April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE

Size: px
Start display at page:

Download "April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE"

Transcription

1 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 itervals ad hypothesis testig Similar material appears i Chapter 4 of your text We ot as iterested i substitutig umbers ito formulas ad lookig up probabilities i tables, but I left the elemetary examples i place 1 Itroductio So, you have a poit estimator, ad a poit estimate Now what? We kow that there is fat chace that you actually hit the mark, that you exactly have that ˆθ = θ Here, we wat to address, usig cofidece itervals: how far off are you? A closely related questio is the followig: I suspect that a coi might be rigged, ad I flip it a hudred times, ad get 75 heads? Now what? Here, we will address this questio i the framework of hypothesis testig I these settigs, sometimes the actual values of the ukow parameters, say meas, are referred to as the populatio mea or the true mea 2 Cofidece itervals 21 Baby (exact) cofidece itervals (for the populatio mea): ormal populatio, kow populatio variace Let X 1,, X be idepedet ormal radom variables all with mea µ ad variace σ 2 Here µ is ukow, but σ is kow Let X be the usual sample mea We kow that Z = X µ σ/ N(0, 1) 1

2 2 TERRY SOO Recall that for α (0, 1), z α is the umber such that P(Z z α ) = α, so that P( z α/2 Z z α/2 ) = 1 α Cosider the followig calculatio 1 α = P( z α/2 Z z α/2 ) ( = P z α/2 X µ ) σ/ z α/2 ( σ = P z α/2 X σ ) µ z α/2 ( = P µ [ σ X zα/2, X σ ] ) + z α/2 ; that is, the probability that the true mea µ lies i the radom iterval give by [ X zα/2 σ, X + zα/2 σ ] is 1 α This calculatio motivates the followig defiitio Suppose we actually observe the values of X 1,, X, ad we fid that X = x The we say that the (determiistic) iterval give by ( x zα/2 σ, x + z α/2 σ ) is a 100(1 α) % (exact, two-sided) cofidece iterval for µ Sometimes a CI is also expressed compactly as Recall that Var X = x ± z α/2 σ σ The term σ is also sometimes called the stadard error of X Let us stress that a cofidece iterval (CI) for µ is computed after we collect data, ad will be a determiistic iterval like (134, 151) The populatio mea µ is a fixed determiistic umber as well, like µ = 123 or µ = 143 Thus, oce the cofidece iterval is computed, whether µ belogs to a cofidece iterval is ot a radom evet It is icorrect to say that µ will lie i a cofidece iterval with probability 1 α; this is oly true for the radom iterval give by our motivatig calculatio Exercise 21 What is the probability that 25 [2, 3]? What is the probability that 25 [3, 5]? Exercise 22 Suppose I kow that the heights of me i Lawrece is ormally distributed with some ukow mea µ, ad I kow the

3 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE3 variace σ 2 = 16cm 2 Suppose we collect a radom sample of 8 me s heights, i cm, i Lawrece, ad have the followig data: Compute a 95 percet CI for µ 178, 173, 176, 156, 190, 175, 170, 190 Solutio We have that CI is give by ( x zα/2 σ, x + z α/2 σ ) ; where = 8, σ = 4, ad α = 005 We eed to compute x ad we eed to determie the value of z 005/2 We have that z 005/2 196 Thus the CI is give by (173228, ) What about whe the variace σ 2 is ukow? Here we use poit estimators for σ 2 For example, S 2 = 1 (X i 1 X) 2 i=1 I the case of coi flips where X i take the values 1 ad 0, we ca also use the radom variable give by X(1 X) 22 Large sample size (CLT approximate) cofidece itervals (for the populatio mea): ukow populatio variace Let X 1,, X be a radom sample (where the X i s may ot ecessary be ormal) The for large, by the cetral limit theorem ad a exercise we already did earlier i the course, we have that Z = X µ S/ N(0, 1); that is P(Z x) is approx give by P(Z x) whe is large (textbooks sometimes state 40) A 100(1 α)% (approx two-sided) cofidece iterval ca be derived i a similar way by replacig σ with a poit estimator i our earlier discussio i Sectio 21 We obtai that the radom iterval give by [ X (zα/2 ) S, X + (z α/2 ) S ] cotais the populatio mea µ with probability approximately 1 α Here, the probability is approximate sice we appealed to a versio of the cetral limit theorem (Sometimes the term S is called the estimated stadard error)

