Confidence Intervals รศ.ดร. อน นต ผลเพ ม Assoc.Prof. Anan Phonphoem, Ph.D. Intelligent Wireless Network Group (IWING Lab)

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1 Cofidece Itervals รศ.ดร. อน นต ผลเพ ม Assoc.Prof. Aa Phophoem, Ph.D. Itelliget Wireless Network Group (IWING Lab) Computer Egieerig Departmet Kasetsart Uiversity, Bagkok, Thailad 1

2 Cofidece Iterval A sample of idepedet observatio x 1,x, x from ukow distributio Wat to fid populatio mea μ How to kow that μ plausible (believable) Poit estimate μ ad its estimated s.e. Cofidece Iterval Iterval that covers parameter (e.g. μ) with rage of cofidece (cofidece level) Ex. 95% of cofidece that μ will be i this rage

3 Chi-Square ( ) Distributio If X is ormal distributio, X N (0,1) Y X is Chi-Square with degree of freedom (d.f) equals to 1 For idepedet X 1, X,,X with is Chi-Square with d.f = Y i X i X N (0,1) For idepedet X 1, X,,X with Y is Chi-Square with d.f = i 1 x i X N (, ) 3

4 Degree of Freedom (d.f) # of data that is ot depeded o others Example: Select 3 radom umbers d.f = 3 Select 3 radom umbers that sum = 10 ca select ay umbers, the last oe is depeded o coditio E.g. Select ad 5, the last must be 3 d.f = Select 3 radom umbers that sum of square(x) = 54 ca select ay 1 umber, the last two are depeded o coditio E.g. Select 7, the rest must be 1 ad d.f = 1 4

5 t-distributio If X is ormal distributio, X N X ad Y are idepedet, variable t (0,1) ad Y t X is t-distributio with d.f = f f For radom from X N (, ), we ca calculate x ad Ad we kow the distributio X N ( 0,1) ( 1) s ( 1 ) s X Therefore, t ( 1) s ( 1) X s is t-distributio with d.f = (-1) 5

6 Cofidece Iterval of μ, kow σ samples are radom X N (0,1) with ukow μ, kow σ (ot practical for study) We ca fid x ad use it to fid Cofidece Iterval of μ From X N (, ), the X N, X ad Z N ( 0,1) 6

7 Cofidece Iterval of μ, kow σ From ormal distributio, suppose we wat to fid μ with the cofidece of 95% (0.95) Therefore, P(-1.96 < Z < 1.96) = z is betwee ±

8 Cofidece Iterval of μ, kow σ P X X X 1.96 X 1.96 X 1.96 X 1.96 X 1.96 Therefore, P X 1.96 X

9 Cofidece Iterval of μ, kow σ Cofidece Level 95% of μ is X 1.96 X 1.96 X X 1.96 ( ) Cofidece Level 95% 9

10 Adjust the Iterval We ca select ay Cofidece Iterval high cofidece iterval wide cofidece iterval legth (ot a good estimatio) To decrease cofidece iterval legth Icrease umber of samples high cost Decrease the cofidece level Popular Cofidece Iterval 90%, 95%, ad 99% For o-ormal distributio Cetral limit theorem > 30 is OK for estimatio 10

11 Example 1 A cylider with ormal distributio Ukow μ, ad kow σ =.009 cm. Radomly select 1 sample cyliders Calculate x 1.91 cm. Fid the cofidece iterval of populatio mea of cylider for this factory with 95% cofidece level Solutio Let μ = populatio mea cofidece iterval = x to

12 More easy symbol z p represets 100 p percetile of ormal Let α is the differet betwee 1.00 ad required value (e.g. require 95% α = 0.05) Cofidece iterval (1- α ) 100% of μ = X z 1 z z represets 95 percetile of ormal represets 99 percetile of ormal 1

13 Example 1 For more cofidece level = 99% α = 0.01 α/ = z cofidece iterval = x z x to Which is wider tha 95% cofidece level (11.77 to ) 13

14 Example Study the salary of employee σ = 10,000 Baht Radomly ask 100 employee Calculate x, 500 Baht Fid the cofidece iterval of populatio mea of salary for this compay with 95% cofidece level Solutio σ = 10,000 σ = 100, x,500, = 100 From cetral limit theorem, 95% cofidece level of μ is x z x , to,

15 Example 3: Natioal Discout, Ic. Natioal Discout has 60 retail outlets throughout the Uited States. Natioal evaluates each potetial locatio for a ew retail outlet i part o the mea aual icome of the idividuals i the marketig area of the ew locatio. Samplig ca be used to develop a iterval estimate of the mea aual icome for idividuals i a potetial marketig area for Natioal Discout. A sample of size = 36 was take. The sample mea, x, is $1,100 The sample stadard deviatio, s, is $4,500 We will use.95 as the cofidece coefficiet i our iterval estimate.

