Confidence intervals. A. Confidence intervals for the population mean µ of normal population with known population standard deviation σ: ).
|
|
- Sheena Spencer
- 5 years ago
- Views:
Transcription
1 Cofidece itervals A. Cofidece itervals for the populatio mea µ of ormal populatio with kow populatio stadard deviatio σ: Usually (if ot always) µ is ukow. The what? What would be a good estimate for µ? Let X 1, X,, X be a radom sample from N(µ, σ). We kow that X N(µ, σ ). Therefore, P z α µ σ z α = 1 α, where z α α are defied as follows: N(0,1) α 1 α α z α 0 + z α The area 1 α is called cofidece level. Whe we costruct cofidece itervals we usually use the followig cofidece levels: 1 α z α The expressio above ca be writte as: ( ) σ σ P x z α µ x + z α = 1 α. (1) 1
2 It is temptig to read this statemet as the probability.... But we should ot! Istead, we ca say that we are 1 α cofidet that µ falls i the iterval x ± z α σ. Why? 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 α. Aswer: I the expressio x ± z α σ the width of the cofidece iterval is give by z α σ (also called margi of error). We wat this width to be equal to E. Therefore, E = z α ( σ z α = 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 populatio mea µ is give by the expressio we foud i part (A): ( ) σ σ P x z α µ x + z α = 1 α. The mea µ falls i the iterval x ± z α σ. Also the sample size determiatio is give by the same formula we foud i part (A): E = z α ( σ z α = 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 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). It ca be show that the ratio X µ s follows the so called t (or Studet s t) distributio with 1 degrees of freedom. We deote it with t 1. Please ote the differece betwee X µ s distributio. ad X µ σ. The latter follows the Z Few commets o the t distributio: Its shape depeds o the degrees of freedom, but it is similar to z. It is cetered at zero, however compared to z, it has more probability i the tails. As the degrees of freedom icrease (see figure below) the t distributio coverges to the z distributo. Let X be a radom variable that follows the t distributio with 1 degrees of freedom. We write X t 1. Below we see few desities of X for differet values of the degrees of freedom. We observe that for 15 degrees of freedom the t is very similar to t already. f(x) N(0, 1) t 15 t 5 t x 6
7 Therefore, P t α ; 1 X µ s t α ; 1 = 1 α where t α ; 1 ad t α ; 1 are defied as follows: t 1 α 1 α α t α 0 + t α The area 1 α is called cofidece level. The values of t α ; 1 ca be foud from the t table. Here are some examples: 1 α t α ; 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 7
8 After some rearragig the expressio above ca be writte as: ( ) s s P x t α ; 1 µ x + t α ; 1 = 1 α () We say that we are 1 α cofidet that µ falls i the iterval: x ± t α ; 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? 8
9 The t distributio table: 9
10 D. Cofidece iterval for the populatio variace σ of ormal distributio: Let X 1, X,, X radom sample from N(µ, σ). It ca be show that the ratio ( 1)S σ follows the so called χ distributio with 1 degrees of freedom, deoted with χ 1. Few commets o the χ distributio: Let X χ 1. The E(X) = 1 ad V ar(x) = ( 1). The radom variable X takes o o-egative values. The distributio is skewed to the right but as the degrees of freedom icrease its shape approaches the ormal distributio (see figure below). Therefore, for large degrees of freedom the χ is approximately the same as N( 1, ( 1)). Shape of the χ distributio: Χ 3 f(x) x Χ 10 f(x) x Χ 30 f(x) x 10
11 Therefore, to costruct a cofidece iterval for σ we use the followig: ( ) P χ ( 1)S α ; 1 χ σ 1 α ; 1 = 1 α where χ α ; 1 ad χ 1 α ; 1 are defied as follows: χ 1 α 1 α α χα χ α 1 Some examples o how to fid the values χ α ; 1 ad χ 1 α ; 1: 1 α χ α ; 1 χ 1 α ; After rearragig the iequality above we get: ( 1)s P σ ( 1)s = 1 α (3) χ 1 α ; 1 χ α ; 1 We say that we are 1 α cofidet that the populatio variace σ falls i the iterval: ( 1)s ( 1)s, χ 1 α ; 1 χ α ; 1 11
12 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? 1
13 The χ distributio table: 13
