Confidence intervals. A. Confidence intervals for the population mean µ of normal population with known standard deviation σ: ).

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1 Cofidece itervals A. Cofidece itervals for the populatio mea µ of ormal populatio with kow stadard deviatio σ: Let X 1, X,, X be a radom sample from N(µ, σ. We kow that X N(µ, σ. Therefore, P z µ σ z = 1, where z are defied as follows: N(0,1 1 z 0 + z The area 1 is called cofidece level. Whe we costruct cofidece itervals we usually use the followig cofidece levels: 1 z The expressio above ca be writte as: ( σ σ P x z µ x + z = 1. (1 It is temptig to read this statemet as the probability.... But we should ot! Istead, we ca say that we are 1 cofidet that µ falls i the iterval x ± z σ. Why? 1

2 Example: Suppose that the legth of iro rods from a certai factory follows the ormal distributio with kow stadard deviatio σ = 0. m but ukow mea µ. Costruct a 95% cofidece iterval for the populatio mea µ if a radom sample of = 16 of these iro rods has sample mea x = 6 m. Sample size determiatio for a give legth of the cofidece iterval: Fid the sample size eeded whe we wat the width of the cofidece iterval to be ±E with cofidece level 1. Aswer: I the expressio x ± z σ the width of the cofidece iterval is give by z σ (also called margi of error. We wat this width to be equal to E. Therefore, E = z ( σ z = σ E. Example: For the example above, suppose that we wat the etire width of the cofidece iterval to be equal to 0.05 m. Fid the sample size eeded. Questio: Is there a 100% cofidece iterval?

3 B. Cofidece itervals for the populatio mea µ with kow populatio stadard deviatio σ: From the cetral limit theorem we kow that whe 30 the distributio of the sample mea X approximately follows: X N(µ, σ Therefore, the cofidece iterval for the populato mea µ is give by the expressio we foud i part (A: ( σ σ P x z µ x + z = 1. The mea µ falls i the iterval x ± z σ. Also the sample size determiatio is give by the same formula we foud i part (A: E = z ( σ z = σ E. Example: A sample of size = 50 is take from the productio of lightbulbs at a certai factory. The sample mea of the lifetime of these 50 lightbulbs is foud to be x = 1570 hours. Assume that the populatio stadard deviatio is σ = 10 hours. a. Costruct a 95% cofidece iterval for µ. b. Costruct a 99% cofidece iterval for µ. c. What sample size is eeded so that the legth of the iterval is 30 hours with 95% cofidece? 3

4 Cofidece itervals - A empirical ivestigatio Two dice are rolled ad the sum X of the two umbers that occured is recorded. The probability distributio of X is as follows: X P (X 1/36 /36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 /36 1/36 This distributio has mea µ = 7 ad stadard deviatio σ =.4. We take 100 samples of size = 50 each from this distributio ad compute for each sample the sample mea x. Preted ow that we oly kow that σ =.4, ad that µ is ukow. We are goig to use these 100 sample meas to costruct 100 cofidece itervals each oe with 95% cofidece level for the true populatio mea µ. Here are the results: Sample x 95%C.I.forµ: x µ x Is µ = 7 icluded? µ 7.57 YES µ 6.97 NO µ 7.5 YES µ 7.1 YES µ 7.37 YES µ 7.5 YES µ 7.87 YES µ 8.9 YES µ 7.61 YES µ 8.03 YES µ 7.73 YES µ 7.75 YES µ 8.09 YES µ 8.09 YES µ 7.47 YES µ 7.61 YES µ 7.87 YES µ 7.37 YES µ 7.77 YES µ 7.71 YES µ 7.65 YES µ 7.85 YES µ 7.47 YES µ 7.61 YES µ 8.77 NO µ 7.67 YES µ 7.73 YES µ 7.49 YES µ 7.63 YES µ 8.13 YES µ 7.71 YES µ 7.73 YES µ 7.73 YES µ 7.47 YES µ 7.79 YES µ 7.85 YES µ 7.75 YES µ 7.91 YES µ 7.49 YES µ 7.93 YES µ 8.01 YES µ 7.9 YES µ 7.77 YES µ 7.65 YES µ 7.65 YES µ 7.73 YES µ 7.81 YES µ 8.17 YES µ 7.75 YES µ 7.99 YES 4

