IntroEcono. Discrete RV. Continuous RV s

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1 ItroEcoo Aoc. Prof. Poga Porchaiwiekul, Ph.D... ก ก Homepage: (c) Poga Porchaiwiekul, Chulalogkor Uiverity Quatitative, e.g., icome, raifall Qualitative, e.g., color Semi-quatitative or emi-qualitative, e.g., coure letter grade, icome group expre order ot magitude (c) Poga Porchaiwiekul, Chulalogkor Uiverity 3 Type of Radom Variable Decriptio of RV uig Probability Ditributio uig Parameter Selected Probability Ditributio (c) Poga Porchaiwiekul, Chulalogkor Uiverity Dicrete RV aume value that do ot form a cotiuou rage, e.g., umber of attedat It poible value are ot limited to oly iteger. Cotiuou RV aume a cotiue rage of value, e.g., temperature (c) Poga Porchaiwiekul, Chulalogkor Uiverity 4

2 Tool to decribe the Ucertaity of a quatitative radom variable Popular tool Probability Ma Fuctio (PMF) Probability Deity Fuctio (pdf) Cumulative Ditributio Fuctio (CDF) Fuctio Algebraic fuctio Graphi fuctio (c) Poga Porchaiwiekul, Chulalogkor Uiverity 5 algebraic pmf of ~ Poio(λ) x,,, λ parameter p( x) e λ x λ x! (c) Poga Porchaiwiekul, Chulalogkor Uiverity 7 For Dicrete RV Biomial Poio For Cotiuou RV Normal ad Stadard Normal Chi-quare (χ ) Studet t F ditributio (c) Poga Porchaiwiekul, Chulalogkor Uiverity 6 Prob. ma λ 3 Sum of all mae x (c) Poga Porchaiwiekul, Chulalogkor Uiverity 8

3 CDF λ 3. ~ N(µ, ) Algebraic pdf of x (c) Poga Porchaiwiekul, Chulalogkor Uiverity 9 f ( x) exp π µ ad are real-valued parameter µ (c) Poga Porchaiwiekul, Chulalogkor Uiverity x Cotiuou Aume ay real value from egative ifiity to poitive ifiity Symmetric bell-haped ditributio Cetered at it populatio mea Diperio repreet degree of ucertaity or radome Area uder pdf Prob. deity µ µ µ µ µ+ (c) Poga Porchaiwiekul, Chulalogkor Uiverity (c) Poga Porchaiwiekul, Chulalogkor Uiverity

4 Populatio Mea, µ µ Populatio Media, Md µ Populatio Mode, Mo µ meaure of locatio Populatio variace, meaure of Populatio Stadard Deviatio, Populatio Skewede, S (meaure of aymmetry) Populatio Kurtoi, K 3 (meaure of peakede) diperio (c) Poga Porchaiwiekul, Chulalogkor Uiverity 3 High peak Thick tail µ Low peak Thi Tail (c) Poga Porchaiwiekul, Chulalogkor Uiverity 5 µ Skewed to the right Poitively kewed Skewed to the left Negatively kewed Z ~ N(,) Algebraic pdf of Z f ( z) exp z π Mo Md µ µ Md Mo No parameter (c) Poga Porchaiwiekul, Chulalogkor Uiverity 4 (c) Poga Porchaiwiekul, Chulalogkor Uiverity 6

5 Area uder pdf Prob. deity Z ~ N(,), Z ~ N(,),, Zm ~ N(,) Defie Z +Z + +Zm idepedet Z.3 ~ χ (m) + + m poitive iteger degree of freedom parameter (c) Poga Porchaiwiekul, Chulalogkor Uiverity 7 (c) Poga Porchaiwiekul, Chulalogkor Uiverity 9. Prob. deity Area uder pdf Populatio Variace m CDF m (c) Poga Porchaiwiekul, Chulalogkor Uiverity 8 (c) Poga Porchaiwiekul, Chulalogkor Uiverity

