UCLA STAT 110B Applied Statistics for Engineering and the Sciences

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1 UCLA STAT 110B Applied Statistics for Egieerig ad the Scieces Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistats: Bria Ng, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig Categorical Data Categorical Data is that which couts the umber of outcomes fallig ito various categories. Biomial Experimet cosists of two categories Multiomial Experimet cosist of more tha two categories Slide 1 Slide Biomial Experimet Biomial Distributio Pdf, E[X], Var[X] idepedet trials Two possible outcomes (S) success ad (F) failure p Probability of success o each trial X Number of successes i trials Slide 3 Slide 4 Multiomial Experimet Multiomial Cot d idepedet trials results i oe of k possible categories labeled 1,, k p i the probability of a trial resultig i the ith category, where p 1 + +p k 1 N i umber of trials resultig i the ith category, where N 1 + +N k The radom variables N 1,,N k have a multiomial distributio p(,..., ) 1 k! 1 p1 1! k! p k k Slide 5 Slide 6

2 Multiomial Cot d Testig Goodess of Fit with Specified Cell Probabilities Expected Value: E[N i ] p i E i Variace: Var [N i ] p i q i Covariace: Cov [N i, N ] -p i p Slide 7 We wish to test whether the cell probabilities are specified by p 1o,, p ko where p 1o + +p ko 1. We will use a test statistic to compare the observed cell cout N i to the expected cell cout uder H o, E i p o i H o :p 1 p 1o, (ad)., (ad) H a : Some p i p i o Slide 8 p k p k o X Test Statistic N Slide 9 E E k ( i i ) i 1 i This is a Pearso s goodess-of-fit statistic Reectio Regio: X > χ α where χ is the chi-squared distributio with k-1 degrees of freedom. Geeral Rule: We wat p io 5 for all cells Example A study is ru to see whether the public favors the costructio of a ew dam. It is thought that 40% favor dam costructio, 30% are eutral, 0% oppose the dam, ad the rest have ot thought about it. A radom sample of 150 idividuals are iterviewed resultig i 4 i favor, 61 eutral, 33 opposed, ad the rest have ot though about it. Does the data idicate that the stated proportios are icorrect? Use α0.01. Slide 10 Example Cot d H o : p 1 0.4, p 0.3, p 3 0., p H a : At least oe probability is ot as specified Test Statistic: X Reectio Regio: X > χ 0.01, Slide 11 Favor Neutral Oppose Uaware Total i pio Ei (4 60) (61 45) (33 30) X (14 15) Sice X > χ 0.01,3, 11.34, we reect H o. Coclude that at least oe of the true proportios differs from that hypothesized Slide 1

3 Goodess of Fit for Distributios (Cotiuous ad Discrete) Uses the cocept of Maximum Likelihood Estimatios (MLE) The rage of a hypothesized distributio is divided ito a set of k itervals (cells). After fidig the MLE of ukow parameters, the cell probabilities are calculated ad the χ test performed Foud i may computer packages - SOCR Testig Normality May test procedures that we have developed rely o the assumptio of Normality. There are may test for Normality of data. Oe uses the ormal to provide cell probabilities for the chi-square goodess-of-fit test. A better test is based o the Normal Probability Plot Slide 13 Slide 14 Testig Normality Cot d Rya-Joier Test Recall: The NPP should be approx liear for ormal data, ad the correlatio coefficiet is a measure of liearity. If r is much less tha oe, we would coclude that the data does t come from a Normal distributio. Slide Order the data x (1),,x (). Compute the ormal percetiles y i Φ 1 i Compute the correlatio coefficiet, R, for the (y i,x (i) ) pairs ad look up the distributio table for the Rya-Joier Statistics, A.1. Slide 16 Rya-Joier Test Example 4. State the Null ad Alterative Hypotheses H o : The populatio is ormal H a : The populatio is ot ormal 5. Specify alpha ad obtai critical values from Table A.1. Compare R to this value Slide 17 Cosider the followig data. Use the Rya- Joier test to test the assumptio of ormality at α ; Raw Data Normal(0,1) radom sample: Slide

