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by: Shlomo Oved

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# Notes for MA2224 MA-UY 2224

Shlomo Oved
NYU

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Covers essentially the whole course
COURSE
Data Analysis
PROF.
Dr. Qian
TYPE
Bundle
PAGES
16
WORDS
KARMA
75 ?

## Popular in Mathematics

This 16 page Bundle was uploaded by Shlomo Oved on Thursday September 8, 2016. The Bundle belongs to MA-UY 2224 at New York University taught by Dr. Qian in Fall 2015. Since its upload, it has received 11 views. For similar materials see Data Analysis in Mathematics at New York University.

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Date Created: 09/08/16
Data  Analysis  Formulas     Midterm 1 ▯ ▯ ▯ (▯▯▯▯) • Sample Variance- ???? = ▯▯▯ (▯▯▯) ▯ (▯▯▯▯)▯ • Sample Standard Deviation- ???? = ▯▯▯ (▯▯▯) c • P(E) (Independent event)= 1-P(E ) ▯(▯∩▯) • Conditional Probability= ????(????/????) = ▯(▯) • Probability= ???? ???? + ????(???? ) = 1 ▯ • Probability= ????(????/????)????(????) + ????(????/???? )????(???? ) ▯ ▯ • Independence= ????(???? ∩ ????)= P(A)*P(B) • Independence= ???? ???? ∪ ???? =P(A)+P(B)-  ????(???? ∩ ????) • Mutually Exclusive= ????(???? ∩ ????)= 0 • Mutually Exclusive= ???? ???? ∪ ???? =P(A)+P(B) • Chebyshev’s Inequality: Inner bound is greater than 1 − ▯ ▯ ▯ ▯ • One sided: outer bound is less than ▯ ▯ ▯▯ • Normal Data Set: • Empirical Rules: • 1. If data is normally distributed, then 68% of data points fall within 1 standard deviation • 2. If data is normally distributed, then 95% of data points fall within 2 standard deviation • 3. If data is normally distributed, then 99.7% of data points fall within 3 standard deviation • Correlation Coefficient ▯ (▯ ▯▯)(▯ ▯▯) • ???? = ▯▯▯ ▯ ▯ (▯▯▯)▯▯ ▯ ▯ ▯ • ???? = ▯▯▯(▯▯▯▯) ▯ (▯▯▯) ▯ ▯ • ???? = ▯▯▯(▯▯▯▯) ▯ (▯▯▯) • If r is positive, then positive correlation (the closer to 1, the stronger the positive correlation) Data  Analysis  Formulas     • If r is negative, then negative correlation (the closer to -1, the stronger the negative correlation) • If r is 0, then no correlation is present • r can only be between 1 and -1 • Combinations and Permutations ▯! • n r= ▯▯▯ ! • nCr= ▯! ▯▯▯ !▯! • Random Variables • P.m.f= Probability Mass Function (Discrete Random Variables) • ???? ???? =P(X=x) for each ???? ∈ ???? ▯ • ???? ???? > 0 •  ▯∈▯???? ???? = 1  (i.e, all the probabilities of the pmf together must add up to ▯ one) ▯ • ???? ???? =  ▯∈▯???? ???? • C.d.f= Cumulative density function (Continuous Random Variables) • ???? ???? = ???? ???? ≤ ???? = ???? ???? = ???? = ????(????) • Can only be less than. If question asks for greater than, than use the below equation: • ???? ???? > ???? = 1 − ???? ???? ≤ ???? • Ex: Derivative of cdf= pdf ???????? ???? ) = ???? ???? ) ▯???? ▯ ???? ???? ???????? = ???? ???? − ????