6 Give Z ~ N(,) ~ χ (m), Defie T Z m T ~ t(m) m poitive iteger degree of freedom parameter (c) Poga Porchaiwiekul, Chulalogkor Uiverity Give ~ χ (m), ~ χ (m), Defie F m m F ~ F(m,m) m umerator degree of freedom parameter m deomiator degree of freedom parameter (c) Poga Porchaiwiekul, Chulalogkor Uiverity 3 Area uder pdf pdf of tudet t m Populatio Variace, T m pdf of tadard ormal Prob. deity Area uder pdf + + m m- (c) Poga Porchaiwiekul, Chulalogkor Uiverity (c) Poga Porchaiwiekul, Chulalogkor Uiverity 4

7 Cae Ukow real probability ditributio Aume probability ditributio fuctio with ukow parameter Etimatio accuracy (c) Poga Porchaiwiekul, Chulalogkor Uiverity 5 to how the ditributio type of graph, e.g., hitogram lie graph bar graph pie chart (c) Poga Porchaiwiekul, Chulalogkor Uiverity 7 Samplig/obervig amplig cheme ample ize or umber of obervatio Preetatio of a data ample graph Calculated tatitic (c) Poga Porchaiwiekul, Chulalogkor Uiverity 6 to etimate the ukow parameter Popular tatitic ample mea ample variace ample kewede ample kurtoi(peakede or tailthicke) (c) Poga Porchaiwiekul, Chulalogkor Uiverity 8

8 Realizatio of the previou obervatio will ot ifluece the reult of the ext obervatio Sample of ize (c) Poga Porchaiwiekul, Chulalogkor Uiverity 9 Etimator of populatio mea (µ ) i i (c) Poga Porchaiwiekul, Chulalogkor Uiverity 3 i are radom variable havig ideti probability ditributio. i are idepedet from each other. Value of i doe ot ifluece Value of j for i j Etimator of populatio variace i Why - ot? ( i ) ( ) (c) Poga Porchaiwiekul, Chulalogkor Uiverity 3 (c) Poga Porchaiwiekul, Chulalogkor Uiverity 3

9 Etimator of populatio SD or Etimator of populatio peakede ( + ) K ( )( )( 3) i 3( ) ( )( 3) Why? i 4 (c) Poga Porchaiwiekul, Chulalogkor Uiverity 33 (c) Poga Porchaiwiekul, Chulalogkor Uiverity 35 Etimator of populatio kewede S k ( )( ) i Why? (c) Poga Porchaiwiekul, Chulalogkor Uiverity 34 i 3 Probability of Calculated Statitic Note that they are radom Aumptio ) ~ N(µ, ) E() µ V() ) Radom amplig (c) Poga Porchaiwiekul, Chulalogkor Uiverity 36

10 ~ N µ, E( ) µ V( ) / Defiitio: i a ubiaed etimator of θ if E Note that i a ubiaed etimator of µ. That i, θˆ ( θ ˆ ) E ( ) θ µ (c) Poga Porchaiwiekul, Chulalogkor Uiverity 37 (c) Poga Porchaiwiekul, Chulalogkor Uiverity 39 ~ o - tadard ( ) ~ χ ( ) E( ) 4 V ( ) (c) Poga Porchaiwiekul, Chulalogkor Uiverity 38 i a ubiaed etimator of Reao for dividig by (-) i to make the etimator ubiaed. Same reao for the formulae of ample kewede ad peakede (c) Poga Porchaiwiekul, Chulalogkor Uiverity 4

11 Deired Propertie of a Etimator Ubiaede Efficiecy or mallet poible variace If caot fid uch a etimator, the ext bet i a coitet etimator aymptotily ubiaed, i.e., bia -> a -> aymptotily dimiihig variace, i.e., variace-> a -> (c) Poga Porchaiwiekul, Chulalogkor Uiverity 4 Subtract with it mea ad divide by it tadard deviatio Z Cal µ ~ N (,) Note that if i kow, Z ca be culated give µ (c) Poga Porchaiwiekul, Chulalogkor Uiverity 43 Etimator i a formula to etimate a parameter Etimate i a culated or oberved value to be ued a a proxy for the parameter (c) Poga Porchaiwiekul, Chulalogkor Uiverity 4 Subtract with it mea ad divide by it ample tadard deviatio t Cal µ ~ t( ) Doe ot eed to kow to culate t. Why? (c) Poga Porchaiwiekul, Chulalogkor Uiverity 44

