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# Statistics for Business Chapters 6, 7 and 8 Cheat Sheet (Final Third of the Class) 285

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This 5 page Study Guide was uploaded by Angie Notetaker on Tuesday March 29, 2016. The Study Guide belongs to 285 at Rutgers University taught by Zinonos in Fall 2015. Since its upload, it has received 46 views. For similar materials see Statistics for Business in Statistics at Rutgers University.

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Date Created: 03/29/16

Chapter 6: Estimation with Confidence Intervals à Summary on page 356 Harry 1 • Identifying and Estimating the Target Parameter o Target Parameter- the unknown population parameter (ex: mean or proportion) that we are estimating o Point Estimator- of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used as an estimate of the target parameter o Confidence Interval- is a formula that tells us how to use the sample data to calculate an interval that estimates the target parameter • Confidence Interval for a Population Mean: Normal Statistic o Confidence Level- is the confidence coefficient (the probability that a confidence interval encloses the population parameter) expressed as a percentage § The confidence interval with confidence coefficient of (1- α) is ???? +/- (Z ) ???? α/2 ???? § Z αs the value where the area α will lie to its right Ex: Z 0.5l be the z- value corresponding to an area of .5- .05= .45 to the right of the mean • N> 30 and population has any distribution X~ n(M ,xσ )xWhere σ isxknown and M is unkxown ▯ ▯ ▯???????? ????~ n(M ,x ▯) so that Z= ▯ ▯ and can be ~n(0,1) ▯ ▯ ▯▯ ▯ ▯ ▯ P( -Zα/2 < ▯▯ < Z )α/2P (???? - Z α/2 ▯ < M <x???? - Z α/2 ▯ ) ▯ ???? Confidence Interval: ???? +/- Z α/2 at confidence level (1- α) % for M x ???? • Confidence Interval for a Population Mean: Student’s t- Statistic • N< 30 and has a normal distribution X~n n(M , σ ) Where M and σ are unknown x x x x ▯ ▯???? {X }−n(x ) T α/2 ▯ ???? Where S= n−1 ▯ ▯▯ ▯ ▯ P( -α/2 < ▯ ▯< t α/2 P (???? - t α/2 < M <x???? + t α/2 ) ▯ ▯ ▯ ???? Confidence Interval: ???? +/- t α/2 at confidence level (1- α) % for M x ???? Let {x} be an i.i.d. sample from X. If n is large, then▯ ▯▯▯ is a place so that ▯ ▯ (???? +/- Zα/2 ) is an approximation (1- α) % confidence interval for M regxrdless ▯ of the distribution of x • Table of Commonly Used Values of Z α/2on page 319 • Large- Sample Confidence Interval for a Population Proportion o We estimate the percentage/ proportion of some group with a certain characteristic o ???? is the probability tat outcome p is chosen ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯ ????X2 ???? ̂= ???? ̂ ???? = ???? ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯ pq If np or nq> 5 then ????~ n( p, ) by CLT n ▯▯▯ pq pq P( -Zα/2 < < Z )α/2P (????- Z α/2 < M <x???? - Z α/2 ) pq n n n 2 Harry ???????? ▯ If n???? and n???? > 15 use: ???? +/- Z α/2 where ????= ????= 1- ???? ???? ▯ ???? ̃???? ̃ ???? + 2 If n???? or n???? < 15 use: ???? ̃+/- Z α/2 where ???? ̃= ????+???? ▯▯▯ • Determining the Sample Size o I want an interval of total length 2h ▯ ▯ (???? - Zα/2 , ???? + Z α/2 ) ▯ ▯ 2Z ▯ = 2h α/2 ▯ • Confidence Interval for a Population Variance 2 ????▯???? ???????? 2 ????▯???? ???????? o A (1- α) % for σ : ???? < ???? < ???? ????????/???? ???? (????▯ ) ???? o Normal Theory: Let Z~ n(0,1) Let Y = x and let Y =x 2 where Y and Y are independent 2 1 2 1 2 Let xi=Z = y ▯ ▯▯▯/▯ ▯▯▯▯ Fy(y)= If x~ n(M, σ) Let {x} be i.id. from x then: 2 ▯ ▯ ▯▯▯▯ 1) S ~ ▯▯▯ ▯ ▯▯ 2) t = ▯ ▯ ~ T n -1 ▯ Confidence Intervals: 1) For asymmetrical distributions ▯ ▯ ▯▯▯ ▯ ▯ P( ???? ▯▯▯,▯/▯ ▯▯ < ???? ▯▯▯,▯/▯)= 1 – α ▯ ▯▯ ▯ = P( > > ) ▯▯▯▯,▯▯ ▯/▯ ▯▯▯ ▯ ▯ ▯ ▯▯▯,▯/▯ ▯ ▯ = P( ▯▯▯ ▯ > σ > ▯▯▯ ▯ ) ▯▯▯▯,▯▯ ▯/▯ ▯▯▯▯,▯/▯ 2) For Symmetrical distributions ▯ ▯ ▯▯ ▯ ▯ P(-????▯▯▯,▯/▯ < ▯ < ???? ▯▯▯,▯/▯) ▯ ▯ ▯ = P(???? - tn-1, α/▯ < M <x???? + t n-1, α▯2)= 1- α Chapter 7: Hypothesis Testing à Summary on page 424 • The Elements of a Test Hypothesis o Null Hypothesis- H 0 o Alternate Hypothesis- H 1 o X~n(M, σ) with σ known We are choosing between 2 statements: H : M> M0 0 H 1 M<M 0 H : M< M 0 0 H 1 M> M 0 H 0 M= M 0 H 1 M ≠ M 0 o Possible Errors: Harry 3 H 0rue H 0alse Pick H ✓ ✗ 0 Don’t Pick H 0 ✗ Type I ✓ Type II Error Error Prob(Type I Error)= α- rejecting H whe0 it is true Prob(Type II Error)= β- accepting H when0it is false We set α to be a pre- specified value à α is the significance level of the test If α is small and if H 0s true, your probable will not reject it o More Hypothesis testing Let x a have density f(x|θ) Where θ is a population parameter Ex: X~ n(M, σ) Let θ ε θ(the set of all points the parameter can be And let A < θ M ε (-∞,∞) σ ε (0, ∞) The goal of hypothesis testing is to pick two competing hypotheses H 0 θ ε A (θ is in the region of A) H 1r H :aθis not in the region A) Decision Rule: If X ε R (if x is in some region R), then I will reject H : 0 X~n(M, σ) with σ known H : M= M H : M= M H : M= M 0 0 0 0 0 0 H 1 M> M 0 H 1 M< M 0 H1: M ≠ M 0 ▯ ▯▯ ▯ ▯ ▯▯ ▯ ▯ ▯▯▯ Z= > Zα Z= < -Zα |Z|= | | > Zα/2 ▯ ▯ ▯ Case I Assumption: (Chapter 7.4- Test about a Population Mean: Normal Z- Statistic) X~n(M, σ) with σ known ▯ A (1- α)% confidence interval for M is (????, +/- Z α/2▯) H 0 M= M 0 H 0 M= M 0 H0: M= M 0 H : M> M H : M< M H : M ≠ M 1 0 1 0 1 0 Z= ▯ ▯▯ ▯> Z Z= ▯ ▯▯ ▯< -Z |Z|= |▯ ▯▯▯ | > Z ▯▯ α ▯▯ α ▯▯ α/2 ▯ ▯ ▯ Case II Assumption: (Chapter 7.5- Test about a Population Mean: t- Statistic ) X~n(M, σ) with M and σ known ▯ A (1- α)% confidence interval for M is (????, +/- t n-1, α/2 ) ▯ H 0 M= M 0 H 0 M= M 0 H0: M= M 0 H 1 M> M 0 H 1 M< M 0 H1: M ≠ M 0 tα> tn-1, α t< -t n-1, α |t|> n-1, α/2 4 Harry Case III Assumption: (Chapter 7.7- Test about a Population Variance) X~n(M, σ) with σ unknown (M is unknown as well) ▯▯▯ ▯ ▯ ▯▯▯ ▯ ▯ A (1- α)% confidence interval for σ is (2 ▯ , ▯ ) ▯▯▯▯,▯/▯ ▯▯▯▯,▯▯▯/▯ 2 2 2 2 2 2 H 0 σ = σ 0 H 0 σ = σ 0 H 0 σ = σ 0 H : σ > σ 2 H : σ < σ 2 H : σ 2≠ σ 2 1 2 0 1 2 0 2 0 2 x> X n-1, α x< X n-1, 1- α X n-1, 1- α/2< X n-1, α/2 Case IV Assumption: N~(0,1) The population X has a finite variance H 0 M= M 0 H 0 M= M 0 H 0 M= M 0 H : M> M H : M< M H : M ≠ M 1 0 1 0 1 0 ▯ ▯▯ ▯ ▯ ▯▯ ▯ ▯ ▯▯ ▯ ▯ > Z α ▯ < -Z α | ▯ | > Zα/2 ▯ ▯ ▯ • Observed Significance Levels: p- Values o P- Value- a value of how extreme the test statistic is under H 0 To solve: ????( Z> z) = α or 2P(Z< -|z|)= α H : M= M H : M= M H : M= M 0 0 0 0 0 0 H 1 M> M 0 H 1 M< M 0 H 1 M ≠ M 0 ????( Z> z) P(Z< z) 2P(Z< -|z|) At a significance level α, you reject H if0α > p o X~ B(p) (0< p< 1) X= {0} and n>1 If y= Σx “ihe amount of 1s” ▯ To estimate ????= ▯ ▯▯ ????~ ????(????, ) ▯ ▯▯ ???????? If n???? and n???? > 15 then ???? +/− Zα/2 (approx. (1- α)% C.I. for p) ▯ H 0 P= P 0 H 0 P= P 0 H 0 P= P 0 H : P> P H : P< P H : P ≠ P 1 0 1 0 1 0 Z > Z α Z < -Z α |Z|= > Zα/2 ????( Z> z) P(Z< z) 2P(Z< -|z|) Chapter 8: Inferences Based on Two Samples • Comparing Two Population Means: Independent Sampling o X~N(M , σ )1 1 Y~N(M , σ2) 2 X is independent of Y What is the estimate for M -M 1 2 =????- ???? E[????- ????] = E[????] – E[????]= M -M 1 2 var (???? + -???? bar) = var(????) + var(????) Note: var(-????)=(-1) var(????) ▯???? ▯???? = ▯+ ▯ ▯ ▯ ▯ ▯ Harry 5 o Confidence Interval for Normal Z- statistic ▯ ????▯ ▯???? ▯ Confidence Interval (????- ????) +/- Z α ▯ + ▯ ▯ ▯ ▯▯ ▯ ▯▯▯ Test Statistic: ???? = ▯????▯▯ ▯????▯ ▯ ▯ ▯ ▯ o Confidence Interval for T- statistic ▯^▯ ▯ ▯^▯ ▯ Confidence Interval: (????- ????) +/- t α/2 + ▯ ▯ ▯ ▯ ▯▯ ▯ ▯▯▯ Test Statistic: t= ▯^▯▯ ▯^▯▯ ▯ ▯ ▯ ▯ ▯ o X~N(M , σ )1 1 Y~N(M , σ2) 2 X is independent of Y If either 1 or n 2r both are small then, ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯▯ ▯ ▯(▯▯▯▯▯) ▯▯ ▯▯ ▯^▯ ▯^▯ ≈???????? Choose ???? = ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯▯ ▯▯ ▯ ▯▯ ▯ ▯▯▯▯ ▯^▯▯ ▯^▯ ▯ Ex: Confidence Interval: (????- ????) +/- t α/2 + ▯ ▯ ▯ ▯ H 0 M 1M =2d 0 H 1 M 1M ≠2d 0 ▯▯ ▯ ▯▯▯ Reject If: > t ▯^▯▯ ▯^▯▯ α/2 ▯▯ ▯ ▯ ▯

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