Flashcards · YMA01 · Topic 8

Unit S2: Statistics 2 flashcards

29 cards on Topic 8, subtopics S2.1 to S2.5. Pick a subtopic or study them all.

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Binomial B(n, p): conditions, probability, mean and variance: Fixed number of trials n; two outcomes; constant probability p; independent trials. P(X = r) = ⁿCᵣ pʳ(1 − p)ⁿ⁻ʳ. Mean np, variance np(1 − p).
Poisson Po(λ): conditions, probability, mean and variance: Events occur singly, independently, at random and at a constant average rate. P(X = r) = e^(−λ) λʳ/r! for r = 0, 1, 2, … Mean λ and variance λ.
Events occur at 1.2 per 4 minutes. What is the distribution for 10 minutes?: Scale the mean: 1.2 ÷ 4 × 10 = 3, so X ~ Po(3). Scale the mean, never the probability.
Poisson approximation to the binomial: When n is large and p is small, B(n, p) ≈ Po(np). Example: B(300, 0.01) ≈ Po(3), so P(X = 2) ≈ e⁻³ × 3²/2 = 0.2240.
Sum of independent Poisson variables: If X ~ Po(a) and Y ~ Po(b) are independent, then X + Y ~ Po(a + b).
X ~ Po(λ) and P(X = 0) = 0.2. Find λ.: e^(−λ) = 0.2, so λ = −ln 0.2 = ln 5.

Conditions for a valid probability density function f(x): f(x) ≥ 0 everywhere and the total area under f is 1. Outside the stated interval f(x) = 0.
Probabilities for a continuous random variable: P(a < X ≤ b) = ∫ from a to b of f(x) dx (area under the pdf). P(X = a) = 0 for every a, so < and ≤ give the same probability.
Link between the cdf F(x) and the pdf f(x): F(x₀) = P(X ≤ x₀) = ∫ from −∞ to x₀ of f(x) dx. f(x) = dF(x)/dx.
E(X) and Var(X) for a continuous random variable: E(X) = ∫ x f(x) dx. Var(X) = ∫ x² f(x) dx − [E(X)]².
Median, quartiles and mode of a continuous random variable: Median m: F(m) = 1/2. Lower and upper quartiles: F(x) = 1/4 and 3/4. Mode: the value of x where f(x) is greatest (it may be an end of the interval).
f(x) = kx³ for 0 ≤ x ≤ 2 (0 otherwise). Find k and the median.: ∫ kx³ dx from 0 to 2 = 4k = 1, so k = 1/4. F(x) = x⁴/16. Median: m⁴/16 = 1/2, m⁴ = 8, m = 8^(1/4) ≈ 1.68.

Continuous uniform X ~ U[a, b]: pdf, mean and variance: f(x) = 1/(b − a) for a ≤ x ≤ b, and 0 otherwise. Mean (a + b)/2. Variance (b − a)²/12.
cdf of U[a, b]: F(x) = 0 for x < a; (x − a)/(b − a) for a ≤ x ≤ b; 1 for x > b.
X ~ U[1, 7]. Find P(X > 5.5) and the 90th percentile.: P(X > 5.5) = (7 − 5.5)/6 = 0.25. 90th percentile: 1 + 0.9 × 6 = 6.4.
Normal approximations to B(n, p) and to Po(λ): B(n, p) ≈ N(np, np(1 − p)) when n is large and p is not near 0 or 1. Po(λ) ≈ N(λ, λ) when λ is large. Both need a continuity correction.
Continuity corrections for P(X ≤ r), P(X < r), P(X ≥ r) and P(X > r): P(X ≤ r) → P(Y < r + 0.5) P(X < r) → P(Y < r − 0.5) P(X ≥ r) → P(Y > r − 0.5) P(X > r) → P(Y > r + 0.5)
X ~ B(150, 0.2). Which normal distribution and corrected event are used for P(X ≤ 24)?: Mean 30, variance 24, so Y ~ N(30, 24). P(X ≤ 24) ≈ P(Y < 24.5), z = −5.5/√24 = −1.123, probability about 0.131.

Population, census and sample: Population: every item or person of interest. Census: every member of the population is observed. Sample: a subset of the population.
Advantages and disadvantages of a census compared with a sample: Census: exact, but slow and costly, and impossible if testing destroys the item. Sample: cheaper and faster, but results vary and a poor frame causes bias.
Sampling unit and sampling frame: Sampling unit: one individual member that can be selected. Sampling frame: the list of all sampling units the sample is chosen from.
Statistic: A quantity calculated from the sample only, with no unknown parameter in it. For a sample X₁, X₂, X₃ with unknown mean μ, the sample mean is a statistic but X₁ − μ is not.
Sampling distribution: The distribution of a statistic over all possible samples of a given size. Example: population {0, 2} equally likely, samples of 2 with replacement: sample mean 0, 1, 2 with probabilities 1/4, 1/2, 1/4.

Null and alternative hypotheses: H₀: the assumed value of the parameter, written with an equals sign (p = p₀ or λ = λ₀). H₁: how the parameter differs: >, < (one-tailed) or ≠ (two-tailed).
Critical region and critical value: Critical region: the values of the test statistic for which H₀ is rejected. Critical value: the boundary of the critical region.
Actual significance level: P(test statistic in the critical region | H₀ true). For a discrete test it is usually below the nominal level.
How is a two-tailed test at the 5% level set up?: H₁ uses ≠ and each tail is given at most 2.5%. Choose each tail of the critical region so its probability is as large as possible without exceeding 2.5%.
X ~ B(14, 0.5), H₁: p < 0.5, 5% level. P(X ≤ 3) = 0.0287, P(X ≤ 4) = 0.0898. Find the critical region.: X ≤ 3, because 0.0287 < 0.05 but 0.0898 > 0.05. Actual significance level 2.87%.
How should a hypothesis test conclusion be written?: State the decision with the comparison ("0.0374 < 0.05, so reject H₀"), then say there is sufficient (or insufficient) evidence at the stated level, in context. Never say H₀ is "accepted" or "proved".