12 Cetral Limit Theorem for () t µ ~ t ( ( ) ) Note: ee the creatio of t-ditributed variable (c) Poga Porchaiwiekul, Chulalogkor Uiverity 45 (c) Poga Porchaiwiekul, Chulalogkor Uiverity 47 Cetral Limit Theorem for () Violatio of ormal ditributio of but till E() µ V() i approximately ormal whe -> or i aymptotily ormal ( µ ) ( ) A µ ~ N (, ) (c) Poga Porchaiwiekul, Chulalogkor Uiverity 46 µ Iterval etimator for µ (cf. Poit etimator) Cocept Cofidece Iterval for µ ± h where h half-width of the cofidece iterval (c) Poga Porchaiwiekul, Chulalogkor Uiverity 48

13 Sample ize (). h decreae a icreae. Cofidece Level. h icreae a cofidece level icreae Note that Cofidece Level (- )% where i the igificace level (choe by the oe who etimate) Stadard cofidece level are 9%, 95% ad 99% Stadard value of are.,.5 ad. (c) Poga Porchaiwiekul, Chulalogkor Uiverity 49 Z Z Z Cal (c) Poga Porchaiwiekul, Chulalogkor Uiverity 5 h µ h t t tcal (c) Poga Porchaiwiekul, Chulalogkor Uiverity 5 (c) Poga Porchaiwiekul, Chulalogkor Uiverity 5

14 µ CI doe ot cover the real µ. Chace i (c) Poga Porchaiwiekul, Chulalogkor Uiverity 53 χ χ ( ) (c) Poga Porchaiwiekul, Chulalogkor Uiverity 55 Cofidece Iterval for Iterval etimator for (cf. Poit etimator) Defie Cofidece Iterval for {L,U} wherel lower limit of the cofidece iterval U upper limit of the cofidece iterval (c) Poga Porchaiwiekul, Chulalogkor Uiverity 54 χ χ (c) Poga Porchaiwiekul, Chulalogkor Uiverity 56

15 (c) Poga Porchaiwiekul, Chulalogkor Uiverity 57 Note that Take ivere χ χ < < χ χ > > (c) Poga Porchaiwiekul, Chulalogkor Uiverity 58 Multiply with Flip over ) ( ) ( χ χ < < ) ( ) ( χ χ > > (c) Poga Porchaiwiekul, Chulalogkor Uiverity 59 (-)% Cofidece Iterval for ),( ) ( χ χ (c) Poga Porchaiwiekul, Chulalogkor Uiverity 6 Cocept Note that alway. I geeral, we are more cocered about the upper ide of the limit. Oe-ided Cofidece Iterval for {,U}

16 (-)% Oe-ided Cofidece Iterval for, ( ) χ χ ( ) (c) Poga Porchaiwiekul, Chulalogkor Uiverity 6 (c) Poga Porchaiwiekul, Chulalogkor Uiverity 63 For (-)% chace that χ < ( ) < < ( ) χ (c) Poga Porchaiwiekul, Chulalogkor Uiverity 6 The Other Oe-ided Cofidece Iterval for { L, } ( ) χ, If we are worried about the lower limit of the cofidece iterval. How to chooe L? (c) Poga Porchaiwiekul, Chulalogkor Uiverity 64

