COUNTING: MATH 111 FORMULA SHEETS Multiplicative Rule: If you draw one element from each of sets with size n1 , n2 , . . . nk , there are n1 n2 n3 · · · nk different results possible. SAMPLES: Sample mean: x = P x n Combinations/Partitions: If you draw n elements from a set of N elements without regard for order, the number of different results is N! N =N Cn = n n! · (N − n)! Sample variance: s2 = P (x − x)2 = n−1 P P 2 ( x) (x2 ) − n n−1 Sample standard deviation: s= √ More generally, if you partition a set of N elements into k groups, with n1 elements in the first group, n2 elements in the second group, etc., the number of different results is N! n 1 ! · n2 ! · · · n k ! s2 Chebyshev’s Rule: For all samples • At least x + 2s; 3 4 • At least x + 3s; 8 9 of the data lie between x − 2s and Permutations: If you draw n elements from a set of N elements and arrange the elements into a distinct of the data lie between x − 3s and order, the number of different results is N Pn • In general, at least 1 − n12 of the data lie between x − ns and x + ns. = N! (N − n)! DISCRETE VARIABLES: Empirical Rule: For roughly normal data sets P Mean (or expected value): µ = E(X) = x · p(x) • Approximately 68% of the data lie between x − s P Variance: σ 2 = E(X − µ)2 = (x − µ)2 · p(x) or and x + s; P 2 2 2 2 σ = E(X ) − µ = ( x · p(x)) − µ2 √ • Approximately 95% of the data lie between x−2s Standard deviation: σ = σ 2 and x + 2s; Binomials: If X is binomial with parameters n and p • Approximately 99.7% of the data lie between x − n then µ = np, σ 2 = npq, and p(x) = px q n−x 3s and x + 3s. x Sample z−score: z = Poisson: If X is a Poisson variable with parameter λ x −λ e then µ = λ, σ 2 = λ, and p(x) = λ x! x−x s Population z−score: z = x−µ σ Hypergeometrics: If X is hypergeometric, where n elements are drawn from a population of size N that PROBABILITY: r(N −r)n(N −n) 2 has r successes initially, then µ = nr , N ,σ = N 2 (N −1) and For any event S, 0 ≤ P (S) ≤ 1 r N −r x n−x Complement Rule: P (AC ) = 1 − P (A) p(x) = N Addition Rule: P (A∪B) = P (A)+P (B)−P (A∩B); n if A and B are disjoint, P (A ∪ B) = P (A) + P (B). Multiplication Rule: P (A ∩ B) = P (A) · P (B|A); if CONTINUOUS VARIABLES: A and B are independent, P (A ∩ B) = P (A) · P (B) Uniform: If X has a uniform distribution over the Conditional Probability: b−a 1 √ interval [a, b], then µ = a+b , and f (x) = b−a 2 , σ = 12 P (A ∩ B) P (A|B) = Exponential: If X has an exponential distribution x P (B) with parameter θ then µ = θ, σ = θ, and f (x) = θ1 e− θ −x Also, P (X ≥ a) = e a Independence: A and B are independent if P (A) = P (A|B) or P (B) = P (B|A) 1 Normal: If X is normal then z = x−µ σ H1 µ 6= µ0 µ > µ0 µ < µ0 Centiles: For normal variable X, the centile is xα = µ + zα σ Reject H0 if |t| > t α2 t > tα t < −tα p−value 2P (T > |t|) P (T > t) P (T < t) SAMPLING DISTRIBUTIONS: One-sample test for p : If pˆ ± 3σpˆ is within the ˆ 0 interval (0,1) then the test statistic is z = p−p σpˆ where p p0 q0 Central Limit Theorem: For a random sample from σpˆ = n a (large) population the sampling ditribution of x is approximately normal. H Reject H if p−value 1 p 6= p0 p > p0 p < p0 Sample mean: For a random sample of n elements from a population with mean µ and standard deviation σ, the sampling distribution has µx = µ and σx = √σn 0 |z| > z α2 z > zα z < −zα 2P (Z > |z|) P (Z > z) P (Z < z) TWO-POPULATION TESTS Normal approximation to binomial: If X is bino√ mial with parameters n and p, and np ± 3 npq is in 0 the interval [0, n],(same as 9q ≤ np and 9p ≤ nq), then Two-sample z−test for means with known σ s or (x1 −x2 )−D0 samples: The test statistic is z = P (a ≤ X ≤ b) = P (a − 1 < X < b + 1) is approxi- large random σ q mately P (a − 12 < Y < b + 12 ) where Y is normal with where σ = (σ1 )2 + (σ2 )2 √ n1 n2 µ = np and σ = npq H1 µ1 − µ2 6= D0 µ1 − µ2 > D0 µ1 − µ2 < D0 CONFIDENCE INTERVALS: For µ with known σ or large random sample: x ± z α2 ( √σn ) Reject H0 if |z| > z α2 z > zα z < −zα p−value 2P (Z > |z|) P (Z > z) P (Z < z) Two-sample t−test for means with unknown σ 0 s and small random samples, where variances are equal: The q (x1 −x2 )−D0 1 test statistic is t = for s = sp n1 + n12 s q (n1 −1)(s1 )2 +(n2 −1)(s2 )2 and sp = and we use t with n1 +n2 −2 For µ with unknown σ and small sample from n + n − 2 degrees of freedom 1 2 a normal population: x ± t α2 ( √sn ) for t with n − 1 degrees of freedom H1 Reject H0 if p−value q α µ − µ = 6 D |t| > t 2P (T > |t|) ˆq 1 2 0 2 For proportion p : If pˆ ± 3 pˆ n is within the interval µ − µ > D t > t P (T > t) q 1 2 0 α pˆ ˆq α µ − µ < D t < −t P (T < t) (0,1),(same as 9ˆ q < nˆ p and 9ˆ p < nˆ q ),then pˆ ± z 2 · n 1 2 0 α Sample size (margin of error b) for µ : z α · σ 2 2 n≥ b Sample size (margin of error b) for p : z α 2 2 n≥ pq b 0 Paired sample t−test: The test statistic is t = xD −D σ σ D where σ = √n and use t with n − 1 degrees of freedom H1 µD = 6 D0 µD > D0 µD < D0 ONE-POPULATION TESTS: Reject H0 if |t| > t α2 t > tα t < −tα p−value 2P (T > |t|) P (T > t) P (T < t) One-sample z−test for µ with known σ or large ran0 Two-sample z−test for proportions: The test statisdom sample: The test statistic is z = x−µ ( √σn ) tic is pˆ1 − pˆ2 z=q H1 Reject H0 if p−value pˆqˆ( n11 + n12 ) µ 6= µ0 |z| > z α2 2P (Z > |z|) 2 µ > µ0 z > zα P (Z > z) where pˆ = nx11 +x +n2 µ < µ0 z < −zα P (Z < z) H1 p1 − p 2 = 6 0 p1 − p 2 > 0 p1 − p 2 < 0 One-sample t−test for µ with unknown σ and small random sample from a normal population: The test 0 statistic is t = x−µ ( √s ) and we use t with n − 1 degrees of n freedom 2 Reject H0 if |z| > z α2 z > zα z < −zα p−value 2P (Z > |z|) P (Z > z) P (Z < z)