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1 Math. Statistics: Statistics? 1 What is Statistics? 1. (collection), (summarization), (analyzing), (presentation) (information) (statistics).., Survey, :, : : QC, 6-sigma, Data Mining(CRM) (Econometrics) :, 1.1 Describe the characteristics of interest in a population: (Descriptive statistics ) Make an inference(estimation and hypothesis testing) about a population based an information contained in a sample and to provide an associated measure of goodness for the inference: (Inferential statistics ) 1. Statistics is a theory of information, with inference making as its objective. A branch of mathematics dealing with the collection, analysis, interpretation, and the presentation of masses of numerical data, Webster Dictionary The branch of the scientific method which deals with data obtained by counting or measuring the properties, Kendall and Stuart The technology of scientific method, the design of experiment and investigation, statistical inference, Mood et al.

1 Describe the characteristics of interest in a population: (Descriptive statistics ) Make an inference(estimation and hypothesis testing) about a population based an information contained in a

2 Math. Statistics: Statistics? Phenomenon( ) Data Collecting( ) Data: numbers with context, numbers that have a story ( ) [ ] Numbers with context Summarizing( ) Analyzing( ), Modeling() Prediction / Forecasting Graphical and Numerical Presenting( ) Information( ): A numerical measure or verbal explanation of the uncertainty about the population of interest. When information is used for understanding or doing something, it is known as knowledge( ). (decision making)

Summarizing( ) Analyzing( ), Modeling() Prediction / Forecasting Graphical and Numerical Presenting( ) Information(

3 Math. Statistics: Statistics? : : 6 Tullius (BC ) 5 ( ) : Caesar (BC 5 ) :. : 17c /, F. Nightingale(1854, Polar Area Diagram, ) (Descriptive Statistics). (: 5) A (census) censura(= ). Statistics status(= )... Fermat & Pascal: Arithmetic Triangle Gauss: Gauss Galton: ( ) Gosset: t- Fisher: (population) (sample) ( ) (Inferential Statistics).?, 017? 100 1,000

(: 5) A 1960-000 (census) censura(= ). Statistics status(= ).

4 Math. Statistics: Statistics? William Thompson-. ( ). / 75%. There is 3 lies in the world: lie, damn lie, and Statistics Statistics can prove everything. Benjamin Disreal- A famous joke in Statistics (exit poll)? BUT,. ( 1)(4%), (33.8%), (16.3%) ( ) : ( ) ( 3) K, ( 4) 30 70%,.? 300%.? 1 (?)??

everything. Benjamin Disreal- A famous joke in Statistics (exit poll)? BUT,.

5 Math. Statistics: Statistics? Statistical Terms (population & sample) (population) (sample). Population(): (,, ),. Sample( ): miniature. (parameter). Sampling frame, census vs. sample survey (/). (measure, metric, quantitative) :,,,,, ( : ). (classified, categorical, non-metric, qualitative) :. (ordinal):, (,, ), (,, ), (A, B, C,..). (nominal):, (,, ). (sampling error) (sampling error)..? ( ).. (SRS: simple random sampling),. z 0.5 / n α / ( ) 95%, 1500 z α 0.5 / n = / % / =

(classified, categorical, non-metric, qualitative) :. (ordinal):, (,, ), (,, ), (A, B, C,..). (nominal):, (,, ).

6 Math. Statistics: Statistics? (,,,, )., (). (parameter) (model,, ). [ 1]. IQ. IQ 10. (IQ)? : IQ µ = 10. µ. t -. [ ]. (,,, ),,. = f ( 1, ).. ( ) 1) (statistical hypothesis), (model) ) 3) (significance). Confirmatory() Data Analysis. (deduction reasoning). Popper(11955) John W. Tukey (EDA: Exploratory Data Analysis). (1) (,, ) (stem-leaf plot, box plot, scatter plot) () ( ) (re-expression = data transformation : log ). (inductive). [ ] IQ IQ IQ

Confirmatory() Data Analysis. (deduction reasoning). Popper(11955). 1977 John W.