4 4 TERRY SOO Suppose we actually observe the values of X 1,, X, ad we fid that X = x ad S = s The we say that the (determiistic) iterval give by s s ) ( x zα/2, x + z α/2 is a 100(1 α)% (approx two-sided) cofidece iterval for µ Exercise 23 Suppose that i a radom sample of 50 kitches with gas cookig appliaces we moitor the CO 2 levels for a oe week period ad fid that the sample mea was (ppm) ad the stadard deviatio was (a) Calculate (approx) a 95 percet (two-sided) cofidece iterval for µ the true average CO 2 level i the populatio of all homes from which the sample was selected (b) Suppose that we assume that the sample stadard deviatio will be o greater tha 175 What sample size would be ecessary to obtai a iterval width o greater tha 50 (ppm) for a cofidece level of 95 percet? Hit: you may use the that if Z N(0, 1), the P( 196 Z 196) 095 ad P(Z 196) 0975 Solutio Sice the sample size is large = 50 > 40, we may use the large sample size cofidece itervals approximatios that are based o the CLT Thus we use the formula: x ± (z α/2 ) s So the required cofidece iterval is (60858, 69974) For the secod part of the questio, ote that the width is give by 2(196) s Sice we kow that s 175, the width is o greater tha 2(196) 175 Thus we eed to solve for i the iequality 50 2(196) 175 We easily obtai that So take = 189 I practice, it may be difficult to obtai large sample sizes I the case where we sample from a ormal distributio, it is ot ecessary to have a large sample size

5 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE5 23 Exact cofidece itervals (for the populatio mea): ormal populatio, ukow populatio variace Let X 1,, X be a radom sample where the X i are ormal radom variables with mea µ ad ukow variace σ 2 The for all 1, we have Z = X µ S/ t 1; i other words Z has t-distributio with 1 degrees of freedom A 100(1 α)% (approx two-sided) cofidece iterval ca be derived i a similar way as before Istead of appealig to the stadard ormal distributio, we appeal to the t-distributio Recall that if T ν t ν, the the umbers t α,ν are such that P(T ν t α,ν ) = α We obtai that the radom iterval give by [ X (tα/2 ) S, X + (tα/2 ) S ] cotais the populatio mea µ with probability 1 α Suppose we actually observe the values of X 1,, X, ad we fid that X = x ad S = s The we say that the (determiistic) iterval give by s s ) ( x tα/2, x + t α/2 is a 100(1 α)% (two-sided) cofidece iterval for µ Exercise 24 From experiece, we kow that test scores for a certai stadardized test ca be modelled usig the ormal distributio; that is, if X is the test score of a radomly sampled studet, the X is ormally distributed with mea µ ad variace σ 2, for some µ ad some (ukow) σ > 0 Suppose that i a simple radom sample of 8 studets we have the followig test scores: 500, 620, 520, 700, 562, 658, 550, 656 Fid a 95 percet cofidece iterval for the true mea µ Solutio We eed to compute x ad s; this ca be doe with your calculator We obtai that x = s = From the tables, we have that t 005/2,7 = 2365 So the required CI is give by ± 6087 or (53488, 65662)

6 6 TERRY SOO 24 Large sample size (CLT approximate) cofidece itervals (for the populatio mea): proportios Let X 1,, X be a simple radom sample where the X i are Beroulli radom variables with parameter p (0, 1) Note that EX i = µ = p Note that X is the proportio of successes or oes that occur i trials I this cotext, we ofte write X = ˆP The observed value of ˆP is ofte deoted by ˆp A versio of the cetral limit theorem implies that Z = ˆP p ˆP (1 ˆP )/ N(0, 1); where the approximatio is good if both p, (1 p) 10 A 100(1 α)% (approx two-sided) cofidece iterval ca be derived i a similar way as before We obtai that the radom iterval give by [ ˆP (1 X (zα/2 ) ˆP ) ˆP (1, X + (zα/2 ) ˆP ) ] cotais the populatio proportio p with probability 1 α Suppose we actually observe the values of X 1,, X, ad we fid that ˆP = ˆp The we say that the (determiistic) iterval give by ( ˆp z α/2 ˆp(1 ˆp) ˆp(1 ˆp) ), ˆp + z α/2 is a 100(1 α)% (approx two-sided) cofidece iterval for p Note that i order to esure that the approximate CI is reliable, we check that ˆp, ˆp(1 ˆp) 10 Example doe i class 25 We wish to ivestigate p, the proportio of people with the disease Deadly-Virus who die withi three years after receivig a ewly discovered treatmet We have take a radom sample of 187 people with Deadly-Virus ad gave them the ew treatmet Three years later, 170 of these patiets have died (a) Fid a 97 percet CI for p (b) Suppose that i the future we wish to carry out a secod study i which we will costruct a 99 percet cofidece iterval which will estimate p to withi 0001 Usig the data give i the setup as a pilot study, estimate the sample size eeded Solutio (a) We have that = 187 ad ˆp = 170 Note that ˆp, ˆp(1 ˆp) 10 We 187 eed to fid z 003/2 From the tables, we have that P(Z 217)