14 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 α X p p(1 p) z α = 1 α, where z α ad z α as o page 1. After rearragig we get: P X p(1 p) z α p X + z α p(1 p) = 1 α. 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 α p ˆp + z α = 1 α. (4) We say that we are 1 α cofidet that p falls i ˆp(1 ˆp) ˆp ± z α Sample size determiatio: Determie the sample size eeded so that the resultig cofidece iterval will have margi of error E with cofidece level 1 α. Aswer: I the expressio ˆp ± z α of error z α E = z α ˆp(1 ˆp) ˆp(1 ˆp). We simply solve for : ˆp(1 ˆp) = z α ˆp(1 ˆp). E the width of the cofidece iterval is give by the margi 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. 14
15 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 α X p p(1 p) z α = 1 α P z α X p p(1 p) ˆp p P p(1 p) p(1 p) z α = 1 α z α = 1 α (ˆp p) P z α = 1 α We obtai a quadratic expressio i p: (ˆp p) z p(1 p) α 0 (1 + z α )p (ˆp + z α )p + ˆp = 0 Solvig for p we get the followig cofidece iterval: ˆp + z α ± z α ˆp(1 ˆp) + z α z α. Whe is large this is the same as (4). (5) 15
16 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. 16
17 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 α + σ σ µ 1 µ x 1 x + z 1 α + 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 α ; 1+ s ( ) µ 1 µ x 1 x + t α ; 1+ s ( ) Where s = ( 1 1)s 1 +( 1)s 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 ) α + (1 x ) p 1 p x 1 x x1 + z 1 (1 x1 x 1 ) α + (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. 17
18 Cofidece itervals - Examples Example 1 A sample of size = 50 is take from the productio of lightbulbs at a certai factory. The sample mea 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? 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 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 8 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. 18
Confidence intervals. A. Confidence intervals for the population mean µ of normal population with known standard deviation σ: ).
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 =
More informationEstimation of a population proportion March 23,
1 Social Studies 201 Notes for March 23, 2005 Estimatio of a populatio proportio Sectio 8.5, p. 521. For the most part, we have dealt with meas ad stadard deviatios this semester. This sectio of the otes
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationA quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population
A quick activity - Cetral Limit Theorem ad Proportios Lecture 21: Testig Proportios Statistics 10 Coli Rudel Flip a coi 30 times this is goig to get loud! Record the umber of heads you obtaied ad calculate
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationMATH/STAT 352: Lecture 15
MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationAgreement of CI and HT. Lecture 13 - Tests of Proportions. Example - Waiting Times
Sigificace level vs. cofidece level Agreemet of CI ad HT Lecture 13 - Tests of Proportios Sta102 / BME102 Coli Rudel October 15, 2014 Cofidece itervals ad hypothesis tests (almost) always agree, as log
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationInferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process.
Iferetial Statistics ad Probability a Holistic Approach Iferece Process Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike
More informationUniversity of California, Los Angeles Department of Statistics. Hypothesis testing
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Elemets of a hypothesis test: Hypothesis testig Istructor: Nicolas Christou 1. Null hypothesis, H 0 (claim about µ, p, σ 2, µ
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig
More informationConfidence Intervals QMET103
Cofidece Itervals QMET103 Library, Teachig ad Learig CONFIDENCE INTERVALS provide a iterval estimate of the ukow populatio parameter. What is a cofidece iterval? Statisticias have a habit of hedgig their
More informationAP Statistics Review Ch. 8
AP Statistics Review Ch. 8 Name 1. Each figure below displays the samplig distributio of a statistic used to estimate a parameter. The true value of the populatio parameter is marked o each samplig distributio.
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationChapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE. Part 3: Summary of CI for µ Confidence Interval for a Population Proportion p
Chapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE Part 3: Summary of CI for µ Cofidece Iterval for a Populatio Proportio p Sectio 8-4 Summary for creatig a 100(1-α)% CI for µ: Whe σ 2 is kow ad paret
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationInstructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?
CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter
More informationBig Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates.
5. Data, Estimates, ad Models: quatifyig the accuracy of estimates. 5. Estimatig a Normal Mea 5.2 The Distributio of the Normal Sample Mea 5.3 Normal data, cofidece iterval for, kow 5.4 Normal data, cofidece
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More information(7 One- and Two-Sample Estimation Problem )
34 Stat Lecture Notes (7 Oe- ad Two-Sample Estimatio Problem ) ( Book*: Chapter 8,pg65) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye Estimatio 1 ) ( ˆ S P i i Poit estimate:
More informationConfidence Intervals รศ.ดร. อน นต ผลเพ ม Assoc.Prof. Anan Phonphoem, Ph.D. Intelligent Wireless Network Group (IWING Lab)
Cofidece Itervals รศ.ดร. อน นต ผลเพ ม Assoc.Prof. Aa Phophoem, Ph.D. aa.p@ku.ac.th Itelliget Wireless Network Group (IWING Lab) http://iwig.cpe.ku.ac.th Computer Egieerig Departmet Kasetsart Uiversity,
More informationTopic 10: Introduction to Estimation
Topic 0: Itroductio to Estimatio Jue, 0 Itroductio I the simplest possible terms, the goal of estimatio theory is to aswer the questio: What is that umber? What is the legth, the reactio rate, the fractio
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
More information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationUnderstanding Dissimilarity Among Samples
Aoucemets: Midterm is Wed. Review sheet is o class webpage (i the list of lectures) ad will be covered i discussio o Moday. Two sheets of otes are allowed, same rules as for the oe sheet last time. Office
More informationHomework 5 Solutions
Homework 5 Solutios p329 # 12 No. To estimate the chace you eed the expected value ad stadard error. To do get the expected value you eed the average of the box ad to get the stadard error you eed the
More informationCONFIDENCE INTERVALS STUDY GUIDE
CONFIDENCE INTERVALS STUDY UIDE Last uit, we discussed how sample statistics vary. Uder the right coditios, sample statistics like meas ad proportios follow a Normal distributio, which allows us to calculate
More informationDirection: This test is worth 250 points. You are required to complete this test within 50 minutes.
Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More informationApril 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE
April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE TERRY SOO Abstract These otes are adapted from whe I taught Math 526 ad meat to give a quick itroductio to cofidece
More informationStat 200 -Testing Summary Page 1
Stat 00 -Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece
More informationData Analysis and Statistical Methods Statistics 651
Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasii/teachig.html Suhasii Subba Rao Review of testig: Example The admistrator of a ursig home wats to do a time ad motio
More informationFrequentist Inference
Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for
More informationUnderstanding Samples
1 Will Moroe CS 109 Samplig ad Bootstrappig Lecture Notes #17 August 2, 2017 Based o a hadout by Chris Piech I this chapter we are goig to talk about statistics calculated o samples from a populatio. We
More informationMathacle. PSet Stats, Concepts In Statistics Level Number Name: Date: Confidence Interval Guesswork with Confidence
PSet ----- Stats, Cocepts I Statistics Cofidece Iterval Guesswork with Cofidece VII. CONFIDENCE INTERVAL 7.1. Sigificace Level ad Cofidece Iterval (CI) The Sigificace Level The sigificace level, ofte deoted
More informationExam II Covers. STA 291 Lecture 19. Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Location CB 234
STA 291 Lecture 19 Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Locatio CB 234 STA 291 - Lecture 19 1 Exam II Covers Chapter 9 10.1; 10.2; 10.3; 10.4; 10.6
More informationAAEC/ECON 5126 FINAL EXAM: SOLUTIONS
AAEC/ECON 5126 FINAL EXAM: SOLUTIONS SPRING 2015 / INSTRUCTOR: KLAUS MOELTNER This exam is ope-book, ope-otes, but please work strictly o your ow. Please make sure your ame is o every sheet you re hadig
More informationDirection: This test is worth 150 points. You are required to complete this test within 55 minutes.
