higher confidence → larger z* → wider interval. More certainty requires a wider net.
Larger sample size with same σ and confidence level gives: — Narrower interval.Larger n → smaller σ/√n (standard error) → smaller margin of error → narrower interval.
A 95% CI is (42, 58). The sample mean is: — 50.CI is symmetric about x̄. x̄ = (42 + 58)/2 = 50.
A 95% CI is (42, 58). The margin of error is: — 8.ME = (58 − 42)/2 = 16/2 = 8. Or: 58 − 50 = 8.
A 99% CI is wider than a 95% CI because: — Larger z* captures more of the distribution.z*(99%) ≈ 2.576 > z*(95%) ≈ 1.960. Larger z* → larger margin of error → wider interval.
The phrase "garbage in, garbage out" in statistics refers to: — How poor data collection undermines valid statistical conclusions.Even perfect statistical analysis cannot overcome flawed data collection. The quality of conclusions is limited by data quality.
Which factor NARROWS a confidence interval? — Larger n.Larger n → smaller SE = σ/√n → smaller ME → narrower interval. More data = more precision.
A random sample from a population means: — Every individual in the population has an equal chance of selection.Random sampling: every individual has an equal (or known) probability of being selected. This allows valid statistical inference.