7 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE The iverse ormal fuctio o your calculator gives P(Z ) 0985 Thus we obtai that the CI is give by ± (b) To estimate the size, we solve for i the equatio 0001 = z 001/2 σ/ We have that the tables give z 001/2 2576, ad sice we are supposed to use the data as a pilot study, we estimate that σ ˆp(1 ˆp), where ˆp = 170/187 Some algebra yields Exercise 26 We have a magicia s coi, ad we wat to ivestigate p (0, 1), the probability that o a flip of the coi it come up heads We wat to costruct a 95 percet cofidece iterval for p (a) Estimate how may flips do we eed to do to estimate p to withi 0001 (b) Suppose I flipped the coi 100 times ad got 23 heads ad 77 tails What is the CI? Solutio Note that z 0025 = table 196 (a) Let X Ber(p), so that V ar(x) = σ 2 = (1 p)(p) We kow that σ 2 1/4; thus σ 1/2 We solve for i the equatio: 0001 = 196(05)/, which gives = (b) We have that ˆp = 023 Thus the CI is give by 023 ± Hypothesis testig 31 Itroductio to hypothesis testig Suppose I wat to determie whether a coi is ufair What if I flipped the coi 30 times, ad foud that I got 29 heads? Would this be eough to covice you that the coi was ufair? What if I oly got 28 heads? Cosider the followig set-up, called hypothesis testig It seems reasoable that without ay evidece to the cotrary, we should assume that the coi is fair; this is called the ull hypothesis, ad is ofte deoted by H 0 The hypothesis that the coi is ot fair, is called the alterate hypothesis, ad is ofte deoted by H a or H 1 Suppose we set the followig criteria: if we flip the coi 30 times ad if we see less tha or equal to 2 heads, or more tha or equal to 28 heads, we deem

8 8 TERRY SOO (or guess) the coi to be ufair, sice otherwise the coi is fair, ad we have witessed a ulikely/extreme evet; this is called the critical rage or rejectio rage (Of course there is othig stoppig a coi from comig up heads 100 times i row; i fact, it would ot be a fair coi if this could ot occur) Let p (0, 1) be the true probability that a flip come up heads Let X Bi(30, p), so that X is the umber of heads i 30 flips Notice that if the coi is fair, the p = 1/2, ad we ca exactly compute all the probabilities associated with X We refer to X as the test statistic, ad we ca express the critical rage as the uio of the evets X 28 ad X 2 I We ca easily compute P(X 2 or X 28) ; this is exactly the probability that the ull hypothesis is correct, but rejected i favour of the alterate hypothesis; whe this occurs, this is called a type 1 error, ad the probability of committig a type I error is referred to the level of sigificace of the test is ofte deoted by a Greek letter α A type II error occurs whe we do ot reject the ull hypothesis, ad it is false; the probability of this evet is deoted by the Greek letter β, ad may be difficult to estimate of compute sice if the ull hypothesis is false, we eed to kow p Notice that if X does ot fall ito the rejectio rage, the that is hardly evidece that p = 1/2 Notice also the calculatio that P(X or X 28) is small is also a calculatio that assumes p = 1/2; maybe p = 0499 I hypothesis testig, we ca oly reject the ull hypothesis or retai it, but we do ot prove it; this is aki to crimial proceedigs i the Uited States, where people are declared o-guilty, but hardly ever is a verdict of iocet give After we collect data, ad have that X = x obs, we will easily see if X falls i the rejectio rage ad whether we reject the ull hypothesis or ot; if it falls ito the rejectio rage the we say that observed test statistic is sigificat A slightly differet, ad perhaps better approach is to compute the P-value The P -value is the lowest level of sigificace at which the observed value of the test statistic is sigificat Suppose that x obs = 3, the we have that P -value = P(X 3 or X 27) = 2P(X 3), where we do the computatio assumig p = 1/2 Notice that by defiitio of the P -value, we reject the ull hypothesis with sigificace level α if ad oly if P -value α Note that the P -value is defied without preselectig a sigificace level Thus the P -value ca be used

9 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE9 to decide whether a ull hypothesis would be rejected with differet sigificace levels For proportioal data we kow that the uderlyig distributio is a Beroulli, ad thus a atural test statistic is the sample mea, which is a Biomial radom variable Note that i the previous discussio if we did ot have the biomial tables or our calculator, we could still appeal to the cetral limit, sice uder the ull hypothesis that p = 1/2, we have that the distributio of Z = X 1/2 05/ 30, ca be approximated by a the distributio a stadard ormal radom variable Z We ca use Z are our test statistic, ad defie the rejectio rage based o Z For example, if we wated a test with a level of sigificace α, we kow that P( z α/2 Z z α/2 ) CLT P( z α Z z α/2 ) = 1 α; thus we could defie a rejectio rage as the uio of { Z < z α/2 } ad { Z > z α/2 } 32 Two-sided test o proportios CLT based We wish to ivestigate p, the proportio of people who will pass a certai math course usig a ew textbook We are iterested i testig with a sigificace of 005 whether the ew textbook will effect the pass-rate Usig the old textbook, it is kow that 70 percet of studets will pass the course Suppose we siged up 123 studets to take this math course usig the ew textbook (a) What should H 0 be? (b) What should H a be? (c) Appealig to the cetral limit theorem, what is a suitable test statistic? (d) What is the rejectio rage? Suppose that the course is over ad 99 studets have passed the course (i) What is the observed value of the test statistic? (ii) What is the P -value? (iii) What are your coclusios? Solutio (a) We take H 0 : p = 07 (b) We take H a : p 07