Term Test 3 (Part A) November 1, 004 Name Math 6 Studet Number Directio: This test is worth 10 poits. You are required to complete this test withi miutes. I order to receive full credit, aswer each problem
More informationENGI 4421 Confidence Intervals (Two Samples) Page 12-01
ENGI 44 Cofidece Itervals (Two Samples) Page -0 Two Sample Cofidece Iterval for a Differece i Populatio Meas [Navidi sectios 5.4-5.7; Devore chapter 9] From the cetral limit theorem, we kow that, for sufficietly
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationOverview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions
Chapter 9 Slide Ifereces from Two Samples 9- Overview 9- Ifereces about Two Proportios 9- Ifereces about Two Meas: Idepedet Samples 9-4 Ifereces about Matched Pairs 9-5 Comparig Variatio i Two Samples
More informationLecture 5. Materials Covered: Chapter 6 Suggested Exercises: 6.7, 6.9, 6.17, 6.20, 6.21, 6.41, 6.49, 6.52, 6.53, 6.62, 6.63.
STT 315, Summer 006 Lecture 5 Materials Covered: Chapter 6 Suggested Exercises: 67, 69, 617, 60, 61, 641, 649, 65, 653, 66, 663 1 Defiitios Cofidece Iterval: A cofidece iterval is a iterval believed to
More informationChapter 23: Inferences About Means
Chapter 23: Ifereces About Meas Eough Proportios! We ve spet the last two uits workig with proportios (or qualitative variables, at least) ow it s time to tur our attetios to quatitative variables. For
More informationOctober 25, 2018 BIM 105 Probability and Statistics for Biomedical Engineers 1
October 25, 2018 BIM 105 Probability ad Statistics for Biomedical Egieers 1 Populatio parameters ad Sample Statistics October 25, 2018 BIM 105 Probability ad Statistics for Biomedical Egieers 2 Ifereces
More informationThis chapter focuses on two experimental designs that are crucial to comparative studies: (1) independent samples and (2) matched pair samples.
Chapter 9 & : Comparig Two Treatmets: This chapter focuses o two eperimetal desigs that are crucial to comparative studies: () idepedet samples ad () matched pair samples Idepedet Radom amples from Two
More informationMBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS
MBACATÓLICA Quatitative Methods Miguel Gouveia Mauel Leite Moteiro Faculdade de Ciêcias Ecoómicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS MBACatólica 006/07 Métodos Quatitativos
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationStat 225 Lecture Notes Week 7, Chapter 8 and 11
Normal Distributio Stat 5 Lecture Notes Week 7, Chapter 8 ad Please also prit out the ormal radom variable table from the Stat 5 homepage. The ormal distributio is by far the most importat distributio
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationMathacle. PSet Stats, Concepts In Statistics Level Number Name: Date:
PSet ----- Stats, Cocepts I Statistics 7.3. Cofidece Iterval for a Mea i Oe Sample [MATH] The Cetral Limit Theorem. Let...,,, be idepedet, idetically distributed (i.i.d.) radom variables havig mea µ ad
More informationTopic 6 Sampling, hypothesis testing, and the central limit theorem
CSE 103: Probability ad statistics Fall 2010 Topic 6 Samplig, hypothesis testig, ad the cetral limit theorem 61 The biomial distributio Let X be the umberofheadswhe acoiofbiaspistossedtimes The distributio
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More informationS160 #12. Review of Large Sample Result for Sample Proportion
S160 #12 Samplig Distributio of the Proportio, Part 2 JC Wag February 25, 2016 Review of Large Sample Result for Sample Proportio Recall that for large sample (ie, sample size is large, say p > 5 ad (1
More informationTests of Hypotheses Based on a Single Sample (Devore Chapter Eight)
Tests of Hypotheses Based o a Sigle Sample Devore Chapter Eight MATH-252-01: Probability ad Statistics II Sprig 2018 Cotets 1 Hypothesis Tests illustrated with z-tests 1 1.1 Overview of Hypothesis Testig..........