16 Example 3: Natioal Discout, Ic. Precisio Statemet There is a.95 probability that the value of a sample mea for Natioal Discout will provide a samplig error of $1,470 or less. Determied as follows: 95% of the sample meas that ca be observed are withi 1.96 of the populatio mea. If, the 4,500 s x x 36

17 Example 3: Natioal Discout, Ic. Iterval Estimate of the Populatio Mea: Ukow Iterval Estimate of is: $1,100 + $1,470 or $19,630 to $,570 We are 95% cofidet that the iterval cotais the populatio mea.

18 May Itervals For a radom set Ca create may itervals (a,b) Depeds o cofidece level For 100 cofidece itervals fidig There are (1-α)*100 times from 100 times that the cofidece itervals will cover the parameter (e.g. μ) 18

19 Example 3 From a ormal distributio with μ = 10, σ = 9 Geerate data 16 sets (8 samplig per set) For each set create cofidece iterval for 95% level of x 19

20 Example 3 Set 1: x With 95% cofidece level (8.16,1.374) cover μ = Set 16: x With 95% cofidece level (10.64,14.781) ot cover μ = 10 Therefore, for 16 sets ot cover: 1/16 = = 6.5% cover: 15/16 = = 93.75%

21 Iterval Estimatio of a Populatio Mea: Small-Sample Case ( < 30) Populatio is Not Normally Distributed The oly optio is to icrease the sample size to > 30 ad use the large-sample iterval-estimatio procedures. Populatio is Normally Distributed ad is Kow The large-sample iterval-estimatio procedure ca be used. Populatio is Normally Distributed ad is Ukow The appropriate iterval estimate is based o a probability distributio kow as the t distributio.

22 t Distributio The t distributio is a family of similar probability distributios. A specific t distributio depeds o a parameter kow as the degrees of freedom. As the umber of degrees of freedom icreases, the differece betwee the t distributio ad the stadard ormal probability distributio becomes smaller ad smaller. A t distributio with more degrees of freedom has less dispersio. The mea of the t distributio is zero.

23 Iterval Estimatio of a Populatio Mea: Small-Sample Case ( < 30) with Ukow Iterval Estimate x t / s where 1 -α = the cofidece coefficiet t α / = the t value providig a area of α / i the upper tail of a t distributio with - 1 degrees of freedom s = the sample stadard deviatio

24 Example 4: Apartmet Rets Iterval Estimatio of a Populatio Mea: Small-Sample Case ( < 30) with Ukow A reporter for a studet ewspaper is writig a article o the cost of off-campus housig. A sample of 10 oe-bedroom uits withi a half-mile of campus resulted i a sample mea of $550 per moth ad a sample stadard deviatio of $60. Let us provide a 95% cofidece iterval estimate of the mea ret per moth for the populatio of oe-bedroom uits withi a half-mile of campus. Assume this populatio to be ormally distributed.

25 Example 4: Apartmet Rets t Value At 95% cofidece, 1 - α =.95, α =.05, ad α / =.05. t.05 is based o - 1 = 10-1 = 9 degrees of freedom. I the t distributio table we see that t.05 =.6. Degrees of Freedom Area i Upper Tail

26 Example 4: Apartmet Rets Iterval Estimatio of a Populatio Mea: Small-Sample Case ( < 30) with Ukow x t / s or $ to $59.9 We are 95% cofidet that the mea ret per moth for the populatio of oe-bedroom uits withi a half-mile of campus is betwee $ ad $59.9.

27 Sample Size for a Iterval Estimate of a Populatio Mea Let E = the maximum samplig error metioed i the precisio statemet. E is the amout added to ad subtracted from the poit estimate to obtai a iterval estimate. E is ofte referred to as the margi of error. We have E z / Solvig for we have ( z ) / E

28 Example 5: Natioal Discout, Ic. Sample Size for a Iterval Estimate of a Populatio Mea Suppose that Natioal s maagemet team wats a estimate of the populatio mea such that there is a 0.95 probability that the samplig error is $500 or less. How large a sample size is eeded to meet the required precisio?

