×

Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

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

by: Angie Notetaker

46

0

5

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

Marketplace > Rutgers University > Statistics > 285 > Statistics for Business Chapters 6 7 and 8 Cheat Sheet Final Third of the Class
Angie Notetaker
Rutgers

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

This document covers the final third of material in the course (chapters 6, 7 and 8) and all concepts taught after exam 2. This material makes up 60% of the final exam while exams 1 and 2 make up 4...
COURSE
PROF.
Zinonos
TYPE
Study Guide
PAGES
5
WORDS
CONCEPTS
KARMA
50 ?

Popular in Statistics

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.

×

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

×

×

What is Karma?

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

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 ▯▯ ▯ ▯ ▯

×

50 Karma

×

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Amaris Trozzo George Washington University

"I made \$350 in just two days after posting my first study guide."

Bentley McCaw University of Florida

Forbes

"Their 'Elite Notetakers' are making over \$1,200/month in sales by creating high quality content that helps their classmates in a time of need."

Become an Elite Notetaker and start selling your notes online!
×

Refund Policy

STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com