17 Two-ided Hypo. Tetig for µ () ( ) χ (c) Poga Porchaiwiekul, Chulalogkor Uiverity 65 i a etimator(rv) of µ It oberved or culated value (etimate) could be differet from the hypotheized value of µ eve though H i true. If it oberved value i ot too far from the hypotheized value, the, we ca accept that H i true. (c) Poga Porchaiwiekul, Chulalogkor Uiverity 67 Hypothei Tetig for µ Prior ifo. ugget the ull hypothei H or a hypotheized value of µ H H : µ : µ H i true util a trog eough evidece (oberved data) ugget that H i ulikely or, alteratively, H i true. H i alo referred to a the Alterative hypothei (c) Poga Porchaiwiekul, Chulalogkor Uiverity 66 Two-ided Hypo. Tetig for µ ( ) How far from the hypotheized value i before H will be rejected? Criterio Set the probability of Type I Error equal to the Sigificat Level () (c) Poga Porchaiwiekul, Chulalogkor Uiverity 68

18 Type I Error Type I Error i the evet that H ha bee rejected while it i, i fact, true. Low ugget that the evidece mut be very trog before we decide to reject H. Remember? igificat level i choe by the teter. tadard level of igificace (c) Poga Porchaiwiekul, Chulalogkor Uiverity 69 Two-ided Hypo. Tetig for µ (4) t Reject H Accept H Reject H t t (c) Poga Porchaiwiekul, Chulalogkor Uiverity 7 Two-ided Hypo. Tetig for µ (3) µ Reject H Accept H Reject H (c) Poga Porchaiwiekul, Chulalogkor Uiverity 7 Two-ided Hypo. Tetig for µ (5) Calculate t x Criterio compare t with t-value from t-table t t t > t > accept > reject H H (c) Poga Porchaiwiekul, Chulalogkor Uiverity 7

19 Oe-ided Hypo. Tetig for µ () Alo led Oe-tailed Prior ifo. ugget that o eed to be worried about the other tail(ay, µ < ). Example H : µ H : µ > Do a right-tailed tet (c) Poga Porchaiwiekul, Chulalogkor Uiverity 73 Oe-ided Hypo. Tetig for µ (3) µ Accept H (c) Poga Porchaiwiekul, Chulalogkor Uiverity 75 Reject H Oe-ided Hypo. Tetig for µ () Eve if the oberved/culated value i le tha or equal to the hypotheized value of µ, there i o good reao to chooe µ > over µ Oly if the oberved value i larger tha the hypotheized value by a large eough margi, the, we ca reject H. (c) Poga Porchaiwiekul, Chulalogkor Uiverity 74 Oe-ided Hypo. Tetig for µ (4) Accept H t Reject H (c) Poga Porchaiwiekul, Chulalogkor Uiverity 76 t

20 Oe-ided Hypo. Tetig for µ (5) Calculate x Criterio compare t with t-value from t-table t t t t > t > accept > reject (c) Poga Porchaiwiekul, Chulalogkor Uiverity 77 H H Left-ided Hypo. Tetig for µ () Calculate x Criterio compare t with t-value from t-table t t > accept H t <t > reject H t (c) Poga Porchaiwiekul, Chulalogkor Uiverity 79 Left-ided Hypo. Tetig for µ () Reject H t Accept H H : µ H : µ < t (c) Poga Porchaiwiekul, Chulalogkor Uiverity 78 Variace i alway o-egative. Rightided tet i mot iteretig ice we are more cocered about too high variace, i.e, H : 6 H : > 6 (c) Poga Porchaiwiekul, Chulalogkor Uiverity 8

21 χ (c) Poga Porchaiwiekul, Chulalogkor Uiverity 8 ( ) Accept H Reject H Calculate χcal ( ) Criterio compare χ Cal with from χ-table χ χ χ > accept > χ > reject 6 χ H H -value (c) Poga Porchaiwiekul, Chulalogkor Uiverity 8

x z Increasing the size of the sample increases the power (reduces the probability of a Type II error) when the significance level remains fixed.

x z Increasing the size of the sample increases the power (reduces the probability of a Type II error) when the significance level remains fixed. ] z-tet for the mea, μ If the P-value i a mall or maller tha a pecified value, the data are tatitically igificat at igificace level. Sigificace tet for the hypothei H 0: = 0 cocerig the ukow mea of a populatio