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationCommon Large/Small Sample Tests 1/55
Commo Large/Small Sample Tests 1/55 Test of Hypothesis for the Mea (σ Kow) Covert sample result ( x) to a z value Hypothesis Tests for µ Cosider the test H :μ = μ H 1 :μ > μ σ Kow (Assume the populatio
More informationChapter 8 Interval Estimation
Iterval Estimatio Learig Objectives 1. Kow how to costruct ad iterpret a iterval estimate of a populatio mea ad / or a populatio proportio.. Uderstad ad be able to compute the margi of error. 3. Lear about
More informationBIOS 4110: Introduction to Biostatistics. Breheny. Lab #9
BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous
More informationStatistics 20: Final Exam Solutions Summer Session 2007
1. 20 poits Testig for Diabetes. Statistics 20: Fial Exam Solutios Summer Sessio 2007 (a) 3 poits Give estimates for the sesitivity of Test I ad of Test II. Solutio: 156 patiets out of total 223 patiets
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More information- E < p. ˆ p q ˆ E = q ˆ = 1 - p ˆ = sample proportion of x failures in a sample size of n. where. x n sample proportion. population proportion
1 Chapter 7 ad 8 Review for Exam Chapter 7 Estimates ad Sample Sizes 2 Defiitio Cofidece Iterval (or Iterval Estimate) a rage (or a iterval) of values used to estimate the true value of the populatio parameter
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More information1 Review of Probability & Statistics
1 Review of Probability & Statistics a. I a group of 000 people, it has bee reported that there are: 61 smokers 670 over 5 960 people who imbibe (drik alcohol) 86 smokers who imbibe 90 imbibers over 5
More informationLesson 7: Estimation 7.3 Estimation of Population Proportio. 1-PropZInterval
Lesso 7: Estimatio 7.3 Estimatio of Populatio Proportio p 1-PropZIterval October 18 Goals Preview Compute a cofidece iterval of the populatio proportio p. Cofidece Iterval for p We would estimate the populatio
More informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More informationChapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010, 2007, 2004 Pearso Educatio, Ic. Comparig Two Proportios Read the first two paragraphs of pg 504. Comparisos betwee two percetages are much more commo
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationAnalysis of Experimental Data
Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both
More informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hard-copy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
More informationChapter 20. Comparing Two Proportions. BPS - 5th Ed. Chapter 20 1
Chapter 0 Comparig Two Proportios BPS - 5th Ed. Chapter 0 Case Study Machie Reliability A study is performed to test of the reliability of products produced by two machies. Machie A produced 8 defective
More informationY i n. i=1. = 1 [number of successes] number of successes = n
Eco 371 Problem Set # Aswer Sheet 3. I this questio, you are asked to cosider a Beroulli radom variable Y, with a success probability P ry 1 p. You are told that you have draws from this distributio ad
More informationIntroduction There are two really interesting things to do in statistics.
ECON 497 Lecture Notes E Page 1 of 1 Metropolita State Uiversity ECON 497: Research ad Forecastig Lecture Notes E: Samplig Distributios Itroductio There are two really iterestig thigs to do i statistics.