10 10 TERRY SOO (c) Let X i be idepedet Beroulli radom variables with parameter p, so that if X i = 1, the the i th has passed the course Set X to be the usual sample mea, ad cosider the test statistic give by Z = X 07 07(03)/123 Note that uder H 0, each X i has mea 07 ad variace 07(03), ad Z N(0, 1) (d) Notice that if Z N(0, 1), the P( z α/2 Z z α/2 ) CLT P( z α/2 Z z α/2 ) = 1 α Thus with α = 005, we take the rejectio rage to be the uio of {Z > 1960} ad {Z < 1960} (i) Z = (ii) P -value CLT 2P(Z > 254) = = (iii) We reject H 0 i favour of H a 33 Oe-sided test o proportios We wish to ivestigate p, the proportio of people with the disease Deadly-Virus who die withi three years after receivig a ewly discovered treatmet We are iterested i testig, with a level of sigificace of 001, whether the ew treatmet will improve life expectacy Without the treatmet, it is kow that 95 percet of the patiets die withi three years Suppose that we siged up 187 patiets to take the ew drug (a) What should H 0 be? (b) What should H a be? (c) Appealig to the cetral limit theorem, what is a appropriate test statistic? (d) What is the rejectio rage? Suppose that after three years 170 of patiets have died (i) What is the observed value of the test statistic? (ii) What is the P -value? (iii) What are your coclusios? Solutio (a) We take H 0 : p = 095 (b) We take H a : p < 095 Here we do ot take H a : p 095, sice we do ot care if p > 095; who wats a less effective drug? (c) Let X i be idepedet Beroulli radom variables with parameter p, so that if X i = 1, the the i th perso has died withi three years Set X to be the usual sample mea, ad cosider the test statistic

11 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE 11 give by Z = X (005)/187 Note that uder H 0, each X i has mea 095 ad variace 095(005), ad Z N(0, 1) (d) Notice that if Z N(0, 1), the P(Z < z α ) CLT P(Z < z α ) = α Thus with α = 001, we take the rejectio rage to be {Z < 2326} (i) Z = (ii) P -value CLT P(Z > 257) = table = (iii) We reject H 0 i favour of H a Let us remark that we have defied the rejectio rage of a test statistic Z to be a evet; i particular, it may be somethig of the form {Z > z α } for some z α R Sometimes, the rejectio rage is defied to be a set of real umbers {z R : z > z α } I what follows, we will treat the cases, whe we do ot assume aythig about the uderlyig distributio, ad where we kow that the uderlyig distributio is ormal 34 CLT based We wish to ivestigate µ, the true mea price of a burger i Kasas city It is kow that i the Uited States the mea price of a burger is 12 dollars, ad we wat to test with a level of sigificace 005 to see whether the price i KC is differet Suppose we will take a radom sample of 81 restaurats i KC (a) What should H 0 be? (b) What should H a be? (c) What is a appropriate test statistic? (d) What is the rejectio rage? Suppose that we foud that the sample mea price was 13 dollars with a sample stadard deviatio of 3 dollars (i) What is the observed value of the test statistic? (ii) What is the P -value? (iii) What are your coclusios? Solutio (a) We take H 0 : µ = 12 (b) We take H a : µ 12 (c) Let (X i ) be a radom sample, where EX i = µ, so that X i is the price of a burger at the ith restaurat Set X to be the usual sample

12 12 TERRY SOO mea, ad S 2 to be the usual sample variace, ad cosider the test statistic give by Z = X 12 S/9 Note that uder H 0, the mea of each X i is 12 Thus, uder H 0, a versio of the cetral limit theorem gives that Z N(0, 1) (d) Notice that if Z N(0, 1), the P( z α/2 Z z α/2 ) CLT P( z α/2 Z z α/2 ) = 1 α Thus with α = 005, we take the rejectio rage to be the uio of {Z > 1960} ad {Z < 1960} (i) We have that Z = 3 (ii) P -value CLT 2P(Z > 3) = (iii) We reject H 0 i favour of H a Exercise 31 Do the previous example with the followig chage: we wat to test to see if a Burger i KC is cheaper Solutio (a) We take H 0 : µ = 12 (b) We take H a : µ < 12 (c) Let (X i ) be a radom sample, where EX i = µ, so that X i is the price of a burger at the ith restaurat Set X to be the usual sample mea, ad S 2 to be the usual sample variace, ad cosider the test statistic give by Z = X 12 S/9 Note that uder H 0, the mea of each X i is 12 Thus, uder H 0, a versio of the cetral limit theorem gives that Z N(0, 1) (d) Notice that if Z N(0, 1), the P(Z < z α ) CLT P(Z < z α ) = α Thus with α = 005, we take the rejectio rage to be {Z < 1645} (i) We have that Z = 3 (ii) P -value CLT P(Z < 3) = (iii) We do ot reject H 0 35 Normal populatio, ukow variace We wish to ivestigate µ, the true mea height (i cm) of wome (aged 18-55) i Lawrece It is kow that i the Uited States the mea height of wome is 165 cm, ad we wat to test with a level of sigificace 0001 to see whether the height i Lawrece is differet Suppose we will take a radom sample of 25 wome i Lawrece, ad assume that

