Flashcards · YMA01 · Topic 7

Unit S1: Statistics 1 flashcards

38 cards on Topic 7, subtopics S1.1 to S1.7. Pick a subtopic or study them all.

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S1.1 Mathematical models in probability and statistics

Statistical model: A simplified mathematical description of a real situation, used to make predictions. A model gives a prediction, never a certainty.
The stages of the modelling cycle: 1. Define the real problem. 2. Make assumptions to simplify it. 3. Build the model and predict. 4. Compare predictions with data. 5. Revise the assumptions and repeat.
How do you comment on a statistical model?: Name an assumption, say why it may fail in this context, and suggest how to improve the model.

S1.2 Representation and summary of data

Histogram with unequal class widths: Unequal widths mean you must use frequency density (frequency ÷ class width) for the bar height. To find a frequency from a bar, multiply height by width.
Box plot: Whisker ends, Q1, the median and Q3 give the five key values. Outliers are marked separately with a cross; a common rule is more than 1.5 × IQR above Q3 or below Q1.
Comparing two data sets: Make one point about location and one about spread, with numbers and context.
Variance and standard deviation from a frequency table: Variance = Σfx²/Σf − (mean)², with mean = Σfx/Σf. Standard deviation = √variance. Variance is in squared units.
Coding y = (x − a)/b: how do you recover the mean and SD of x?: Mean of x = b × (mean of y) + a. SD of x = b × (SD of y). The a does not affect the spread. Example: y = (x − 300)/20 with mean 2.5, SD 0.8 gives mean 350, SD 16.
Positive and negative skew from quartiles and from mean and median: Positive skew: Q2 − Q1 < Q3 − Q2 and mean > median (tail to the right). Negative skew: Q2 − Q1 > Q3 − Q2 and mean < median. Quote the numbers that support your choice.

S1.3 Probability

The addition (sum) law for two events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Conditional probability P(B | A): P(B | A) = P(A ∩ B)/P(A). Rearranged: P(A ∩ B) = P(A) × P(B | A).
Mutually exclusive events compared with independent events: Mutually exclusive: cannot both happen, P(A ∩ B) = 0. Independent: one happening does not change the probability of the other, P(A ∩ B) = P(A)P(B).
How do you test whether A and B are independent?: Compare P(A) × P(B) with P(A ∩ B). Equal means independent. Example: P(A) = 0.5, P(B) = 0.4, P(A ∪ B) = 0.8 gives P(A ∩ B) = 0.1, but P(A)P(B) = 0.2, so not independent.
P(A) = 0.45, P(B) = 0.3 and P(neither) = 0.4. Find P(A ∩ B).: P(A ∪ B) = 1 − 0.4 = 0.6. P(A ∩ B) = 0.45 + 0.3 − 0.6 = 0.15.
Bag A (probability 0.5) gives red with probability 0.2; bag B (0.5) gives red with 0.6. Find P(bag A | red).: P(red) = 0.5 × 0.2 + 0.5 × 0.6 = 0.4. P(A | red) = 0.1/0.4 = 0.25.

S1.4 Correlation and regression

Formulae for Sxx, Syy and Sxy: Sxx = Σx² − (Σx)²/n Syy = Σy² − (Σy)²/n Sxy = Σxy − (Σx)(Σy)/n
The product moment correlation coefficient r: r = Sxy/√(Sxx Syy), with −1 ≤ r ≤ 1. Near 1: strong positive linear correlation; near −1: strong negative; near 0: no linear relationship. Correlation does not prove causation, and outliers can distort r.
Least squares regression line of y on x: y = a + bx, with b = Sxy/Sxx and a = ȳ − b x̄. The line passes through (x̄, ȳ).
How do you interpret the gradient b of a regression line?: The change in y for each one-unit increase in x, in context and with units.
Interpolation compared with extrapolation: Interpolation (x inside the data range) is reasonably reliable. Extrapolation (outside the range) is unreliable because the pattern may change. Do not use the y on x line to predict x from y.
How does coding affect r?: Adding constants or multiplying by positive constants leaves r unchanged. Multiplying one variable by a negative number changes the sign of r.

S1.5 Discrete random variables

Conditions for a valid probability function: Each p(x) lies between 0 and 1, and Σ p(x) = 1.
Cumulative distribution function, and P(X ≥ a) for a whole-number variable: F(x₀) = P(X ≤ x₀). P(X ≥ a) = 1 − F(a − 1); P(X > a) = 1 − F(a); P(X < a) = F(a − 1).
E(X) and Var(X) for a discrete random variable: E(X) = Σ x p(x). E(X²) = Σ x² p(x). Var(X) = E(X²) − (E(X))².
E(aX + b) and Var(aX + b): E(aX + b) = aE(X) + b. Var(aX + b) = a²Var(X). Adding b does not change the variance.
X takes 0, 1, 2, 3 with probabilities 0.1, 0.2, 0.4, 0.3. Find E(X) and Var(X).: E(X) = 0.2 + 0.8 + 0.9 = 1.9. E(X²) = 0.2 + 1.6 + 2.7 = 4.5. Var(X) = 4.5 − 1.9² = 0.89.
p(x) = k(x + 1) for x = 0, 1, 2, 3. Find k.: k(1 + 2 + 3 + 4) = 10k = 1, so k = 1/10.

S1.6 Discrete distributions

Discrete uniform distribution on 1, 2, …, n: probability, mean and variance: P(X = x) = 1/n. E(X) = (n + 1)/2. Var(X) = (n² − 1)/12.
When is a discrete uniform model suitable?: When a variable takes n separate values and each is equally likely, such as the score on a fair die.
X is uniform on the whole numbers 3 to 8. Find E(X) and Var(X).: Let Y = X − 2, uniform on 1 to 6: E(Y) = 3.5, Var(Y) = 35/12. E(X) = 5.5, Var(X) = 35/12 (a shift does not change the variance).
X is uniform on 1 to 6 and Y = 3X + 1. Find E(Y) and Var(Y).: E(X) = 3.5, Var(X) = 35/12. E(Y) = 3(3.5) + 1 = 11.5. Var(Y) = 9 × 35/12 = 26.25.
X is uniform on 1 to n and Var(X) = 4. Find n.: (n² − 1)/12 = 4, so n² = 49 and n = 7. Then E(X) = 4.

S1.7 The Normal distribution

What does X ~ N(μ, σ²) mean, and what shape does it have?: Normal with mean μ and variance σ² (the second value is the variance, so divide by σ when standardising). Bell-shaped and symmetric about μ; mean = median = mode = μ; total area 1.
Standardising, and writing P(Z > z) and P(Z < −z) using Φ: Z = (X − μ)/σ, with Z ~ N(0, 1), and Φ(z) = P(Z ≤ z). For z > 0: P(Z > z) = 1 − Φ(z), P(Z < −z) = 1 − Φ(z), P(Z > −z) = Φ(z).
X ~ N(40, 25). Find P(X < 46), given Φ(1.2) = 0.8849.: σ = 5. z = (46 − 40)/5 = 1.2. P(X < 46) = Φ(1.2) = 0.8849.
How do you find a value k from a probability (inverse normal)?: Sketch and shade. Write the probability as Φ(z) = p, find z from the table (negative if the left tail is below 0.5), then k = μ + zσ.
How do you find μ and σ from two probability statements?: Find the z-value for each, with the correct sign. Write x₁ = μ + z₁σ and x₂ = μ + z₂σ, subtract to find σ, then substitute for μ.
Symmetry shortcut: P(X < a) = P(X > b): Then μ is halfway between them: μ = (a + b)/2.