STATS 200: Introduction to Statistical Inference. Lecture 1: Course introduction and polling

Transcription

1 STATS 200: Itroductio to Statistical Iferece Lecture 1: Course itroductio ad pollig

2 U.S. presidetial electio projectios by state (Source: fivethirtyeight.com, 25 September 2016)

3 Pollig Let s try to uderstad how pollig ca be used to determie the popular support of a cadidate i some state (say, Iowa). Key quatities: N = 3,046,355 populatio of Iowa p = 1 p = # people who support Hillary Clito N # people who support Doald Trump N We kow N but we do t kow p.

4 Pollig Let s try to uderstad how pollig ca be used to determie the popular support of a cadidate i some state (say, Iowa). Key quatities: N = 3,046,355 populatio of Iowa p = 1 p = # people who support Hillary Clito N # people who support Doald Trump N We kow N but we do t kow p. Questio #1: What is p?

5 Pollig Let s try to uderstad how pollig ca be used to determie the popular support of a cadidate i some state (say, Iowa). Key quatities: N = 3,046,355 populatio of Iowa p = 1 p = # people who support Hillary Clito N # people who support Doald Trump N We kow N but we do t kow p. Questio #1: What is p? Questio #2: Is p > 0.5?

6 Pollig Let s try to uderstad how pollig ca be used to determie the popular support of a cadidate i some state (say, Iowa). Key quatities: N = 3,046,355 populatio of Iowa p = 1 p = # people who support Hillary Clito N # people who support Doald Trump N We kow N but we do t kow p. Questio #1: What is p? Questio #2: Is p > 0.5? Questio #3: Are you sure?

7 Simple radom sample Suppose we poll a simple radom sample of = 1000 people from the populatio of Iowa. This meas: Perso 1 is chose at radom (equally likely) from all N people i Iowa. The perso 2 is chose at radom from the remaiig N 1 people. The perso 3 is chose at radom from the remaiig N 2 people, etc. Or equivaletly, all ( ) N = N!!(N )! possible sets of people are equally likely to be chose. The we ca estimate p by ˆp = # sampled people who support Hillary Clito

8 Simple radom sample Say 540 out of the 1000 people we surveyed support Hillary, so ˆp = Does this mea p = 0.54? Does this mea p > 0.5?

9 Simple radom sample Say 540 out of the 1000 people we surveyed support Hillary, so ˆp = Does this mea p = 0.54? Does this mea p > 0.5? No! Let s call our data X 1,..., X : { 1 if perso i supports Hillary X i = 0 if perso i supports Doald The ˆp = X 1 + X X. The data X 1,..., X are radom, because we took a radom sample. Therefore ˆp is also radom.

10 Uderstadig the bias ˆp is a radom variable it has a probability distributio. We ca ask: What is E[ˆp]? What is Var[ˆp]? What is the distributio of ˆp?

11 Uderstadig the bias ˆp is a radom variable it has a probability distributio. We ca ask: What is E[ˆp]? What is Var[ˆp]? What is the distributio of ˆp? Each of the N people of Iowa is equally likely to be the i th perso that we sampled. So each X i Beroulli(p), ad E[X i ] = p. [ ] X X E[ˆp] = E = 1 (E[X 1] E[X ]) = p Iterpretatio: The average value of ˆp is p. We say that ˆp is ubiased.

12 Uderstadig the variace For the variace, recall that for ay radom variable X, Var[X ] = E[X 2 ] (E[X ]) 2 Let s compute E[ˆp 2 ]: [ (X1 ) ] E[ˆp X 2 ] = E = 1 2 E [ X X 2 + 2(X 1 X 2 + X 1 X X 1 X ) ]

13 Uderstadig the variace For the variace, recall that for ay radom variable X, Let s compute E[ˆp 2 ]: [ (X1 ) ] E[ˆp X 2 ] = E Var[X ] = E[X 2 ] (E[X ]) 2 = 1 2 E [ X X 2 + 2(X 1 X 2 + X 1 X X 1 X ) ] = 1 ( ) ) ( 2 E[X1 2 ] + 2 E[X 1 X 2 ] 2 = 1 E[X 1 2 ] + 1 E[X 1X 2 ]

14 From the previous slide: Uderstadig the variace E[ˆp 2 ] = 1 E[X 2 1 ] + 1 E[X 1X 2 ] Sice X 1 is 0 or 1, X 1 = X 2 1. The E[X 2 1 ] = E[X 1] = p. Q: Are X 1 ad X 2 idepedet?

15 From the previous slide: Uderstadig the variace E[ˆp 2 ] = 1 E[X 2 1 ] + 1 E[X 1X 2 ] Sice X 1 is 0 or 1, X 1 = X 2 1. The E[X 2 1 ] = E[X 1] = p. Q: Are X 1 ad X 2 idepedet? A: No. E[X 1 X 2 ] = P[X 1 = 1, X 2 = 1] = P[X 1 = 1] P[X 2 = 1 X 1 = 1] We have: P[X 1 = 1] = p, P[X 2 = 1 X 1 = 1] = Np 1 N 1

16 Uderstadig the variace Var[ˆp] = E[ˆp 2 ] (E[ˆp]) 2 = 1 p + 1 p ( Np 1 N 1 ) p 2

17 Uderstadig the variace Var[ˆp] = E[ˆp 2 ] (E[ˆp]) 2 = 1 p + 1 ( Np 1 p N 1 ( 1 = 1 1 N 1 ) p + ) p 2 ( 1 ) N N 1 1 p 2

18 Uderstadig the variace Var[ˆp] = E[ˆp 2 ] (E[ˆp]) 2 = 1 p + 1 ( Np 1 p N 1 ( 1 = 1 1 N 1 = N (N 1) p + ) p + N (N 1) p2 ) p 2 ( 1 ) N N 1 1 p 2