13 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE 13 their heights are ormally distributed with the same mea ad variace (a) What should H 0 be? (b) What should H a be? (c) What is a appropriate test statistic? (d) What is the rejectio rage? Suppose that we foud that the sample mea height was 171 cm with a sample stadard deviatio of 10 cm (i) What is the observed value of the test statistic? (ii) What is the P -value? (iii) What are your coclusios? (iv) Would your reject H 0 with a sigificace level of 0005? (v) Costruct a 999 percet CI for µ Solutio (a) We take H 0 : µ = 165 (b) We take H a : µ 165 (c) Let X i be idepedet ormal radom variables all with mea µ, so that X i is the height of the ith wome Set X to be the usual sample mea, ad S 2 to be the usual sample variace, ad cosider the test statistic give by T = X 165 S/5 Note that uder H 0, the mea of each X i is 165 Thus, uder H 0, we have that T t 24 (d) Notice that if T t 24, the P( t α/2,24 T t α/2,24 ) = 1 α Thus with α = 0001, we take the rejectio rage to be the uio of {T > 3745} ad {T < 3745} (i) We have that T = 3 (ii) Note that P(T > 2797) = table 0005 ad P(T > 3091) = table Thus P -value table 0005 The calculator gives P(T > 3) = calc , so that P -value = calc (iii) We do ot reject H 0 (iv) No, sice P -value 0005 (v) We kow that the CI is give by x ± s t α/2, 1,

14 14 TERRY SOO where α = 0001, x = 171, s = 10, = 25, ad t α/2, 1 = table 3745 Thus the CI is (16351, 17849) Exercise 32 Treat the example i Sectio 35, assumig we kow that the true stadard deviatio is σ = 10 Solutio The oly differece is that we get the advatage that we ca defie Z = X /2 to be the test statistic, ad the uder H 0, we have that Z N(0, 1) Istead of t α/2 we get to use z α/2 The rejectio rage is give by the uio of {Z > 3290} ad {Z < 3290} The observed value of the test statistic remais 3, ad we still do ot reject the ull hypothesis However, the P -value = 2P(Z > 3) = table 00026, is smaller, so much so that we would reject H 0 with a sigificace level of 0005 The required CI also is smaller, ad is give by x ± σ z α/2, where α = 0001, x = 171, σ = 10, = 25, ad z α/2 = table 3290 Thus the CI is (16442, 17758) 4 The coectio betwee hypothesis testig ad cofidece itervals There is a close coectio to betwee hypothesis testig ad cofidece itervals Cosider the case where we are samplig from a ormal distributio with ukow variace, ad are iterested i the true mea µ Give observed data (x 1,, x ), a 100(1 α) percet two-sided cofidece iterval for µ is give by ( x s t α/2, 1, x + s t α/2, 1 ) Let µ 0 be some umber If we wat to test the ull hypothesis that µ = µ 0, agaist the alterate hypothesis that µ µ 0, with a sigificace level of α, the with the test statistic T = X µ 0 S/, the rejectio rage is give by the uio of { T > tα/2, 1 } ad { T < tα/2, 1 }

15 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE 15 Some easy algebra yields that whe we observe the value T = t, we have that t > t α/2, 1 or t < t α/2, 1, if ad oly if µ 0 ( x s t α/2, 1, x + s t α/2, 1 ) I other words, we reject H 0 if ad oly if µ 0 does ot lie i the correspodig cofidece iterval Exercise 41 The coectio described i Sectio 4 does ot hold i the case of proportioal data where build a cofidece iterval ad test statistic by appealig to the cetral limit theorem Explai Solutio The differece is that i the case of proportios, uder the ull hypothesis say p = p 0, we kow that the variace is p 0 (1 p 0 ), but i the cofidece iterval settig, we use a estimator for the variace ˆp(1 ˆp)

A quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population

A 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 information

BIOS 4110: Introduction to Biostatistics. Breheny. Lab #9

BIOS 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 information

Common Large/Small Sample Tests 1/55

Common 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 information

Data Analysis and Statistical Methods Statistics 651

Data 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 information

Chapter 8: Estimating with Confidence

Chapter 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 information

MATH/STAT 352: Lecture 15

MATH/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 information

Class 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Class 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 information

Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.