5 Sample x 95%C.I.forµ: x µ x Is µ = 7 icluded? µ 7.1 YES µ 7.81 YES µ 7.31 YES µ 8.13 YES µ 8.01 YES µ 7.95 YES µ 7.3 YES µ 8.39 NO µ 7.33 YES µ 7.47 YES µ 7.75 YES µ 7.5 YES µ 7.97 YES µ 7.77 YES µ 7.35 YES µ 7.65 YES µ 7.61 YES µ 7.45 YES µ 7.87 YES µ 7.57 YES µ 7.09 YES µ 7.15 YES µ 7.79 YES µ 7.57 YES µ 7.91 YES µ 7.7 YES µ 7.95 YES µ 7.85 YES µ 7.43 YES µ 7.73 YES µ 7.67 YES µ 7.75 YES µ 7.85 YES µ 7.93 YES µ 7.55 YES µ 6.95 NO µ 7.73 YES µ 7.33 YES µ 7.85 YES µ 7.53 YES µ 7.63 YES µ 7.93 YES µ 7.35 YES µ 7.43 YES µ 7.97 YES µ 7.71 YES µ 8.01 YES µ 7.39 YES µ 7.31 YES µ 7.97 YES We observe that four cofidece itervals amog the 100 that we costructed fail to iclude the true populatio mea µ = 7 (about 5%. It is also clear from this experimet why we should ever use the word probability to iterpret a cofidece iterval. Cosider for example the first sample. Our cofidece iterval is 6.3 µ Does it make sese to say the probability is 95% that µ = 7 falls betwee 6.3 ad 7.57? Of course the probability is 1 here. Look at sample. The resultig cofidece iterval is 5.63 µ Here the probability that µ = 7 icluded i this iterval is 0. Therefore, the probability is either 0 or 1. The cofidece iterval either icludes or ot the populatio mea µ. We say: we are 95% cofidet that µ falls i the iterval we just costructed. 5

6 C. Cofidece itervals for the populatio mea of ormal distributio whe the populatio stadard deviatio σ is ukow: Let X 1, X,, X be a radom sample from N(µ, σ. We kow that X µ s P t ; 1 X µ s t ; 1 = 1 t 1. Therefore, where t ; 1 ad t ; 1 are defied as follows: t 1 1 t 0 + t The area 1 is called cofidece level. The values of t ; 1 ca be foud from the t table. Here are some examples: 1 t ; Note: The sample stadard deviatio is computed as follows: i=1 (x i x s = 1 or easier usig the shortcut formula. s = 1 [ x i ( ] i=1 x i 1 i=1 6

7 After some rearragig the expressio above ca be writte as: ( s s P x t ; 1 µ x + t ; 1 = 1 ( We say that we are 1 cofidet that µ falls i the iterval: x ± t ; 1 s. Example: The daily productio of a chemical product last week i tos was: 785, 805, 790, 793, ad 80. a. Costruct a 95% cofidece iterval for the populatio mea µ. b. What assumptios are ecessary? 7

8 D. Cofidece iterval for the populatio variace σ of ormal distributio: Let X 1, X,, X radom sample from N(µ, σ. We kow that ( 1S σ ( P χ ; 1 ( 1S σ χ 1 ; 1 = 1 χ 1. Therefore, where χ ; 1 ad χ 1 ; 1 are defied as follows: χ 1 1 χ χ 1 Some examples o how to fid the values χ ; 1 ad χ 1 ; 1: 1 χ ; 1 χ 1 ; After rearragig the iequality above we get: ( 1s P σ ( 1s = 1 (3 χ 1 ; 1 χ ; 1 We say that we are 1 cofidet that the populatio variace σ falls i the iterval: ( 1s ( 1s, χ 1 ; 1 χ ; 1 8

9 Commet: Whe the sample size is large the χ 1 distributio ca be approximated by N( 1, ( 1. Therefore, i this situatio, the cofidece iterval for the variace ca be computed as follows: s 1 + z σ s 1 z Example: A precisio istrumet is guarateed to read accurately to withi uits. A sample of 4 istrumet readigs o the same object yielded the measuremets 353, 351, 351, ad 355. Fid a 90% cofidece iterval for the populatio variace. What assumptios are ecessary? Does the guaratee seem reasoable? Note: if the populatio is ot ormal the coverage is poor (the χ is ot robust. I these situatios (samplig from o-ormal populatios a asymptotically distributio-free cofidece iterval for the variace ca be obtaied usig the large sample theory result: ( (s σ N 0, µ 4 σ 4 or, (s σ µ4 σ 4 N(0, 1 where, µ 4 = E(X µ 4 is the fourth momet of the distributio. Of course, µ 4 is ukow ad will be estimated by the fourth sample momet m 4 = 1 i=1 (X i X 4. The cofidece iterval for the populatio variace is computed as follows: s z m4 s 4 σ s + z m4 s 4 9