(????) ▯ (F(x)=cdf, f(x)=pdf) • P.d.f= Probability density function (Continuous Random Variables) • ???? ???? ≥ 0,???? ∈ ???? ▯ • ???? ???? ???????? = 1 ▯ • ???? ???? ≤ ???? ≤ ???? = ???? ???? ???????? ▯ • Ex: Data  Analysis  Formulas     ???? + 1,                 − 1 ≤ ???? ≤ 0 • ????????????  ???? ???? = 1 − ????,                            0 ≤ ???? ≤ 1 Midterm 2 • For r.v. X: • ???? ???? = ????????????????????????????????  ????????????????????/???????????????????????????? • ???? ???? = ▯????????(????) for discrete case ▯ • ???? ???? = ▯▯ ???????? ???? ???????? for continuous case • ???? ???????? = ???? ∗ ???? ???? • ???? ???????? + ???? = ???? ∗ ???? ???? + ???? • ???? ???????? + ???? = ???? ∗ ???? ???? ▯ + ???? • ???? ???? ▯ = ▯???? ????(????) for discrete case ▯ ▯ ▯ • ???? ???? = ▯▯ ???? ???? ???? ???????? for continuous case • For r.v. X: • ???????????? ???? = ???? ???? ▯ − ???? ▯ • ???????????? ???? = ???? (???? − ????) ▯ ???? ???? • ???? ???? > ???? ???? ???? • ????(???? ) ≠ ???? • ???????????? ???????? + ???? = ???????????? ???????? = ???? ????????????(????) • ???????????? 2???? = 4???????????? ???? • ???????????? ???? + ???? ≠ ???????????? ???? + ???????????? ???? • ???????????? −???? = ???????????? ???? • Covariance and Variance of Sums of Random Variables • If x and y are independent then the joint pdf of x and y: ???? ????,???? = ▯ ???? ????▯???? • ???????? ????,???? = ???? ???????? − ???? ???? ▯ ▯ • ????  ????????????  ????  ????????????  ????????????????????????????????????????????  ????ℎ????????  ???????? ????,???? = 0 • ???? ????,???? = ???? ????▯ ▯ • ????????  ????????????????????????????: • ???????????? ???? + ???? = ???????????? ???? + ???????????? ???? + 2???????? ????,???? ▯ ▯ • ???????????? ???????? + ???????? = ???? ???????????? ???? + ???? ???????????? ???? + 2???????? ∗ ????????(????,????) • When independent: • ???????????? ???? + ???? = ???????????? ???? + ???????????? ???? Data  Analysis  Formulas     • ???????????? ???????? + ???????? = ???? ???????????? ???? + ???? ???????????? ????▯ • ???? ???????? = ???? ???? ???? ???? • ???? ???? ????▯ ▯ = ???? ???? ???? ???? ▯ • Chebyshev’s Inequality: • ????????????????????  ????????????????????: ▯▯ ▯ • ???? ???? − ???? ≥ ???? ≤ ▯= ???? ???? − ???? ≥ ???????? ≤ ▯ ▯ ▯ ▯ • ???? = ▯ ,???? = ????????, ▯ = ▯ ▯ (▯▯) ▯▯ • ????????????????????  ????????????????????: ▯▯ ▯ • ???? ???? − ???? ≤ ???? ≥ 1 − ▯▯=  ???? ???? − ???? ≤ ???????? ≥ 1 − ▯▯ Distributions • Bernoulli r.v.- Only two possible outcomes: Success or Failure ????,???? = 1  ???????????????????????????? • ???? ???? = ???? = 1 − ????,???? = 0  ???????????????????????????? ▯ ▯▯▯ • ???? ???? = ???? ???? • ???? ???? = ???? • ???????????? ???? = ???????? • Binomial r.v.-Repetition of a Bernoulli trial (with success probability “p”) n times. X= number of successes, the we say x has binomial distribution with parameters n and p. • ????~???????????? ????,????    ???? = 0,1,2,….???? • ???? ???? = ▯ ???? ????