Statistical 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 information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY 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 information

Overview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions

Overview. 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 information

Topic 9: Sampling Distributions of Estimators

Topic 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 information

HYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018

HYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018 HYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018 We are resposible for 2 types of hypothesis tests that produce ifereces about the ukow populatio mea, µ, each of which has 3 possible

More information

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara

Econ 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 information

Topic 9: Sampling Distributions of Estimators

Topic 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 information

Tests of Hypotheses Based on a Single Sample (Devore Chapter Eight)

Tests 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 information

Introduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 3

Introduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 3 Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd- Numbered Ed- of- Chapter Exercises: Chapter 3 (This versio August 17, 014) 015 Pearso Educatio, Ic. Stock/Watso

More information

Math 140 Introductory Statistics

Math 140 Introductory Statistics 8.2 Testig a Proportio Math 1 Itroductory Statistics Professor B. Abrego Lecture 15 Sectios 8.2 People ofte make decisios with data by comparig the results from a sample to some predetermied stadard. These

More information

Topic 9: Sampling Distributions of Estimators

Topic 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 information

Direction: This test is worth 150 points. You are required to complete this test within 55 minutes.

Direction: 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 information

Recall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and non-users, x - y.

Recall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and non-users, x - y. Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad o-users, x - y. Such studies are sometimes viewed

More information

Math 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency

Math 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency Math 152. Rumbos Fall 2009 1 Solutios to Review Problems for Exam #2 1. I the book Experimetatio ad Measuremet, by W. J. Youde ad published by the by the Natioal Sciece Teachers Associatio i 1962, the

More information

Expectation and Variance of a random variable

Expectation 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 information

Discrete Mathematics for CS Spring 2008 David Wagner Note 22

Discrete 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

Frequentist Inference

Frequentist 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 information

Stat 200 -Testing Summary Page 1

Stat 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 information

1 Inferential Methods for Correlation and Regression Analysis

1 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 information

Agreement of CI and HT. Lecture 13 - Tests of Proportions. Example - Waiting Times

Agreement 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 information

STAC51: Categorical data Analysis

STAC51: Categorical data Analysis STAC51: Categorical data Aalysis Mahida Samarakoo Jauary 28, 2016 Mahida Samarakoo STAC51: Categorical data Aalysis 1 / 35 Table of cotets Iferece for Proportios 1 Iferece for Proportios Mahida Samarakoo

More information

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4

MATH 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 information

Last Lecture. Wald Test

Last Lecture. Wald Test Last Lecture Biostatistics 602 - Statistical Iferece Lecture 22 Hyu Mi Kag April 9th, 2013 Is the exact distributio of LRT statistic typically easy to obtai? How about its asymptotic distributio? For testig

More information

Stat 319 Theory of Statistics (2) Exercises

Stat 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 information

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals

7-1. 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 information

Simulation. Two Rule For Inverting A Distribution Function

Simulation. 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 information

This is an introductory course in Analysis of Variance and Design of Experiments.

This 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 information

Lecture 6 Simple alternatives and the Neyman-Pearson lemma

Lecture 6 Simple alternatives and the Neyman-Pearson lemma STATS 00: Itroductio to Statistical Iferece Autum 06 Lecture 6 Simple alteratives ad the Neyma-Pearso lemma Last lecture, we discussed a umber of ways to costruct test statistics for testig a simple ull

More information

Chapter 13: Tests of Hypothesis Section 13.1 Introduction

Chapter 13: Tests of Hypothesis Section 13.1 Introduction Chapter 13: Tests of Hypothesis Sectio 13.1 Itroductio RECAP: Chapter 1 discussed the Likelihood Ratio Method as a geeral approach to fid good test procedures. Testig for the Normal Mea Example, discussed

More information

Sampling Distributions, Z-Tests, Power

Sampling Distributions, Z-Tests, Power Samplig Distributios, Z-Tests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace

More information

Since X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain

Since X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain Assigmet 9 Exercise 5.5 Let X biomial, p, where p 0, 1 is ukow. Obtai cofidece itervals for p i two differet ways: a Sice X / p d N0, p1 p], the variace of the limitig distributio depeds oly o p. Use the

More information

1 Constructing and Interpreting a Confidence Interval

1 Constructing and Interpreting a Confidence Interval Itroductory Applied Ecoometrics EEP/IAS 118 Sprig 2014 WARM UP: Match the terms i the table with the correct formula: Adrew Crae-Droesch Sectio #6 5 March 2014 ˆ Let X be a radom variable with mea µ ad

More information

This chapter focuses on two experimental designs that are crucial to comparative studies: (1) independent samples and (2) matched pair samples.

This 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 information

Chapter 6 Sampling Distributions

Chapter 6 Sampling Distributions Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to

More information

Mathematical Notation Math Introduction to Applied Statistics

Mathematical Notation Math Introduction to Applied Statistics Mathematical Notatio Math 113 - Itroductio to Applied Statistics Name : Use Word or WordPerfect to recreate the followig documets. Each article is worth 10 poits ad ca be prited ad give to the istructor

More information

STATISTICAL INFERENCE

STATISTICAL INFERENCE STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample

More information

Section 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis

Section 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 information

Random Variables, Sampling and Estimation

Random 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 information

University of California, Los Angeles Department of Statistics. Hypothesis testing

University 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 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

- 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 information

Confidence intervals summary Conservative and approximate confidence intervals for a binomial p Examples. MATH1005 Statistics. Lecture 24. M.

