Flashcards · H240 · Topic 2

Statistics flashcards

39 cards on Topic 2, subtopics 2.01 to 2.05. Pick a subtopic or study them all.

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2.01 Statistical sampling

Population, sample and census: Population: everything you want information about. Sample: the part actually observed. Census: observing every member of the population.
Sampling frame: A complete list of the population from which a sample is drawn.
Bias in sampling: A systematic tendency for a sample to differ from the population, for example because some members could never be chosen.
How do you take a simple random sample?: 1. Make a frame and number it 1 to N. 2. Generate random numbers in range. 3. Ignore repeats and out-of-range values. 4. Stop at n members.
Opportunity sampling: method and main weakness: Use whoever is available. It is not random, so it is biased to the place and time of sampling.
Systematic sampling: method, with a population of 600 and a sample of 40: Choose every kth member from a random start. Here k = 600/40 = 15, with a random start from 1 to 15. A pattern in the list can clash with k.
Stratified, cluster and quota sampling in one line each: Stratified: a proportional random sample from each group. Cluster: whole groups chosen at random. Quota: fixed numbers from each group, chosen by the interviewer (not random).

2.02 Data presentation and interpretation

Histogram: what is plotted on the vertical axis?: Frequency density = frequency ÷ class width. The area of each bar is proportional to the frequency.
Two definitions of a possible outlier: More than 1.5 × IQR below Q₁ or above Q₃. More than 2 standard deviations from the mean.
Q₁ = 22 and Q₃ = 30. Is 45 a possible outlier (1.5 × IQR rule)?: IQR = 8, so 1.5 × IQR = 12 and the upper fence is 30 + 12 = 42. 45 > 42, so yes.
Which measures resist extreme values?: The median and the IQR. The mean and standard deviation are affected by extreme values.
How do you describe a scatter diagram?: Direction (positive, negative, none), strength (strong, weak), shape (linear or curved), and unusual features (outliers, separate groups), all in context.
Explanatory and response variables: Explanatory: plotted on x, thought to influence the other. Response: plotted on y, the variable you predict.
Interpolation vs extrapolation: Interpolation: predicting for x inside the range of the data (more reliable). Extrapolation: predicting outside the range (unreliable).
Does strong correlation show that one variable causes the other?: No. Correlation does not imply causation; a third (confounding) variable may influence both.

2.03 Probability

Mutually exclusive vs independent: Mutually exclusive: cannot happen together, P(A ∩ B) = 0. Independent: one does not affect the other, P(A ∩ B) = P(A) × P(B). Events with non-zero probabilities cannot be both.
Addition rule for any two events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Conditional probability: P(A | B) = P(A ∩ B)/P(B). Along a tree: P(A ∩ B) = P(A) × P(B | A).
How do you test whether A and B are independent?: Compare P(A ∩ B) with P(A) × P(B) (or P(B | A) with P(B)). Equal means independent. State the conclusion in a sentence.
The quickest way to find P(at least one): 1 − P(none).
With and without replacement on a tree diagram: Without replacement, the second-stage probabilities depend on the first outcome. With replacement (or from a very large population) they stay the same.

2.04 Statistical distributions

P(X = x) = k(2x − 1) for x = 1, 2, 3. Find k and P(X ≥ 2).: k + 3k + 5k = 1, so k = 1/9. P(X ≥ 2) = (3 + 5)/9 = 8/9.
Conditions for a binomial model B(n, p): A fixed number of trials n; each trial has two outcomes; the trials are independent; the probability of success p is constant.
Binomial: P(X = x), mean and variance: P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ, q = 1 − p. Mean np; variance npq; standard deviation √(npq).
Shape and key features of the normal distribution N(μ, σ²): Bell-shaped and symmetrical about μ, total area 1. Points of inflection at μ ± σ. P(X = a) = 0, so P(X < a) = P(X ≤ a).
Roughly what proportion of a normal distribution lies within 1, 2 and 3 standard deviations of the mean?: Within μ ± σ: about two-thirds. Within μ ± 2σ: about 95%. Within μ ± 3σ: almost all.
Standardising, and how to find x from a probability: Z = (X − μ)/σ, so X = μ + zσ, with Z ~ N(0, 1). Inverse normal: enter the area to the left of x.
Effect of changing μ and σ on the normal curve: Changing μ moves the curve left or right without changing its shape. A larger σ gives a wider, lower curve; the area stays 1.
X ~ B(n, p) with n large. Which normal distribution could approximate X?: N(np, npq). For example B(200, 0.3) is approximately N(60, 42).

2.05 Statistical hypothesis testing

H₀ and H₁: H₀ always has "=". The direction of H₁ comes from the wording of the question, not the data.
Significance level and p-value: The level is chosen before; the p-value comes from the data.
Critical value and test statistic: The critical value is a boundary; the test statistic is the sample result.
Type I and Type II: Type I is wrongly rejecting a true H₀. Type II is failing to reject a false H₀. H240 itself only requires that the significance level is the probability of incorrectly rejecting H₀; Type II is extension.
σ and σ/√n: Use σ for individual values, σ/√n for sample means.
Binomial test: which tail probability is the p-value for H₁: p > p₀, p < p₀ and p ≠ p₀?: p > p₀: P(X ≥ x) = 1 − P(X ≤ x − 1). p < p₀: P(X ≤ x). p ≠ p₀: compare each tail with half the level (or double the smaller tail).
Normal test for a mean: distribution under H₀ and test statistic: X̄ ~ N(μ₀, σ²/n), so z = (x̄ − μ₀)/(σ/√n).
H₀: μ = 60, H₁: μ < 60, σ = 5, n = 25, 5% level. Find the critical region for X̄.: Under H₀, X̄ ~ N(60, 1²). Critical region: X̄ < 60 − 1.645 × 1 = 58.36 (2 d.p.).
Testing for correlation with Pearson’s product-moment correlation coefficient: H₀: ρ = 0; H₁: ρ > 0, ρ < 0 or ρ ≠ 0 (ρ is the population coefficient). Compare the sample r with the critical value from the table for n pairs and the significance level (or compare a given p-value with the level), then conclude in context. The data are assumed to come from a bivariate normal distribution.
What does the value of r tell you?: −1 ≤ r ≤ 1. The sign gives the direction; the size gives how close the points lie to a straight line (not the gradient). r near 0 means no linear pattern, though a curve is still possible.