▯▯▯ ▯ • ???? ???? = ???????? • ???????????? ???? = ???????????? • When using Poisson to estimate Binomial (n large and p small), ???? = ???????? • Hyper-Geometric • N objects of 2 distinct types, n1 and n2 • n1+n2=N • Randomly select n from N (n≤N) • Let x= number of type 1 objects selected • ????~???????????????????????????? ????,????1,????2 : ▯▯ ▯▯ • ???? ???? = ▯ ▯▯▯ ▯ ▯ Data  Analysis  Formulas     ▯▯ • ???? ???? = ???? ▯ • Poisson Distribution • Can be used to estimate a binomial when n is large and p is small. ???? = ???????? ▯ • ???? ???? = ???? ▯▯▯ ,???? = 0,1,2…,???? ▯! • ???? ???? = ???? • ???????????? ???? = ???? • Uniform Distribution • ????~???? ????,???? • ???? ???? = ▯ ▯▯▯ • ???? ???? = ▯▯▯ ▯ (▯▯▯) • ???????????? ???? = ▯▯ • Normal Distribution ▯ • ????~???? ????,???? ▯▯▯ • ????????  ???????????????????????????????????????????? = ???? > ???????? < ▯ • ????ℎ????????  ????????????  ????????????????????????  ????????????????????????????????????????????????  ???????????????????? • Exponential Distribution • ???? ???? = ???????? ▯▯▯ ▯ • ???? ???? = ▯ • ???????????? ???? = ▯ ▯▯ • "Lack  of  memory" − ???? ???? > ???? = ????(???? > ???? + ????|???? > ????) • Distributions arising from Normal • Chi-squared distributions with r degrees of freedom • ???? ???? = ???? ???? ▯ • ???? ???? = ???? • ???????????? ???? = 2???? • See Chi-squared table • T-distributions with n degrees of freedom ▯ • ???? ▯ ▯ ▯ ▯ Data  Analysis  Formulas     • ????  ????????  ????????????????????????????????????  ????????????????????  ???? = 0 ▯ • See T-distribution table Midterm 3 Central Limit Theorem • Sample Mean Distribution ▯ ▯ • ????~????(????, ▯ ) ▯ • ????~????(????  ????,???????? ) Sampling Distribution from a Normal Population ▯▯ • ????~????(????, ▯ ) ▯ ▯ • ???? ???? = ???? ▯▯▯ ∗▯▯ ▯ • ▯ = ???? ▯▯▯ ▯▯ ▯ ▯ • ???? ???? = ???? + ???? ▯ • ????  ????????????  ????  ????????????  ????????????????????????????????????????????  ????.????. • ????ℎ????????????????????:???? ,???? ▯…???? ▯ ???????? ▯????  ????????????????????????  ????????????????????????  ????????????????  ???? ????,???? ▯ , • ????ℎ????????   ▯▯▯ • 1. ▯ ~Ζ ▯ ▯▯▯ • 2.    ▯ ~???? ▯▯▯ ▯ 2 2 • 3. ????2 ~ ????−1 ∗????  ????????  ????  ????) ????−1 ????2 ???? ????2 • Parameter Estimation • Interval Estimates Using T Table • ????    ???????????????????????????????????????? • Case 1: ???? = 1.645 ▯ .▯▯ • ???? ,▯ ,▯????  ????▯????  ????(????,???? ) ????.▯▯▯ = 1.96 • Goal: to estimate ???? • Point estimator: ???? ???? = 2.326 ▯ .▯▯ • Sampling distribution of the statistic: ????~????(????, ▯ ) ▯ ????.▯▯▯ = 2.576 ▯▯▯ • ▯ ~Ζ ▯ • 100 1 − ???? %  ????????????????????????????????????????  ????????????????????????????????  ????????????  ????: Data  Analysis  Formulas     • (???? ± ???? *???? ???? )= ???? − ???? ???? ???? • ????????????  ????????????????????  ???????? − ????????????????????  ????????????????????  ????????  ????????????  ????????????????????  ????????????????????  ????????????????????????????????????????  ???????????????????????????????? ???? • ???? < ???? + ???? * ???? ???? • ????????????????????  ????????????????????  ????????  ????????????  ????????????????????  ????????????????????  ????????????????????????????????????????  ???????????????????????????????? ???? • ???? > ???? − ???? * ???? ???? • Case 2 (Small Sample Problem- n<30) • ???? ,▯ ,▯????  ????▯????  ????(????,???? ), ????  ????????  ???????????????????????????? • Goal: to estimate ???? • Point estimator: ???? ▯ ▯ • Sampling distribution of the statistic: ????~????(????, ▯) ▯▯▯ • ▯ ~????▯▯▯ ▯ • 100 1 − ???? %  ????????????????????????????????????????  ????????????????????????????????  ????????????  ????: ???? ???? • (???? ± ???? ????▯????,????* ????) )= ???? − ???? • Case 3: (Large sample problem, unknown distribution) ▯ ▯ ▯ • ???? ,▯ ,▯????  ????▯????  ????(????,???? ), ????  ????????  ????????????????????????????,(  ???? < ∞) • Goal: to estimate ???? • Point estimator: ???? ▯ • Sampling distribution of the statistic: ????~????(????,▯ ) ▯ ▯▯▯ • ▯ ~???? ▯ ???? • (???? ± ???? *???? ) )= ???? − ???? ???? ???? • Finding a Number C: • Based on Cases 1-3, and either ???? > ????, ???? < ???? ▯ • Use ????  ????????????   ▯ • If ???? > ???? use – • If ???? < ???? use + Data  Analysis  Formulas     • ????  ????????????????????????????????????   ▯▯▯ ∗▯▯ ▯▯▯ ∗▯▯ • < ???? <   ▯ ▯ ▯ ▯ ▯ ▯ ▯▯▯▯ ▯▯▯,▯▯▯   ▯ ▯ • ▯▯▯ ∗▯ < ???? < ▯▯▯ ∗▯   ▯▯ ▯ ▯▯ ▯ ▯▯▯▯ ▯▯▯,▯▯▯   • Finding a Number V: • If V<???? ▯ ▯ ▯▯▯ ∗▯ • ▯ ▯   ▯▯▯,▯▯▯ • If V>????   ▯ ▯▯▯ ∗▯ • ▯ ▯   ▯▯▯,▯ Sampling from a finite population • Population: 1, 2,3……N, (ex:N=60) • Number of supporters= n (ex: n=20) ▯ • Randomly pick 1 voter, p(support)= ▯ • Estimating the difference in means of 2 Normal Populations: ▯ ▯ • Case 1 (???? an▯ ???? are ▯nown) ▯ • ???? ~▯ ???? ,???? ▯ ▯ • ???? ~▯(???? ,???? )▯ • Goal: to estimate ???? −▯???? ▯ ▯▯ ▯▯▯ • Sampling distribution of ???? − ????~???? ???? − ???? , ▯ ▯ + ▯ ▯ ▯▯▯ • ▯ ▯ ~???? ▯▯▯▯▯ ▯ ▯ • 100 1 − ???? %  ????????????????????????????????????????  ????????????????????????????????  ????????????  ???? − ???? :▯ ▯ ???? ???? ???? ???? ???????? • ???? − ???? ± ???? ???? ???? + ???? ???? Data  Analysis  Formulas     • Case 2 (Large Sample Case, ???? is unknown) • ???? ▯???? ???? ,????▯ ▯ ▯ • ???? ▯????(???? ,????▯) • Goal: to estimate ????▯− ???? ▯ ▯ ▯▯ • Sampling distribution of ???? − ????~???? ???? − ???? , ▯▯ + ▯ ▯ ▯ ▯ ▯ ▯▯▯ • ▯~???? ▯▯ ▯▯ ▯▯▯ • • 100 1 − ???? %  ????????????????????????????????????????  ????????????????????????????????  ????????????  ???? − ???? : ▯ ▯ ???????? ???????? • ???? − ???? ± ???? ???? ????+ ???? ???? ???? ???? • Case 3 (Small Sample Case, both underlying distributions Normal, ▯ ▯ variance unknown but equal, ???? =▯???? ) ▯ ▯ • ???? ▯???? ???? ,????▯ ▯ • ???? ▯????(???? ,????▯) • Goal: to estimate ????▯− ???? ▯ ▯ ▯ • Sampling distribution of ???? − ????~???? ???? −▯???? ,????▯ ▯ + ▯ ▯ • Pooled Sample Variance: ▯ ▯▯▯ ∗▯▯▯ ▯▯▯ ∗▯▯ • ???????? = ▯▯▯▯▯ ▯▯▯ • ▯ ▯~???? ▯▯▯▯▯ ▯▯ ▯▯▯ • 100 1 − ???? %  ????????????????????????????????????????  ????????????????????????????????  ????????????  ???? ▯ ???? :▯ ???? ???? • ???? − ???? ± ???? ????▯????▯????,???? ∗ ???????? + ???? ???? ???? • Approximate CI for mean of a Bernoulli r.v. ????,???? = 1 • ???? ???? = ???? = 1 − ????,???? = 0 • ???? = ???? • ???? = ???????? ▯ • Point Estimator: ???? = ▯ Data  Analysis  Formulas     ▯▯ • Sample distribution of ????~????(????, ▯) ▯▯▯ • ▯~???? ▯ • 100 1 − ???? %  ????????????????????????????????????????  ????????????????????????????????  ????????????????: ???????? • ???? ± ???? ???? ???? ???? • ∗ ????????????????????????????  ????????????????????  ????????????????  ???? =.????  ????????????  ???? =.???? • ????????????  ????????  ????????????????  ????????  ????????????????????????????????  ???? Hypothesis Testing Type 1 Error= ???? Type 2 Error =???? Find type 2 error by finding Critical Region, then calculating ????(????????????????????  ???????????????????? < ???? < ????????????????????  ????????????????????) by standardizing for Z. • 1. ???? ▯????????????????  ????????????????????ℎ????????????????,????????????????????????  ????????????, ????ℎ????  ????????????????????,????????????????????????  ????????????????????ℎ???????????????? • ???? :▯????????????????????????????????  ????????????????????ℎ????????????????,????ℎ????????????????????????????  ????????  ????????????????????????  ????????????,????????????????????????????????  ????????  ???? ▯ • 2.????????????????  ????????  ????????????????  ????????: • ????−value • ????????????????????????????  ???????????????????????? • “Confidence Interval” • (Use t, when ???? ≤ 30, and ????  ????????????????????????????) • (Use Z, when n>30, and ???? unknown/known) • P-Value: • The probability that a random sample is at least as extreme as the one observed, assuming the null hypothesis (???? ▯ is true. • Reject ???? i▯ p-value is less than ???? • Can’t reject ???? ▯ if p-value is greater than ???? • • 2 Sided: ▯▯▯▯ • 2 ∗ ???? ???? > ▯ ▯ • ???????? Data  Analysis  Formulas     ▯▯▯ ▯ • 2 ∗ ???? ???? < ▯ ▯ • 1 Sided: ▯▯▯ ▯ • ???? ???? > ▯ ▯ • ???????? • ???? ???? < ▯▯▯ ▯ ▯ ▯ • ???????? • ???? ???? ▯▯▯▯▯▯  ▯▯  ▯▯▯▯▯▯▯ ???????? >  ???? ▯▯▯ • ???? = ▯▯▯ ▯▯▯ ▯ ▯ • Critical Region • If the sample is in the critical region, reject null hypothesis▯(???? ), or if ???? is in CR, reject null hypothesis (????▯) • Two Sided: ▯  ▯▯  ▯ ▯  ▯▯  ▯ • ???? > ???? + ▯  ∗▯ or ???? < ???? −▯????  ∗ ▯ ▯ ▯ ▯ • ???? > ???? + ???? ▯ ∗ ▯  or      ???? < ???? − ???? ▯ ∗ ▯   ▯ ▯▯▯ ▯ ▯ ▯ ▯▯▯ ▯ ▯ • 1 Sided: • ???? > ???? + ???? ∗ ▯  ▯▯  ▯ ▯ ▯ ▯ ▯  ▯▯  ▯ • ???? <  ???? ▯ ???? ∗ ▯ ▯ • ???? >  ???? + ???? ∗ ▯   ▯ ▯▯▯ ,▯ ▯ ▯   • ???? <  ???? ▯  ???? ▯▯▯ ,▯∗ ▯ • “Confidence Interval”- • Create confidence interval depending on the case. • If  ????▯is an element of the interval, can’t reject▯???? • If  ???? is not an element of the interval, reject ???? ▯ ▯ Testing the equality of means of 2 Normal populations ▯ ▯ • Case 1 (???? a▯d ???? are▯known) • ???? ~???? ???? ,???? ▯ ▯ ▯ • ???? ~▯(???? ,???? ▯ ▯ Data  Analysis  Formulas     • ???? :▯ =▯???? ▯ • ???? ▯???? >▯???? ▯ • ???? ▯???? <▯???? ▯ • ???? ▯???? ≠▯???? ▯ ▯▯▯ • ~???? ▯▯ ▯▯ ▯▯ ▯ ▯ ▯ ▯▯ ▯▯ • Sampling distribution of ???? − ????