10 E. Cofidece iterval for the populatio proportio p: Let X 1, X,, X be a radom sample from the Beroulli distributio with probability of success p. Costruct a cofidece iterval for p. We kow that whe is large: X p p(1 p N(0, 1 Therefore, P z X p p(1 p z = 1, where z ad z as o page 1. After rearragig we get: P X p(1 p z p X + z p(1 p = 1. The ratio x is the poit estimate of the populatio p ad it is deoted with ˆp = x. The problem with this iterval is that the ukow p appears also at the ed poits of the iterval. As a approximatio we ca simply replace p with its estimate ˆp = x. Fially the cofidece iterval is give below: ˆp(1 ˆp ˆp(1 ˆp P ˆp z p ˆp + z = 1. (4 We say that we are 1 cofidet that p falls i ˆp(1 ˆp ˆp ± z Sample size determiatio: Determie the sample size eeded so that the resultig cofidece iterval will have margi of error E with cofidece level 1. Aswer: I the expressio ˆp ± z of error ˆp(1 ˆp E = z ˆp(1 ˆp. We simply solve for : ˆp(1 ˆp the width of the cofidece iterval is give by the margi = z ˆp(1 ˆp. E However the value of ˆp is ot kow because we have ot selected our sample yet. If we use ˆp = 0.5 we will obtai the largest possible sample size. Of course if we have a idea about its value (from aother study, etc. we ca use it. 10

11 Example: At a survey poll before the electios cadidate A receives the support of 650 voters i a sample of 100 voters. a. Costruct a 95% cofidece iterval for the populatio proportio p that supports cadidate A. b. Fid the sample size eeded so that the margi of error will be ±0.01 with cofidece level 95%. Aother formula for the cofidece iterval for the populatio proportio p: A more accurate cofidece iterval ca be obtaied as follows: P z X p p(1 p z = 1 P z X p p(1 p ˆp p P p(1 p p(1 p z = 1 z = 1 (ˆp p P z = 1 We obtai a quadratic expressio i p: (ˆp p z p(1 p 0 (1 + z p (ˆp + z p + ˆp = 0 Solvig for p we get the followig cofidece iterval: ˆp + z ± z ˆp(1 ˆp + z z. Whe is large this is the same as (4. (5 11