29 Example 5: Natioal Discout, Ic. Sample Size for Iterval Estimate of a σ Populatio Mea z α/ 500 At 95% cofidece, z.05 = Recall that = 4,500 Solvig for we have (1.96 ) (500 ( 4500 ) ) We eed to sample 31 to reach a desired precisio of $500 at 95% cofidece.

30 Iterval Estimatio of a Populatio Proportio Iterval Estimate p z / p (1 p ) where: 1 -α is the cofidece coefficiet z α/ is the z value providig a area of α/ i the upper tail of the stadard ormal probability distributio p is the sample proportio

31 Example 6: Political Sciece, Ic. Iterval Estimatio of a Populatio Proportio Political Sciece, Ic. (PSI) specializes i voter polls ad surveys desiged to keep political office seekers iformed of their positio i a race. Usig telephoe surveys, iterviewers ask registered voters who they would vote for if the electio were held that day. I a recet electio campaig, PSI foud that 0 registered voters, out of 500 cotacted, favored a particular cadidate. PSI wats to develop a 95% cofidece iterval estimate for the proportio of the populatio of registered voters that favors the cadidate.

32 Example 6: Political Sciece, Ic. Iterval Estimate of a Populatio Proportio p z / p (1 p ) where: = 500, p = 0/500 =.44, z α/ = ( ) PSI is 95% cofidet that the proportio of all voters that favors the cadidate is betwee.3965 ad.4835.

33 Sample Size for a Iterval Estimate of a Populatio Proportio Let E = the maximum samplig error metioed i the precisio statemet. We have E z / p (1 p ) Solvig for we have ( z ) p (1 p ) / E

34 Example 7: Political Sciece, Ic. Sample Size for a Iterval Estimate of a Populatio Proportio Suppose that PSI would like a.99 probability that the sample proportio is withi.03 of the populatio proportio. How large a sample size is eeded to meet the required precisio?

35 Example 7: Political Sciece, Ic. Sample Size for Iterval Estimate of a Populatio Proportio At 99% cofidece, z.005 =.576. Note: ( z / ) E p (1 p ) (.576 (. 03 ( We used 0.44 as the best estimate of p i the above expressio. If o iformatio is available about p, the 0.5 is ofte assumed because it provides the highest possible sample size. If we had used p = 0.5, the recommeded would have bee ) ) )(. 56 )

36 Cofidece Iterval of μ, ukow σ samples are radom X N with ukow μ, ukow σ (0,1) Cofidece Level 95% of μ is X s For > 30, good estimate < 30, ot good estimate of σ For < 30, Cofidece Level 95% of μ is X t s 36

37 t α for 95% cofidece level Sample size () t α Note: t fast icreasig for small t slow icreasig for large For, t z (t ormal distributio) 37

38 Example 8 8 samples radomly select from populatio Ukow μ with x 7.91, s = 0.67 Fid 95% cofidece level of cofidece Itervals of μ Solutio x t α= 0.05, d.f = -1 = 8-1 =7, t 0.05 =.36 s With 95% cofidece level that μ will be i the cofidece iterval (7.35,8.47) 38

39 Example 9 49 studets radomly select from 3 rd IUP studets Ukow μ with average grade x.3, s = 0.7 Fid 95% cofidece level of cofidece Itervals of μ Solutio =49, ukow σ s > 30 use z (ormal) x z With 95% cofidece level that μ will be i the cofidece iterval (.1,.5) s

40 Estimatig Populatio Total Sometimes, we iterested i Total populatio more tha populatio mea Total N i 1 X i NX ; N = # populatio Cofidece Iterval (1-α) 100% of Total is N X Nt ( 1 ) s N X Nt ( 1 ) s N N 1 for fpc factor fpc factor: Fiite populatio correctio factor N >>, factor ~ 1 40

41 Example 10 For a maize field, Area = 5,000 Rai Radomly select 1,000 Rai The average total output x 1076 Tug, s =73 Tug/Rai Solutio Total output of 5,000 Rai ca be approximate with N X Nt ( 1 ) 5,000 (1,076 s ) N N 1 5,000 (1.984 ) ,000 5, ,380,000 68, Total output of maize field is i the iterval (5,111,865.14, 5,648, ) 41

42 Referece สถ ต พ นฐาน ส าหร บน กพ ฒนาส งคม, ว น ส พ ชวณ ชย สมจ ต ว ฒนาชยาก ล และเบญจมาศ ต ลยน ต ก ล, คณะว ทยาศาสตร และเทคโนโลย ม.ธรรมศาสตร, มค.547 4