Statistics (statistica) - formulario

CONTINUITY CORRECTION

problem approximating a discrete r. v. (binomial) with aat most : ≤ continuous r. v. (normal)

step oneat least : ≥ solution : Pr(T = k) = Pr(k - 0.5 ≤ T ≤ k + 0.5)

Pr(T ≥ k) = Pr(T ≥ k-0.5)

Pr(T ≤ k) = Pr(T ≤ k+0.5).

INFERENTIAL STATISTICS- ESTIMATOR DISTRIBUTION

¯ estimator for µ ¯ estimator for p

ESTIMATE P. PARAMETER

σ² STANDARD ERROR : Population: Bernoulli× µ¯ N(µ, ) - σ p(1-p)- n s σ2 2 SE(¯) = SD(¯) = SE(P ) = SE(¯) =^n

A-S X2 2n-1 nnp (or ×) P- -P (or ¯) Bern. & CLT n increases → 0n- S estimator for σ

CONFIDENCE INTERVAL FOR µ CONFIDENCE INTERVAL FOR p 2 2↙ ↘ ↓≠ sample s.d. n µ)2σ KNOWN, µ UNKNOWN σ UNKNOWN : σ → S 2 p(1-p) Σ (X ii=1S2* p ± z ·N^ µ knownσ² =1-α/2¯ - µ n¯ ~ N(µ, ) nT = ~ T→ always between 0 and 1→

n-1n 2√S²/n nΣ (X ¯)ii=1S2¯ - µ not µ known =T = t-dist. n 1 d.f−Z = ~ N(0, 1) n−1 n-→ σ/√n S S* Pr(¯-t ≤ µ ≤ ¯-t = 1 - α· · )n-1; 1-α/2 n-1; 1-α/2√n √n- s s.d. est.· = mean est. ± ·× ± t tn-1; 1-α/2 n-1; 1-α/2 -[ → test statistic ]√n √n- [- ; +] GENERAL RULE C.I. level 1-α for θ : p. estimate ± quantile of estimator · stand. error [- ; +]α1- /2 only with 2 bound2θ = µ : point estimate × θ = µ : point estimate p^maquantile = ZNORMAL DATA CLT(n big nough)α = - % + 1 1-α/2 p(1-p)standard error =σ known → quantile = Z quantile = Z1-α/2 1-α/2 nSt.error = σ/√n St.error = σ/√ nσ not known → quantile = t quant. est. · stand. error = margin errorn ≥ 50[2

tails n-1; 1-α/2

Standard error = S /√n

margin error = length/2x n

not known : p = 0.5

length CI = amplitude = 2 · ME

Inference is made on μ, not σ, hence not on n- infos are on the population not on the random sample

wider interval = wider confidence interval

To cut the standard error in half, need to sample 2² = 4 times more the number of people we had in the initial sample.