4 Example Corr(N(0,1), Data) H o : Data is Normal For higher cofidece, smaller Type I error α, we eed smaller Correlatios, R(N,D) R ~ Rya-Joier (α,) RJ(0.01,4) RJ(0.10,4) R 1 Sice R o R o > Critical Value Strog Correlatio Ca t Reect H o Slide 19 α Ascedig Order Stats: N(0,1) Data Testig Homogeeity of Populatios *We wish to compare I multiomial populatios, each with J categories. * Take i samples from the ith populatio Let N be the umber of observatios from the i th populatio i the th category. Hece, Σ N i Place the data i a I x J table Slide 0 Table Category 1. J Total J 1. 1 J. Pop I I1 I. IJ I. Total.1...J Correspodig to each cell, there is a cell probability p probability ad outcome for the i th populatio falls ito the th category, where Σ p 1 Category 1. J 1 p11 p1. p1j p1 p pj Pop I pi1 pi. pij Slide 1 Slide Test H o : p 1 p p I, 1,,J H a : Some p p i Uder H o, the commo cell probability p is estimated by pˆ Test Cot d The estimated expected cell frequecy is Eˆ pˆ The test statistic is X i rows colums i ( Eˆ ) Eˆ Reectio Regio: X > χ α with d.f. (I-1)(J-1) Slide 3 Slide 4

5 Testig for Associatio * Idividuals are categorized by two categorical variables. We wish to determie whether these variables are associated. * Row Categories A 1,,A I Colum Categories B 1,,B J Total umber of observatios the umber of idividuals classified as A i ad B Hece, ΣΣ H o : P(A i B ) P(A i )P(B ) for all i, H a : Some P(A i B ) P(A i )P(B ) Slide 5 Slide 6 Expected Frequecy: ˆ E Test Statistic: X i rows colums x ( Eˆ ) Eˆ 0.5 The Chi-square distributio df df 4 df 7 prob (prob) df df 10 Reectio Regio: X > χ α with d.f (I-1)(J-1) Slide Slide 8 Lotto after 399 umbers have bee draw Do some umbers appear more frequetly i LOTTO? TABLE Frequecy of Wiig Numbers i LOTTO 1. (7). (10) 3. (8) 4. (9) 5. (13) 6. (8) 7. (1) 8. (16) 9. (11) 10. (6) 11. (13) 1. (10) 13. (9) 14. (11) 15. (11) 16. (6) 17. (11) 18. (13) 19. (6) 0. (13) 1. (7). (9) 3. (8) 4. (1) 5. (6) 6. (4) 7. (10) 8. (8) 9. (14) 30. (1) 31. (11) 3. (1) 33. (9) 34. (11) 35. (6) 36. (8) 37. (14) 38. (10) 39. (15) 40. (10) (Expected freq.) Figure Number o ball Frequecy of LOTTO wiig umbers Lotto after 399 umbers have bee draw Do some umbers appear more frequetly i LOTTO? Number-rage: [1:40] Number of balls selected at each draw: 7 Number of samples: 57 Total umber of balls selected: 57*7399, Expected value of each umber: 399/ Observed χ statistics is x df P-value Coclusio: No evidece for departure from the ull hypothesis. Slide 9 Slide 30

6 A Example, Researchers i a Califoria commuity have asked a sample of 175 automobile owers to select their favorite from three popular automotive magazies. Of the 111 import owers i the sample, 54 selected Car ad Driver, 5 selected Motor Tred, ad 3 selected Road & Track. Of the 64 domestic-make owers i the sample, 19 selected Car ad Driver, selected Motor Tred, ad 3 selected Road & Track. At the 0.05 level, is import/domestic owership idepedet of magazie preferece? What is the most accurate statemet that ca be made about the p-value for the test? Slide 31 First, arrage the data i a table. Car ad Motor Road & Driver (1) Tred () Track (3) Totals Import (Imp) Domestic (Dom) Totals Secod, compute the expected values ad cotributios to χ for each of the six cells. The to the hypothesis test... Slide 3 Car ad Motor Road & Driver (1) Tred () Track (3) Import (Imp): O E χ cotributio Domestic (Dom) : O E χ cotributio Σχ cotributios I. Hypotheses: H 0 : H 1 : II. Reectio Regio: α 0.05 df (r 1)(k 1) ( 1) (3 1) 1 If χ > 5.991, reect H 0. Type of magazie ad auto owership are idepedet. Type of magazie ad auto owership are ot idepedet. Do Not Reect H 0 Reect H χ Slide 33 Slide 34 III. Test Statistic: χ IV. Coclusio: Sice the test statistic of falls beyod the critical value of 5.991, we reect the ull hypothesis with at least 95% cofidece. V. Implicatios: There is eough evidece to show that magazie preferece is ot idepedet from import/domestic auto owership. p-value: I a cell o a Microsoft Excel spreadsheet, type: CHIDIST(6.747,). The aswer is: p-value Slide 35

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