More informationContinuous Data that can take on any real number (time/length) based on sample data. Categorical data can only be named or categorised
Questio 1. (Topics 1-3) A populatio cosists of all the members of a group about which you wat to draw a coclusio (Greek letters (μ, σ, Ν) are used) A sample is the portio of the populatio selected for
More informationSample Size Determination (Two or More Samples)
Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie
More informationS160 #12. Sampling Distribution of the Proportion, Part 2. JC Wang. February 25, 2016
S160 #12 Samplig Distributio of the Proportio, Part 2 JC Wag February 25, 2016 Outlie 1 Estimatig Proportio Usig Itervals Cofidece Iterval for the Populatio Proportio iclicker Questios 2 JC Wag (WMU) S160
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationConfidence Intervals
Cofidece Itervals Berli Che Deartmet of Comuter Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Referece: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chater 5 & Teachig Material Itroductio
More informationChapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010 Pearso Educatio, Ic. Comparig Two Proportios Comparisos betwee two percetages are much more commo tha questios about isolated percetages. Ad they are more
More informationBIOSTATISTICS. Lecture 5 Interval Estimations for Mean and Proportion. dr. Petr Nazarov
Microarray Ceter BIOSTATISTICS Lecture 5 Iterval Estimatios for Mea ad Proportio dr. Petr Nazarov 15-03-013 petr.azarov@crp-sate.lu Lecture 5. Iterval estimatio for mea ad proportio OUTLINE Iterval estimatios
More informationSection 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis
Sectio 9.2 Tests About a Populatio Proportio P H A N T O M S Parameters Hypothesis Assess Coditios Name the Test Test Statistic (Calculate) Obtai P value Make a decisio State coclusio Sectio 9.2 Tests
More information24.1 Confidence Intervals and Margins of Error
24.1 Cofidece Itervals ad Margis of Error Essetial Questio: How do you calculate a cofidece iterval ad a margi of error for a populatio proportio or populatio mea? Resource Locker Explore Idetifyig Likely
More informationConfidence intervals summary Conservative and approximate confidence intervals for a binomial p Examples. MATH1005 Statistics. Lecture 24. M.
MATH1005 Statistics Lecture 24 M. Stewart School of Mathematics ad Statistics Uiversity of Sydey Outlie Cofidece itervals summary Coservative ad approximate cofidece itervals for a biomial p The aïve iterval
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More information2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2
Chapter 8 Comparig Two Treatmets Iferece about Two Populatio Meas We wat to compare the meas of two populatios to see whether they differ. There are two situatios to cosider, as show i the followig examples:
More informationSimple Random Sampling!
Simple Radom Samplig! Professor Ro Fricker! Naval Postgraduate School! Moterey, Califoria! Readig:! 3/26/13 Scheaffer et al. chapter 4! 1 Goals for this Lecture! Defie simple radom samplig (SRS) ad discuss
More informationComparing Two Populations. Topic 15 - Two Sample Inference I. Comparing Two Means. Comparing Two Pop Means. Background Reading
Topic 15 - Two Sample Iferece I STAT 511 Professor Bruce Craig Comparig Two Populatios Research ofte ivolves the compariso of two or more samples from differet populatios Graphical summaries provide visual
More informationInterval Estimation (Confidence Interval = C.I.): An interval estimate of some population parameter is an interval of the form (, ),
Cofidece Iterval Estimatio Problems Suppose we have a populatio with some ukow parameter(s). Example: Normal(,) ad are parameters. We eed to draw coclusios (make ifereces) about the ukow parameters. We
More informationConfidence Level We want to estimate the true mean of a random variable X economically and with confidence.
Cofidece Iterval 700 Samples Sample Mea 03 Cofidece Level 095 Margi of Error 0037 We wat to estimate the true mea of a radom variable X ecoomically ad with cofidece True Mea μ from the Etire Populatio
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationEconomics Spring 2015
1 Ecoomics 400 -- Sprig 015 /17/015 pp. 30-38; Ch. 7.1.4-7. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 1-7 of Groeber text ad all relevat lectures
More informationClass 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 23 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 2017 by D.B. Rowe 1 Ageda: Recap Chapter 9.1 Lecture Chapter 9.2 Review Exam 6 Problem Solvig Sessio. 2
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationThis exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
More informationSTAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)
STAT 350 Hadout 9 Samplig Distributio, Cetral Limit Theorem (6.6) A radom sample is a sequece of radom variables X, X 2,, X that are idepedet ad idetically distributed. o This property is ofte abbreviated
More informationChapter two: Hypothesis testing
: Hypothesis testig - Some basic cocepts: - Data: The raw material of statistics is data. For our purposes we may defie data as umbers. The two kids of umbers that we use i statistics are umbers that result
More informationST 305: Exam 3 ( ) = P(A)P(B A) ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) σ X 2 = σ a+bx. σ ˆp. σ X +Y. σ X Y. σ X. σ Y. σ n.
ST 305: Exam 3 By hadig i this completed exam, I state that I have either give or received assistace from aother perso durig the exam period. I have used o resources other tha the exam itself ad the basic
More information