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A Level Statistics
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Binomial Distribution – Probability Calculations
Calculate P(X = k) for binomial
\[ P(X = 5) \text{ where } X \sim B(12, 0.3) \]
Calculate exact probability of k successes.
Calculate P(X ≤ k) cumulative
\[ P(X \leq 4) \text{ for binomial} \]
Cumulative probability up to k.
Calculate P(X < k) strict inequality
\[ P(X < 6) = P(X \leq 5) \]
Strictly fewer than k successes.
Calculate P(X ≥ k) using complement
\[ P(X \geq 7) = 1 – P(X \leq 6) \]
k or more successes.
Calculate P(X > k) strict inequality
\[ P(X > 5) = 1 – P(X \leq 5) \]
Strictly more than k successes.
Calculate P(a ≤ X ≤ b) range
\[ P(3 \leq X \leq 7) \]
Probability in a given range.
Binomial Distribution – Setup and Modelling
State distribution from context
\[ X \sim B(n, p) \]
Extract n and p from word problem.
Identify binomial condition that fails
\[ \text{Fixed n, Independent, Constant p} \]
Identify violated condition.
Calculate for repeated experiments
\[ P(\text{exactly 2 from 4}) \]
Small n scenarios.
Normal Distribution – Probability Calculations
Find P(X < a) for normal
\[ P(X < 65) \text{ where } X \sim N(50, 100) \]
Probability below a value.
Find P(X > a) for normal
\[ P(X > 72) = 1 – P(X < 72) \]
Probability above a value.
Find P(a < X < b) for normal
\[ P(45 < X < 60) \]
Probability between two values.
Find x given P(X < x) = p
\[ P(X < x) = 0.95 \Rightarrow x = ? \]
Inverse normal calculation.
Standardise using z = (x – μ)/σ
\[ z = \frac{x – \mu}{\sigma} \]
Convert value to z-score.
Normal Distribution – Finding Parameters
Find σ given μ and probability
\[ P(X < 80) = 0.9 \Rightarrow \sigma = ? \]
Determine SD from probability.
Find μ given σ and probability
\[ P(X < 75) = 0.8413 \Rightarrow \mu = ? \]
Determine mean from probability.
Set up simultaneous equations
\[ \frac{x_1 – \mu}{\sigma} = z_1 \]
Form equations from two conditions.
Solve simultaneous for μ and σ
\[ \text{Find } \mu \text{ and } \sigma \]
Solve to find both parameters.
Normal Approximation to Binomial
Calculate mean and variance
\[ \mu = np, \quad \sigma^2 = np(1-p) \]
Parameters for normal approximation.
Apply continuity correction
\[ P(X \geq 50) \approx P(Y > 49.5) \]
Correct for discrete to continuous.
Calculate with normal approx
\[ B(150, 0.6) \approx N(90, 36) \]
Complete approximation process.
Hypothesis Testing – Binomial
State hypotheses (proportion)
\[ H_0: p = 0.3, \quad H_1: p > 0.3 \]
Write H₀ and H₁ for proportion test.
Find critical region (one-tailed)
\[ X \geq k \text{ at 5%} \]
Identify boundary for rejection.
Find critical regions (two-tailed)
\[ X \leq a \text{ or } X \geq b \]
Both tails for rejection.
Calculate actual significance level
\[ P(X \leq k_1) + P(X \geq k_2) \]
Sum tail probabilities.
State conclusion from test
\[ \text{Reject } H_0 \text{? Yes/No} \]
Determine if value in critical region.
Hypothesis Testing – Normal Mean
State hypotheses for mean test
\[ H_0: \mu = 500, \quad H_1: \mu < 500 \]
Write H₀ and H₁ for mean.
Calculate test statistic for mean
\[ z = \frac{\bar{x} – \mu_0}{\sigma / \sqrt{n}} \]
Compute z from sample data.
Calculate p-value for mean test
\[ \text{p-value} = P(Z > z) \]
Find probability as extreme.
Compare with critical value
\[ z > 1.645 \Rightarrow \text{Reject } H_0 \]
State decision from comparison.
Venn Diagrams and Probability
Calculate P(A∪B)
\[ P(A \cup B) = P(A) + P(B) – P(A \cap B) \]
Inclusion-exclusion formula.
Find P(A∩B) from union
\[ P(A \cap B) = P(A) + P(B) – P(A \cup B) \]
Rearrange to find intersection.
Calculate P(A’) complement
\[ P(A’) = 1 – P(A) \]
Event not occurring.
Three-set Venn diagram
\[ P(\text{exactly two}) \]
Add pairwise intersections.
Set notation from words
\[ \text{“A but not B”} \rightarrow A \cap B’ \]
Translate verbal to notation.
Conditional Probability and Independence
Calculate P(A|B)
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Conditional probability formula.
Test for independence
\[ P(A \cap B) = P(A) \times P(B) \text{?} \]
Check product rule.
Identify mutually exclusive
\[ P(A \cap B) = 0 \Rightarrow \text{M.E.} \]
Recognise from intersection.
Independence from conditional
\[ P(A|B) = P(A) \Rightarrow \text{Indep.} \]
Alternative test for independence.
Correlation and Regression
Identify correlation type
\[ \text{Strong positive, weak negative…} \]
Describe from pattern.
Hypothesis test for PMCC
\[ H_0: \rho = 0, \quad |r| > \text{critical} \]
Compare r with critical value.
Interpret gradient
\[ y = 12 + 2.5x \Rightarrow \text{gradient} = 2.5 \]
Meaning of gradient in context.
Interpret intercept
\[ y = 15 + 3x \Rightarrow \text{when } x=0 \]
Value of y-intercept.
Summary Statistics from Coded Data
Calculate mean from Σx and n
\[ \bar{x} = \frac{\sum x}{n} \]
Apply mean formula.
Calculate SD from Σx and Σx²
\[ \sigma = \sqrt{\frac{\sum x^2}{n} – \bar{x}^2} \]
Standard deviation formula.
Transform mean using coding
\[ y = x – a \Rightarrow \bar{y} = \bar{x} – a \]
Effect of coding on mean.
SD unchanged by translation
\[ y = x – a \Rightarrow \sigma_y = \sigma_x \]
Translation doesn’t change SD.
Timer (Optional)
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