Confidence 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 information

Continuous Data that can take on any real number (time/length) based on sample data. Categorical data can only be named or categorised

Continuous 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 information

STAT431 Review. X = n. n )

STAT431 Review. X = n. n ) STAT43 Review I. Results related to ormal distributio Expected value ad variace. (a) E(aXbY) = aex bey, Var(aXbY) = a VarX b VarY provided X ad Y are idepedet. Normal distributios: (a) Z N(, ) (b) X N(µ,

More information

Lecture 5: Parametric Hypothesis Testing: Comparing Means. GENOME 560, Spring 2016 Doug Fowler, GS

Lecture 5: Parametric Hypothesis Testing: Comparing Means. GENOME 560, Spring 2016 Doug Fowler, GS Lecture 5: Parametric Hypothesis Testig: Comparig Meas GENOME 560, Sprig 2016 Doug Fowler, GS (dfowler@uw.edu) 1 Review from last week What is a cofidece iterval? 2 Review from last week What is a cofidece

More information

Chapter 11: Asking and Answering Questions About the Difference of Two Proportions

Chapter 11: Asking and Answering Questions About the Difference of Two Proportions Chapter 11: Askig ad Aswerig Questios About the Differece of Two Proportios These otes reflect material from our text, Statistics, Learig from Data, First Editio, by Roxy Peck, published by CENGAGE Learig,

More information

Interval Estimation (Confidence Interval = C.I.): An interval estimate of some population parameter is an interval of the form (, ),

Interval 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 information

Comparing Two Populations. Topic 15 - Two Sample Inference I. Comparing Two Means. Comparing Two Pop Means. Background Reading

Comparing 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 information

Lecture Notes 15 Hypothesis Testing (Chapter 10)

Lecture Notes 15 Hypothesis Testing (Chapter 10) 1 Itroductio Lecture Notes 15 Hypothesis Testig Chapter 10) Let X 1,..., X p θ x). Suppose we we wat to kow if θ = θ 0 or ot, where θ 0 is a specific value of θ. For example, if we are flippig a coi, we

More information

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced

More information

6 Sample Size Calculations

6 Sample Size Calculations 6 Sample Size Calculatios Oe of the major resposibilities of a cliical trial statisticia is to aid the ivestigators i determiig the sample size required to coduct a study The most commo procedure for determiig

More information

1036: Probability & Statistics

1036: Probability & Statistics 036: Probability & Statistics Lecture 0 Oe- ad Two-Sample Tests of Hypotheses 0- Statistical Hypotheses Decisio based o experimetal evidece whether Coffee drikig icreases the risk of cacer i humas. A perso

More information

Parameter, Statistic and Random Samples

Parameter, 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 information

5. Likelihood Ratio Tests

5. Likelihood Ratio Tests 1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,

More information

Chapter 23: Inferences About Means

Chapter 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 information

KLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions

KLMED8004 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 information

Sample Size Determination (Two or More Samples)

Sample 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 information

MA238 Assignment 4 Solutions (part a)

MA238 Assignment 4 Solutions (part a) (i) Sigle sample tests. Questio. MA38 Assigmet 4 Solutios (part a) (a) (b) (c) H 0 : = 50 sq. ft H A : < 50 sq. ft H 0 : = 3 mpg H A : > 3 mpg H 0 : = 5 mm H A : 5mm Questio. (i) What are the ull ad alterative

More information

NO! This is not evidence in favor of ESP. We are rejecting the (null) hypothesis that the results are

NO! This is not evidence in favor of ESP. We are rejecting the (null) hypothesis that the results are Hypothesis Testig Suppose you are ivestigatig extra sesory perceptio (ESP) You give someoe a test where they guess the color of card 100 times They are correct 90 times For guessig at radom you would expect

More information

Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date: Confidence Interval Guesswork with Confidence

Mathacle. 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 information

Inferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process.

Inferential 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 information

Statistics 511 Additional Materials

Statistics 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 information

Kurskod: TAMS11 Provkod: TENB 21 March 2015, 14:00-18:00. English Version (no Swedish Version)

Kurskod: TAMS11 Provkod: TENB 21 March 2015, 14:00-18:00. English Version (no Swedish Version) Kurskod: TAMS Provkod: TENB 2 March 205, 4:00-8:00 Examier: Xiagfeg Yag (Tel: 070 2234765). Please aswer i ENGLISH if you ca. a. You are allowed to use: a calculator; formel -och tabellsamlig i matematisk

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 9

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 9 Hypothesis testig PSYCHOLOGICAL RESEARCH (PYC 34-C Lecture 9 Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I

More information

Module 1 Fundamentals in statistics

Module 1 Fundamentals in statistics Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly

More information

z is the upper tail critical value from the normal distribution

z is the upper tail critical value from the normal distribution Statistical Iferece drawig coclusios about a populatio parameter, based o a sample estimate. Populatio: GRE results for a ew eam format o the quatitative sectio Sample: =30 test scores Populatio Samplig

More information

Class 27. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Class 27. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700 Class 7 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 013 by D.B. Rowe 1 Ageda: Skip Recap Chapter 10.5 ad 10.6 Lecture Chapter 11.1-11. Review Chapters 9 ad 10

More information

Chapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.