~???? 0, ▯ + ▯ • Case 2 (???? and ???? are unknown, large sample) ▯ ▯ • ???? ~???? ???? ,???? ▯ ▯ ▯ • ???? ~????(???? ,???? ) ▯ ▯ ▯ • ???? :???? = ???? ▯ ▯ ▯ • ???? ▯???? >▯???? ▯ • ???? ▯???? <▯???? ▯ • ???? ▯???? ≠▯???? ▯ ▯▯▯ • ▯▯ ▯▯~???? ▯▯ ▯ ▯ ▯ ▯ ▯▯ ▯ • Sampling distribution of ???? − ????~???? 0, + ▯ ▯ ▯ ▯ • Case 3 (???? ▯nd ???? a▯e unknown, small sample) ▯ • ???? ▯???? ???? ,????▯ • ???? ~????(???? ,???? ) ▯ ▯ ▯ • ????????????????????????  ???? = ???? ▯ ▯ ▯ • ???? :???? = ???? ▯ ▯ ▯ • ???? :???? > ???? ▯ ▯ ▯ • ???? :???? < ???? ▯ ▯ ▯ • ???? :???? ≠ ???? ▯ ▯ ▯ ▯▯▯ • Sampling distribution of ▯ ▯~???? ▯▯▯▯▯ ▯▯ ▯ ▯ ▯ ▯▯▯ • ???? ▯▯▯= ▯▯ ▯▯▯ ▯ ▯ ▯ ▯ ▯▯▯ ▯▯ ▯(▯▯▯)(▯▯) • ???????? = ▯▯▯▯▯ • ???? − ???????????????????? = ????(???? ▯▯▯▯▯ < ???????? > ????▯▯▯) Data  Analysis  Formulas     • Dependent data/ paired study Paired t-Test: • Find difference of both groups of data ???? = ???? − ????  ????????  ???? − ???? • Use the differences as your new data ▯ ▯ ▯ ▯ ▯ ???? ▯???? ▯ 0 • ???? :▯ ????▯= 0 ???? ▯???? ▯ ≠ ???????? < ???????? > 0   • ???? :▯ ????▯< ???????? > ????????   ≠ 0 ▯ − ???? ▯ ▯▯▯ ????????????????????  ???? ▯   ~???? ▯▯▯   • ????????????????????  ???? :▯  ▯ ~???? ▯▯▯   ????▯ ▯ √ ???? Hypothesis tests concerning variance of a normal population • ???? ,???? ,…???? ~????(????,???? ) ▯ ▯ ▯ ▯ ▯ ▯ • 1.???? :▯ =▯???? ▯ ▯ ▯ • ???? :▯ ????▯< ???? ▯ • ???? ▯ ???? ▯ ▯ ▯ • ???? ▯ ???? ▯ (▯▯▯)▯▯ ▯ • 2.????????????????????  ???? :▯ ▯▯ ~???? ▯▯▯ ▯ ▯ • ????????????????????????????  ???? ▯▯▯  ????????.▯▯▯   Hypothesis testing for Proportions • Y and N are given • ???? :▯ = ???? • ???? :▯ < ???????? > ???????? ≠ ???? ▯ • ???? = ▯ ▯▯ • ????????????????????  ???? :▯~????????????(????,????) → ????~????(????????,????????????) → ????~????(????, ▯ ) • ????????  ????????????????  ???? − ????????????????????: • ????~????(????, ????????) ???? • 1. ???? ???? < ???????? > ???? • ???? ???? < ???????? > ???? ± .5 ▯ • ???? ???? < ???????? > ▯ Data  Analysis  Formulas     ▯▯▯ • ????(???? < ???????? > ) ▯▯ ▯ • ????~???? ????????,???????????? • 2. ???? ???? < ???????? > ???? • ???? ???? < ???????? > ???? ± .5 ▯▯▯▯ • ????(???? < ???????? > ) ▯▯▯ • Compare 2 proportions • ???? :▯ ???? ▯ ???? ▯ • ???? :▯ < ▯ ▯ • ???? > ???? ▯ ▯ • ???? ≠▯???? ▯ • ????????????????????  ???? :???? − ???? ~????(0, ▯▯▯▯▯ ∗ 1 − ▯▯▯▯ ▯ ∗ ▯ + ▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯▯▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯ • ???? − ???????????????????? = ???? < ???????? > ▯ ▯ ▯▯▯▯▯ ∗ ▯▯▯▯▯▯▯ ∗ ▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯ Chapter 9- Regression • ???? ,???? ,…???? ????????????????????????????????????????????  ????????????????????????????????,????????????????????????  ???????????????????????????????? → ▯ ▯ ▯ ????(????????????????????????????????????  ????????????????????????????????,????????????????????????  ????????????????????????????????) • ???? = ???? + ▯ ???? +▯ ▯???? + ⋯▯ ▯ ???? + ???? ▯ ▯ • ???? = ????????????????????????  ????????????????????,???? ???? = 0 • ????????  ???? > 1,????????????????????????????????????????????????  ???????????????????????????????????????? • ???? = 1,????????????????????????  ???????????????????????????????????????? • ???? :▯ ???? = 0 (No regression) • ???? :???? ≠ 0 ▯ • Least square estimators of regression parameters • r=1, ???? = ???? + ???????? + ???? • Estimator for ????: A • Estimator for ????: B • ???? = ???? + ???????? ▯ ▯ ▯ ▯ • Sum of Squares (S.S.)= ▯▯▯ ????▯− ???? ▯ = ▯▯▯ ???? ▯ (???? + ???????? ) ▯ Data  Analysis  Formulas     • ???? = ???? − ???????? ▯ • ???? = ▯▯▯▯▯ ▯▯▯▯▯ ▯ ▯▯▯▯▯ ▯ ▯▯▯ • Distribution of the estimators (B- sampling distribution of β) • ???? = ???? + ???????? + ???? • Assume: ????~???? 0,???? ▯ • Distribution for ????~???? ???? + ????????,???? ▯ = ???? ▯???? ???? + ???????? ,???? ▯ ▯ ▯ ▯▯▯▯▯▯▯)∗▯▯ • ???? = ▯ ▯▯▯▯▯ ▯ ▯▯▯ • ???? ???? = ???? ▯ • ???????????? ???? = ▯ ▯▯▯▯▯ ▯ ▯ • ????~????(0,???? ) ▯ ▯ • ????.???? =▯ ▯▯▯ ????▯− ???? + ???????? ▯ • Residuals: ???? =▯???? − ▯???? + ???????? ) ▯ ▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯ ▯ • ▯▯ ~???? (▯▯▯) ▯.▯▯ ▯ • ~???? (▯▯▯) ▯▯ ▯.▯ • ???? ▯ ▯ = ???? − 2 ▯.▯ • ???? =▯ ▯ ▯▯▯ ▯▯▯ • ▯   ????????????????????????????????  ????????????????????????????????????  ????????  ???? ~???? ▯▯▯ ▯ ▯▯▯ ▯▯▯ • ▯.▯ (????????????????????????????????  ????????????????????  ????????  ????)~???? ▯▯▯ ▯ ▯▯▯ ∗▯▯▯ • CI for ????: ▯▯ • ???? ± ???? ▯ ▯ ▯▯▯,▯ (▯▯▯)▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ • ???? − ???????????????? = ▯▯▯▯▯▯▯▯  ▯▯▯▯▯ • ????????????????????  ????????????????????, • ???????????????????????????????????????? = ???????? − ???????? ???? ???? • ???????????????????????????????? = ???????? ???? • ???????????????????? = ???????? ???? Data  Analysis  Formulas     The coefficient of determination and the correlation coefficient • ???? ▯▯ = ▯ (????▯− ????)(???? −▯????) = ▯ ????▯ ▯− ???????????? ▯▯▯ ▯ ▯▯▯ • ???? ▯▯ = ▯▯▯(????▯− ????) = ▯▯▯ ????▯▯− ???? ???? ▯ ▯ ▯ ▯ ▯ ▯ • ???? ▯▯ = ▯▯▯(????▯− ????) = ▯▯▯ ????▯− ???????? ▯▯▯ • ???? = ▯▯▯ • ???? = ???? − ???????? ▯ ▯▯▯∗▯▯▯▯▯▯▯ • ???????? =▯ ▯ ▯▯ ▯ ▯▯▯ • ????????????????????????????????????????????  ????????  ???????????????????????????????????????????????????? = ???? = ▯ ▯▯ ▯  ????????  1 −▯▯ ▯▯▯ ▯▯▯ • ???? ▯▯????????  ????ℎ????  ????????????????????  ???????????????????? • ???????? ▯????????  ????ℎ????  ????????????????????????  ???????????????????? • ???? − ????????  ????????  ????ℎ????  ????????????????????  ????ℎ????????  ????????????  ????????  ????????????????????????????????????  ????????  ????ℎ????  ????????????????????  ???????????????????????????????? ▯▯ ▯ Analysis of Residual • If residuals aren’t randomly distributed (positive and negative) data are not fitted with regression line.

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