12 Survey poll - a example: Below we see part of a survey poll from Afghaista. The etire survey ca be accessed at: Frustratio With War, Problems i Daily Life Sed Afghas Support for U.S. Efforts Tumblig ABC News/BBC/ARD Natioal Survey of Afghaista ANALYSIS by GARY LANGER Feb. 9, 009 The Uited States, its NATO allies ad the govermet of Hamid Karzai are losig ot just groud i Afghaista but also the hearts ad mids of the Afgha people. A ew atioal public opiio poll i Afghaista by ABC News, the BBC ad ARD Germa TV fids that performace ratigs ad support levels for the Kabul govermet ad its Wester allies have plummeted from their peaks, particularly i the past year. Widespread strife, a resurget Taliba, strugglig developmet, soarig corruptio ad broad complaits about food, fuel, power ad prices all play a role. The effects are remarkable: With expectatios for security ad ecoomic developmet umet, the umber of Afghas who say their coutry is headed i the right directio has dived from 77 percet i 005 to 40 percet ow fewer tha half for the first time i these polls. I 005, moreover, 83 percet of Afghas expressed a favorable opiio of the Uited States uheard of i a Muslim atio. Today just 47 percet still hold that view, dow 36 poits, acceleratig with a 18-poit drop i U.S. favorability this year aloe. For the first time slightly more Afghas ow see the Uited States ufavorably tha favorably. The umber who say the Uited States has performed well i Afghaista has bee more tha halved, from 68 percet i 005 to 3 percet ow. Ratigs of NATO/ISAF forces are o better. Just 37 percet of Afghas ow say most people i their area support Wester forces; it was 67 percet i 006. Ad 5 percet ow say attacks o U.S. or NATO/ISAF forces ca be justified, double the level, 13 percet, i 006. Nor does the electio of Barack Obama hold much promise i the eyes of the Afgha public: While two i 10 thik he ll make thigs better for their coutry, early as may thik he ll make thigs worse. The rest either expect o chage, or are waitig to see. This survey is ABC s fourth i Afghaista sice 005, part of its ogoig "Where Thigs Stad" series there ad i Iraq. It was coducted i late December ad early Jauary via face-to-face iterviews with a radom atioal sample of 1,534 Afgha adults i all 34 of the coutry s provices, with field work by the Afgha Ceter for Socio-Ecoomic ad Opiio Research i Kabul. The survey comes at a critical time for the coflict i Afghaista, as the Uited States begis early to double its deploymet of troops there, addig as may as 30,000 to the 3,000 already preset, ad, uder the ew Obama admiistratio, to rethik its troubled strategy. (Said Vice Presidet Joe Bide: "We ve iherited a real mess." While Afghas likely will welcome a ew strategy, they re far cooler o ew troops: Cotrary to Washigto s plas, just 18 percet say the umber of U.S. ad NATO/ISAF forces i Afghaista should be icreased. Far more, 44 percet, wat the opposite a decrease i the level of these forces. (ISAF stads for Iteratioal Security Assistace Force, the U.N.-madated, NATO-led multiatioal force i Afghaista. SECURITY The failures to date to hold groud ad provide effective security are powerful factors i Afgha public opiio. Far fewer tha i past years say Wester forces have a strog presece i their area (34 percet, dow from 57 percet i 006, or crucially see them as effective i providig security (4 percet, dow from 67 percet. Amid widespread experiece of warfare gu battles, bombigs ad air strikes amog them the umber of Afghas who rate their ow security positively has dropped from 7 percet i 005 to 55 percet today ad it goes far lower i high-coflict provices. I the coutry s beleaguered Southwest (Helmad, Kadahar, Nimroz, Uruzga ad Zabul provices oly 6 percet feel secure from crime ad violece; i Helmad aloe, just 14 percet feel safe. Civilia casualties i U.S. or NATO/ISAF air strikes are a key complait. Sevety-seve percet of Afghas call such strikes uacceptable, sayig the risk to civilias outweighs the value of these raids i fightig isurgets. Ad Wester forces take more of the blame for such casualties, a public relatios advatage for ati-govermet forces: Forty-oe percet of Afghas chiefly blame U.S. or NATO/ISAF forces for poor targetig, vs. 8 percet who maily blame the isurgets for cocealig themselves amog civilias. Give that view, more Afghas ow blame the coutry s strife o the Uited States ad its allies tha o the Taliba. Thirty-six percet mostly blame U.S., Afgha or NATO forces or the U.S. or Afgha govermets for the violece that s occurrig, up by 10 poits from 007. Fewer, 7 percet, ow maily blame the Taliba, dow by 9 poits. Afghaista s cetral ad provicial govermets have a stroger presece ad greater public cofidece tha Wester forces but they, too, have suffered. I 005, still celebratig the Taliba s ouster i November 001, 83 percet of Afghas approved of the work of Presidet Karzai ad 80 percet approved of the atioal govermet overall. Today those have slid to 5 ad 49 percet respectively. (Karzai s expected to ru for re-electio i August. Ad fewer tha half rate their provicial govermet positively. IMPACT Crucially, the Kabul govermet ad its Wester allies do better where they are see as havig a strog presece ad as beig effective i providig security, as well as i areas where reported coflict is lower. Where security is weaker or these groups have less presece, their ratigs declie sharply. For example, amog people who say the cetral govermet, the provicial govermet or Wester forces have a strog local presece, 58, 57 ad 46 percet, respectively, approve of their performace. Where the presece of these etities is see as weak, however, their respective approval ratigs drop to just 31, ad 5 percet... Descriptio of the methodology used for this survey poll (from ABC News Website: METHODOLOGY This ABC News/BBC/ARD poll is based o i-perso iterviews with a radom atioal sample of 1,534 Afgha adults from Dec. 30, 008 to Ja. 1, 009. The results have a.5-poit error margi. Field work by the Afgha Ceter for Socio-Ecoomic ad Opiio Research i Kabul, a subsidiary of D3 Systems Ic. of Viea, Va. 1