Chapter 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 information

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:

STA 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 information

One-Sample Test for Proportion

One-Sample Test for Proportion Oe-Sample Test for Proportio Approximated Oe-Sample Z Test for Proportio CF Jeff Li, MD., PhD. November 1, 2005 c Jeff Li, MD., PhD. c Jeff Li, MD., PhD. Oe Sample Test for Proportio, 1 I DM-TKR Data,

More information

AP Statistics Review Ch. 8

AP 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 information

This exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.

This 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 information

Hypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance

Hypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?

More information

Statistics 20: Final Exam Solutions Summer Session 2007

Statistics 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 information

LESSON 20: HYPOTHESIS TESTING

LESSON 20: HYPOTHESIS TESTING LESSN 20: YPTESIS TESTING utlie ypothesis testig Tests for the mea Tests for the proportio 1 YPTESIS TESTING TE CNTEXT Example 1: supervisor of a productio lie wats to determie if the productio time of

More information

Y i n. i=1. = 1 [number of successes] number of successes = n

Y 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 information

Chapter 4 Tests of Hypothesis

Chapter 4 Tests of Hypothesis Dr. Moa Elwakeel [ 5 TAT] Chapter 4 Tests of Hypothesis 4. statistical hypothesis more. A statistical hypothesis is a statemet cocerig oe populatio or 4.. The Null ad The Alterative Hypothesis: The structure

More information

If, for instance, we were required to test whether the population mean μ could be equal to a certain value μ

If, for instance, we were required to test whether the population mean μ could be equal to a certain value μ STATISTICAL INFERENCE INTRODUCTION Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I oesample testig, we essetially

More information

Topic 6 Sampling, hypothesis testing, and the central limit theorem

Topic 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 information

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen) Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................

More information

Final Examination Solutions 17/6/2010

Final 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 information

Chapter 22: What is a Test of Significance?

Chapter 22: What is a Test of Significance? Chapter 22: What is a Test of Sigificace? Thought Questio Assume that the statemet If it s Saturday, the it s the weeked is true. followig statemets will also be true? Which of the If it s the weeked,

More information

1 Constructing and Interpreting a Confidence Interval

1 Constructing and Interpreting a Confidence Interval Itroductory Applied Ecoometrics EEP/IAS 118 Sprig 2014 WARM UP: Match the terms i the table with the correct formula: Adrew Crae-Droesch Sectio #6 5 March 2014 ˆ Let X be a radom variable with mea µ ad

More information

Properties and Hypothesis Testing

Properties and Hypothesis Testing Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.

More information

ENGI 4421 Confidence Intervals (Two Samples) Page 12-01

ENGI 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 information

Binomial Distribution

Binomial 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

Chapter 5: Hypothesis testing

Chapter 5: Hypothesis testing Slide 5. Chapter 5: Hypothesis testig Hypothesis testig is about makig decisios Is a hypothesis true or false? Are wome paid less, o average, tha me? Barrow, Statistics for Ecoomics, Accoutig ad Busiess

More information

Rule of probability. Let A and B be two events (sets of elementary events). 11. If P (AB) = P (A)P (B), then A and B are independent.

Rule of probability. Let A and B be two events (sets of elementary events). 11. If P (AB) = P (A)P (B), then A and B are independent. Percetile: the αth percetile of a populatio is the value x 0, such that P (X x 0 ) α% For example the 5th is the x 0, such that P (X x 0 ) 5% 05 Rule of probability Let A ad B be two evets (sets of elemetary

More information

Worksheet 23 ( ) Introduction to Simple Linear Regression (continued)

Worksheet 23 ( ) Introduction to Simple Linear Regression (continued) Worksheet 3 ( 11.5-11.8) Itroductio to Simple Liear Regressio (cotiued) This worksheet is a cotiuatio of Discussio Sheet 3; please complete that discussio sheet first if you have ot already doe so. This

More information

The standard deviation of the mean

The standard deviation of the mean Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider

More information

October 25, 2018 BIM 105 Probability and Statistics for Biomedical Engineers 1

October 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 information

Lesson 2. Projects and Hand-ins. Hypothesis testing Chaptre 3. { } x=172.0 = 3.67

Lesson 2. Projects and Hand-ins. Hypothesis testing Chaptre 3. { } x=172.0 = 3.67 Lesso 7--7 Chaptre 3 Projects ad Had-is Project I: latest ovember Project I: latest december Laboratio Measuremet systems aalysis I: latest december Project - are volutary. Laboratio is obligatory. Give

More information

Lecture 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.

Lecture 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 information

ST 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 ( ) = 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