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# COMPLETE Statistics Notes - Part 7 (got 4.0 in the course)

9 Pages
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School
Boston College
Department
Economics
Course
ECON 1151
Professor
All
Semester
Spring

Description
Final—Sampling Distributions, C.I.’s, Hypothesis Tests Keely Henesey Normal Distribution Standard Normal Distribution Random Variable X ∈(−∞,∞) Z∈(−∞,∞) 2 Denoted By… X N μ,σ ) Z N (0,1) E X =μ E Z]=0 Properties 2 Var X =σ Var(Z)=1 Cumulative Density Function F( 0=P (≤ x 0 F( 0=P (≤ z 0) Range Probabilities P(a5 Standard Normal  − Central Limit Theorem Holds Approximation X−μ p−P Z= Z= σX σ̂p μ±z α/2X P± zα/2σ̂p Acceptance Interval 1−α →ProbabilityThat SampleMeanisIncluded∈theInterval {α →ProbabilityThatSampleMeanis NOT Included∈theInterval } General Guidelines for C.L.T Final—Sampling Distributions, C.I.’s, Hypothesis Tests Keely Henesey Population Distribution Normally  Symmetric  Uniform  Highly Skewed Distributed Distribution Distribution Necessary n  for a  Normal Sample Mean  Any n n [20,25 ] n≥25 n≥50 Distribution Sample Variance Distribution Chi­Square Distribution of Sample & Population Variance 2 1 2 2 Random Variable s = n−1 ∑ (xi−́x) χ ∈ 0,∞ ) 2 2 2 E s =σ E [ vv Properties 2 (σ2) 2 Var [s2= Var [ ]vv (n−1) ONLY If Population is Normally Distributed Chi–Square  2 2 2 Approximation χ2 = (n−1)s = ∑ (xi−́x) = 1 x −x =2 xi−́x n−1 σ2 σ 2 σ2∑ (i ) ∑ ( σ Sampling Distributions Uniform Distribution Exponential Distribution Random Variable X ∈ [a,b ] T ∈(0,∞) Probability Density  f (x)= 1 f t =λe −λt Function b−a Cumulative Probability  F ( )f (x)[ x −a] ( ) −λt Density Function 0 0 F t =1−e a+b E [X]= 2 E [X = [1/λ] Properties 2 Var(X )= (b−a) Var X = 1/λ 2] 12 Final—Sampling Distributions, C.I.’s, Hypothesis Tests Keely Henesey Bernoulli Distribution Normal Approximation Binomial Distribution Proportion Random Variable P Denotes… Probability of success in a single triProbability of x successes in n independent trialstion p of successes in n  independent trials Random  X P= X Variable n Conditions n=1 nP(1−P)>5 nP(1−P)<5 nP(1−P)>5 Success:P(1)=P Possible  Outcomes for X FailureP(0)=1−P Mean μ XE [X =P μX=E X =nP μP=E [P]=P 2 2 2 2 2 2 P 1−P ) Variance σ XE X[( X) ]P(1−P) σX=E X[μ X)]nP(1−P) σP=E P[μ P) ] n X−nP n! x (n−x) Z= X−P Calculating  P(x)=P(X=x) Z= P(x)= P (1−P) P 1−P ) Probabilities √nP(1−P) x!(n−x)! √ n Final—Sampling Distributions, C.I.’s, Hypothesis Tests Keely Henesey Confidence Level 90 94 95 96 97 98 99 α 0.100 0.06 0.05 0.04 0.03 0.02 0.01 zα/2 1.645 1.88 1.96 2.05 2.17 2.33 2.58 100(1−α)  Confidence Interval Mean of Normal Population  σ σ  Known x± z α/2 1. SRS √ n 2. Normal Pop. (CLT) 3. Sampling With Replacement  σ   x±t s Unknown n−1,α/√n (N≥10n) Difference Between Means, Dependent Samples  ́ sd (Dependent Samples) d±t n−1,α/√n 2 2 σ x σy σ  Known (x−́y)±z α/2 n + n √ x y Difference Between Means  σ   sp sp (Independent Samples) Unknown  x−́ y)±tnx+y −2,α /2 + (Equal) √n x n y σ   sx s2y Unknown (x−́y)±t v,α /2 + (Unequal) √ nx n y Mean of Normal Population  p 1−p ̂ ) 1. SRS p±t n−1,α/2 2. nP(1−P)>5 √ n Difference Between Proportions (Large,  ̂ ̂ p x(̂ px) py(−̂ p y Independent Samples) (p xp ±y) α/2√ n + n x y 2 2 Confidence  n−1 s) χ 2 α = α n−1,α /2 ( n−12) 2 2 2 α χ n−1,(1−α /2) P χ n−1zα x́−μ H 1μ>μ 0 σ/ √ P z> 0 ¿ H 0μ≤ μ 0 ( σ/√n ) x>x cμ +0 σα n√ ́−μ 0 H 0μ=μ 0 σ/ √ z 2∙P z< x−μ 0 0 0 1 0 σ/√n ∣ α /2 ( σ/√n ) Mean of Normal Population  ( σ  Unknown) Null Hypothesis ( Alternative Hypothesis ( H 0 ) H 1 ) Reject H 0  if… p­Value ́−μ H 0μ=μ 0 0>t n−1,α x−μ ¿ H 1μ>μ 0 s/√n P t n−1 0 ( s/√n ) H 0μ≤μ 0 x>x cμ +0 n
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