19 Uderstadig the variace Var[ˆp] = E[ˆp 2 ] (E[ˆp]) 2 = 1 p + 1 ( ) Np 1 p p 2 N 1 ( 1 = 1 ) ( 1 1 p + N 1 = N (N 1) p + N (N 1) p2 = p(1 p) N p(1 p) = N 1 N N 1 1 ( 1 1 ) N 1 ) p 2

20 Uderstadig the variace Var[ˆp] = p(1 p) ( 1 1 ) N 1 Whe N is much bigger tha, this is approximately p(1 p), which would be the variace if we sampled people i Iowa with replacemet. (I that case ˆp would be a Biomial(, p) radom variable divided by.) The factor 1 1 N 1 is the correctio for samplig without replacemet. For N = 3,046,355, = 1000, ad p 0.54, the stadard deviatio of ˆp is Var[ˆp]

21 Uderstadig the samplig distributio Fially, let s look at the distributio of ˆp. Suppose p = We ca use simulatio to radomly sample X 1,..., X from Np people who support Hillary ad N(1 p) people who support Doald, ad the compute ˆp. Doig this 500 times, here s a histogram of the 500 (radom) values of ˆp that we obtai: Histogram of p_hat Frequecy p_hat

22 Uderstadig the samplig distributio ˆp looks like it has a ormal distributio, with mea 0.54 ad stadard deviatio Why? Heuristically, if N is much larger tha, the X 1,..., X are almost idepedet. If is also reasoably large, the the distributio of (ˆp p) = (X 1 p) (X p) is approximately N (0, p(1 p)) by the Cetral Limit Theorem. So ˆp is approximately N (p, p(1 p) ).

23 A cofidece statemet Recall that 95% of the probability desity of a ormal distributio is withi 2 stadard deviatios of its mea. ( , ) = (0.508, 0.572) is a 95% cofidece iterval for p. I particular, we are more tha 95% cofidet that p > 0.5.

24 Fudametal priciple We will assume throughout this course: Data is a realizatio of a radom process.

25 Fudametal priciple We will assume throughout this course: Data is a realizatio of a radom process. Why? Possible reasos: 1. We itroduced radomess i our experimetal desig (for example, pollig or cliical trials) 2. We are actually studyig a radom pheomeo (for example, coi tosses or dice rolls) 3. Radomess is a modelig assumptio for somethig we do t uderstad (for example, errors i measuremets)

26 Statistical iferece Statistical iferece = Probability 1 Probability: For a specified probability distributio, what are the properties of data from this distributio? Example: X 1,..., X 10 iid N (2.3, 1). What is P[X1 > 5]? What is the distributio of 1 10 (X X 10 )?

27 Statistical iferece Statistical iferece = Probability 1 Probability: For a specified probability distributio, what are the properties of data from this distributio? Example: X 1,..., X 10 iid N (2.3, 1). What is P[X1 > 5]? What is the distributio of 1 10 (X X 10 )? Statistical iferece: For a specified set of data, what are properties of the distributio(s)? Example: X 1,..., X 10 iid N (θ, 1) for some θ. We observe X 1 = 3.67, X 2 = 2.24, etc. What is θ?

28 Goals I statistical iferece, there is usually ot a sigle right aswer. For a give iferetial questio, what is a good (best?) method of aswerig that questio usig data? How do we compare differet methods for aswerig the same questio? How do we uderstad the error/ucertaity i our aswer? How do we uderstad the depedece of our aswer o our modelig assumptios?

29 Iferece tasks Hypothesis testig: Askig a biary questio about the distributio. (Is p > 0.5?) Estimatio: Determiig the distributio, or some characteristic of it. (What is our best guess for p?) Cofidece itervals: Quatifyig the ucertaity of our estimate. (What is a rage of values to which we re reasoably sure p belogs?)

30 Course logistics Webpage: stats200.staford.edu All course iformatio (syllabus, office hours), lecture otes/slides, ad homeworks will be posted here. Grades ad other restricted cotet will be posted o Staford Cavas. (There s a lik i the above page.)

31 Prerequisites Probability theory (STATS 116 or equivalet) Multivariable calculus (MATH 52 or equivalet) Homework assigmets will iclude simple computig exercises askig you to perform small simulatios, create histograms ad plots, ad aalyze data. You may use ay laguage (e.g. R, Pytho, Matlab) ad will be graded oly o your results, ot o the quality of your code. Your TA Alex Chi will teach a Itroductio to R sectio, time ad place TBD. The first couple homework assigmets will also walk you through how to do these thigs i R.

32 Requiremets Homework: Due Wedesdays at the start of class. First homework due ext Wedesday, October 5. Collaboratio: You ca work together o homework, but you must submit your ow write-up, i your ow words ad usig your ow code for the programmig exercises. Please idicate at the top of your write-up the ames of the studets with whom you worked. Exams: Oe midterm, oe fial (both closed-book).

33 Notes ad textbook Lectures will switch betwee slides ad blackboard; I ll post slides/otes olie after class. Readigs are assiged from Joh A. Rice, Mathematical Statistics ad Data Aalysis: Teachig two separate courses, oe o theory ad oe o data aalysis, seems to me artificial. Rice

34 For referece: Morris H. DeGroot ad Mark J. Schervish, Probability ad Statistics Larry Wasserma, All of Statistics: A cocise course i statistical iferece

35 Studets who aalyze data, or who aspire to develop ew methods for aalyzig data, should be well grouded i basic probability ad mathematical statistics. Usig facy tools like eural ets, boostig, ad support vector machies without uderstadig basic statistics is like doig brai surgery before kowig how to use a bad-aid. Wasserma