13 Other cofidece itervals Cofidece iterval for the differece betwee two populatio meas µ 1 µ whe σ 1, σ are kow: σ x 1 x z 1 + σ σ µ 1 µ x 1 x + z 1 + σ 1 1 Where x 1, x are the sample meas of two samples idepedetly selected from two populatios with meas µ 1, µ ad variaces σ 1, σ respectively. Cofidece iterval for the differece betwee two ormal populatio meas µ 1 µ whe σ 1 = σ but ukow: x 1 x t ; 1+ s ( µ 1 µ x 1 x + t ; 1+ s ( Where s = ( 1 1s 1 +( 1s 1 + is the pooled sample variace (the estimate of the true but ukow commo populatio variace σ. This cofidece iterval is based o the fact that ( 1+ s χ σ 1 +. Cofidece iterval for the differece betwee two populatio proportios p 1 p : x 1 x x1 z 1 (1 x1 x 1 + (1 x p 1 p x 1 x x1 + z 1 (1 x1 x 1 + (1 x Where x 1 is the umber of successes amog 1 trials with probability of success p 1, ad x is the umber of successes amog trials with probability of success p. Cofidece iterval for the ratio of two ormal populatio variaces σ 1 : σ s 1 s 1 F 1 ; 1 1, 1 σ 1 σ s 1 F s 1 ; 1, 1 1 Where s 1, s are the sample variaces based o two idepedet samples of size 1, selected from two ormal populatios N(µ 1, σ 1 ad N(µ, σ. 13

14 Cofidece itervals - Examples Example 1 A sample of size = 50 is take from the productio of lightbulbs at a certai factory. x = 1570 hours. Assume that the populatio stadard deviatio is σ = 10 hours. The sample mea is foud to be a. Costruct a 95% cofidece iterval for µ. b. Costruct a 99% cofidece iterval for µ. c. What sample size is eeded so that the legth of the iterval is 30 hours with 95% cofidece? Example The UCLA housig office wats to estimate the mea mothly ret for studios aroud the campus. A radom sample of size = 36 studios is take from the area aroud UCLA. The sample mea is foud to be x = $900. Assume that the populatio stadard deviatio is σ = $150. a. Costruct a 95% cofidece iterval for the mea mothly ret of studios i the area aroud UCLA. b. Costruct a 99% cofidece iterval for the mea mothly ret of studios i the area aroud UCLA. c. What sample size is eeded so that the legth of the iterval is $60 with 95% cofidece? Example 3 We wat to estimate the populatio proportio of studets that are Democrats at UCLA. A sample of size = is selected. There are Democrats i the sample. a. Costruct a 95% cofidecre iterval for the populatio proportio p of studets that are Democrats at UCLA. What do you observe? b. What is the sample size eeded i order to obtai a ±% margi of error? Example 4 A precisio istrumet is guarateed to read accurately to withi uits. A sample of 4 istrumet readigs o the same object yielded the measuremets 353, 351, 351, ad 355. Fid a 90% cofidece iterval for the populatio variace. What assumptios are ecessary? Does the guaratee seem reasoable? Example 5 A chemical process must produce, o the average, 800 tos of chemical per day. The daily yields for the past week are 785, 805, 790, 793, ad 80 tos. Do these data provide evidece that the average productio of this chemical is ot 800 tos? Use 90% cofidece level. Example 6 A chemist has prepared a product desiged to kill 60% of a particular type of isect. What sample size should be used if he desires to be 95% cofidet that he is withi 0.0 of the true fractio of isects killed? Example 7 Suppose that two idepedet radom samples of 1 ad observatios are selected from ormal populatios with meas µ 1, µ ad variaces σ 1, σ respectively. Fid a cofidece iterval for the variace ratio σ 1 σ with cofidece level 1. Example 8 The sample mea X is a good estimator of the populatio mea µ. It ca also be used to predict a future value of X idepedetly selected from the populatio. Assume that you have a sample mea x ad a sample variace s, based o a radom sample of measuremets from a ormal populatio. Costruct a predictio iterval for a ew observatio x, say x p. Use 1 cofidece level. Hit: Start with the quatity X p X ad the use the defiitio of the t distributio. Example 9 Let 10.5, 11.3, 1.8, 9.6, 5.3 the times i secods eeded for dowloadig 5 files o your computer from a course website. If we assume that this sample we selected from a ormal distributio, costruct a 98% cofidece iterval for the populatio me µ. Also, costruct a 99% cofidece iterval for the populatio variace σ. Example 10 The sample mea lifetime of 1 = 100 light bulbs was foud to be equal x 1 = 1500 hours. After ew material was used i the productio, aother sample of size = 100 light bulbs was selected ad gave x = 1600 hours. If assume that the stadard deviatio is σ = 150 hours i both case, costruct a 95% cofidece iterval for the differece i the populatio meas, µ 1 µ. 14