7 Math. Statistics: Statistics? 7 [ ] IQ IQ IQ ( ) (random experiment). ( :element) (object). w X ( w) = x (discrete), (continuous) (finite), (infinite).,,,,.,. : (metric, measurable, quantitative) ( non-metric, classified, categorical, qualitative). (ordinal) (nominal). IQ,, (A, B, C, D, F).,. : (casual relationship) ( ) (exploratory variable) (independent) (dependent) (response).

8 Math. Statistics: Statistics? 8 Y, X.,. (object) (date) (probability distribution function) (domain), (probability) (range) ( ) (, prob. density function) f (x) ( p(x) )..,,, (parameter) (statistics) f ( θ ). f. (parameter). x i ~ f ( x : θ ) iid ( x1, x,..., xn ) f ( x; θ ) : ~ (population) (sample) ( x 1, x,..., xn) θ ( ) µ, σ, p. (θˆ ) x, s, pˆ ( x1, x,..., xn). ( µ ), ( σ ) ( x x = i ) n ( x X ) s = i. (estimate) n 1

density function) f (x) ( p(x) )..,,,. 1.5.4 (parameter) (statistics) f ( θ ). f. (parameter). x i ~ f ( x : θ ) iid ( x1, x,.

9 Math. Statistics: Statistics? 9 (estimator).. (????).. (population). (, ).. (parameter) (model). p θ = p. ( ). ( Corr( X, Y ) = 0? ) (census)., (sample). (subset). miniature. ( ), ( )., p ˆ = 4 / 00 = 0.1. (statistic).,,. miniature ( ) (estimate). 1%.. ˆ # of " yes" θ = pˆ =. n (point estimate). (interval estimate) ( θ, θ ). (distribution) (sampling distribution). ˆp pq ˆˆ ˆˆ., pˆ ~ app N( p, ). 100(1 α)% ( ˆ pq pq p z1 α /, pˆ + z1 α / ). n n n L U

,,. miniature ( ) (estimate). 1%.. ˆ # of " yes" θ = pˆ =. n (point estimate). (interval estimate) ( θ, θ ).

10 Math. Statistics: Statistics? 10 95%? % 95%?.?. 95% ( 5%).. (.).. ( : H0 : p = p0 ) pˆ p0 ~ app N(0,1). pq 0 0 / n... 1 (type 1 error, α, ) 1 (significant level) 5%. pˆ p0 pq 0 0 / n : ( ) (p-value). ( ),. two-sided(, ~, ) 1/. t. F χ.

11 Math. Statistics: Statistics? ( ) (random variable) ( ). X, Y, Z.. EXAMPLE ( ) 4 (IQ, ). : X, IQ 130, 114, x i ( x, x,..., xn 1 ) :Y, Gender,, x i ( x 1, x,..., xn ) ( )Y=1,, Y=0,., IQ ,, (probability density function). (domain, x ), ( f( x )) (range, y ).? ).( ) (Central Limit Theorem), t-. f (x) F(x

.., xn 1 ) :Y, Gender,, x i ( x 1, x,..., xn ) ( )Y=1,, Y=0,., IQ. 1.5.

12 Math. Statistics: Statistics? (sample) (random) ( ) ( ). (random sample). iid f ( x). iid : independently and identically distributed x i ~ (independently):. (identically): ( x i : ) f (x). (sampling distribution). ( x 1, x,..., xn ) f( x; θ )( θ ) µ () x µ t -. s / n. t - (,, ), χ - (, ), F - (, ). ( x 1, x,..., xn ) (sample distribution). -, - () (outlier). CLT σ Normal( µ, ) n

(identically): ( x i : ) f (x). (sampling distribution). ( x 1, x,.

확률과통계 강의자료-1.hwp

확률과통계 강의자료-1.hwp 1. 통계학이란? 1.1 수학적 모형 실험 또는 증명을 통하여 자연현상을 분석하기 위한 수학적인 모형 1 결정모형 (deterministic model) - 뉴톤의 운동방정식 : - 보일-샤를의 법칙 : 일정량의 기체의 부피( )는 절대 온도()에 정비례하고, 압력( )에 반비례한다. 2 확률모형 (probabilistic model) - 주사위를 던질 때