STATISTICAL METHODS FOR BUSINESS

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1 STATISTICAL METHODS FOR BUSINESS UNIT 7: INFERENTIAL TOOLS. DISTRIBUTIONS ASSOCIATED WITH SAMPLING Distributios associated with the samplig process Iferetial processes ad relevat distributios.

2 UNIT 7. GOALS To describe the chi-square ad Studet s t distributios. To calculate probabilities ad percetiles (quatiles). To apply the mai pivotal statistics used i iferetial processes o the mea, the proportio ad the variace of a populatio.

3 STATISTICAL METHODS FOR BUSINESS UNIT 7: INFERENTIAL TOOLS. DISTRIBUTIONS ASSOCIATED WITH SAMPLING Distributios associated with the samplig process.

4 Probability models associated with samplig Normal distribució Chi-squared distributio Studet's t distributio N ( μ, σ ) χ 2 t

5 Chi-squared distributio X 1,..., X INDEPENDIENT RVs with distributio N(0,1): Z X 2 χ 2 i i 1 Z has a chi-squared distributio with degrees of freedom

6 Degrees of freedom Expressio RANDOM VARIABLES CONSTRAINTS Degrees of freedom i=1 X i 2 X 1, X 2,..., X Noe ( X i X ) 2 X 1, X 2,..., X i=1 or X 1 X, X 2 X,..., X X i= 1 X i oe = X ( X i X )=0 i=1-1 The umber of degrees of freedom i a expressio may be iterpreted as the umber of values that may be arbitrarily chose.

7 Chi-squared distributio Reproductivity Give two idepedet RVs X 2 Y 2 m It holds: 2 m X Y The chi-squared model is REPRODUCTIVE with respect to the umber of degrees of freedom.

8 Fisher s Theorem Give a simple radom sample (X 1,..., X ) from a N( ) populatio, it holds: The distributio of the radom variable is chi-squared with -1 degrees of freedom. X The sample mea ad the sample variace S 2 are idepedet radom variables. ( 1) S 2 σ 2 S 2 = ( X i X ) 2 i=1 1 SAMPLE VARIANCE

9 Studet's t distributio X ad Y INDEPENDIENT RVs The t = X Y X N(0,1) t 2 Y RV distributed as a Studet s t with freedom degrees

10 STATISTICAL METHODS FOR BUSINESS UNIT 7: INFERENTIAL TOOLS. DISTRIBUTIONS ASSOCIATED WITH SAMPLING Iferetial processes ad relevat distributios.

11 Pivotal statistics associated with iferetial processes POPULATION X Parameter Simple r.s. (X 1, X 2,...,X ) Estimator T(X 1, X 2,...,X ) STANDARDIZATION Pivot d T

12 Pivotal statistics The estimator T summarizes the iformatio that the sample cotais o the ukow parameter. The pivot is obtaied by stadardizig the estimator. A pivot should have a kow probability distributio, ot depedig o ukow parameters.

13 Estimator for the populatio mea i=1 X = X i E ( X )=μ SAMPLE MEAN Var ( X )= σ2 σ X = σ Ubiased estimator Stadard error If X N(, ) the X N( μ, σ ) If the distributio of X is ukow ad large X N( μ, σ )

14 Iferece o the populatio mea ESTIMATOR X E ( X )=μ Var ( X )= σ2 PIVOT d X = X E ( X ) Var ( X ) = X μ σ

15 Iferece o the populatio mea What is the probability distributio of the pivot? NORMAL POPULATION kow NON-NORMAL POPULATION kow ad large Reproductivity of ormal model N(0,1) Cetral Limit Theorem d X = X μ σ Ukow NORMAL POPULATION ukow t -1 d X = X μ S

16 Estimator for the populatio proportio (X 1,, X ) a s.r.s. with X i beig Beroulli distributed: X i =1 If the characteristic uder study is preset. P(X i =1)=p X i =0 Otherwise. P(X i =0)=1-p p= X SAMPLE PROPORTION X: Number of elemets i the sample that have the characteristic of iterest. X B(,p ) E ( p)= E ( X ) = p =p Var ( p )= Var ( X ) = p(1 p ) = p(1 p ) 2 2 Ubiased estimator

17 Iferece o the populatio proportio PIVOT d p = p p p(1 p) large N(0,1) Cetral Limit Theorem d p = p(1 p p p) 1 p ukow

18 Estimator for the populatio variace S 2 = ( X i X ) 2 i=1 Plug-i estimator E (S 2 )= 1 σ 2 S 2 = E( S ) ( X i X ) 2 i=1 1 SAMPLE VARIANCE E S E( S) ( 1) ( 1) Biased estimator Ubiased estimator

19 Iferece o the populatio variace 2 ( 1 )S PIVOT d 2= S σ 2 2 χ 1 NORMAL POPULATION Fisher Theorem

20 What parameters are we iterested i? Mea Proportio Variace p What estimators to use? Sample mea Sample proportio Sample variace X ˆp S Normal Studet's t Normal Chi-squared What probability distributios are relevat?

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