Chapter 6 Part 5. Confidence Intervals t distribution chi square distribution. October 23, 2008

Size: px
Start display at page:

Download "Chapter 6 Part 5. Confidence Intervals t distribution chi square distribution. October 23, 2008"

Transcription

1 Chapter 6 Part 5 Cofidece Itervals t distributio chi square distributio October 23, 2008 The will be o help sessio o Moday, October 27.

2 Goal: To clearly uderstad the lik betwee probability ad cofidece itervals. Skills: Be able to calculate (1 - α)% cofidece iterval for a sample mea both for the case that the populatio variace is kow ad ot kow. Be able to explai what a cofidece iterval does ad does ot mea. Cotets: Cofidece iterval usig t distributio Page 7 Defiitio of chi square distributio Page 13 Stata commads: ci tde(df, t) ttail(df,t) ivttail(df, p)

3 Cofidece iterval for the mea where the radom variable X has ukow mea μ ad ukow stadard deviatio If is ot kow, the we will use the sample stadard deviatio s to estimate $ (i.e. s = ). So istead of havig Z = X μ ( / ) where Z - N(0,1), we have (where S is the radom variable represetig the stadard deviatio). t = X μ ( S / ) This is the t distributio (also call Studet s t distributio). Like the ormal distributio, the t distributio is a probability desity fuctio. Probability will be related to area. We will come back to cofidece itervals but first we eed to kow somethig about the t distributio i order to kow how to calculate the cofidece itervals. The t distributio, like the ormal distributio, is actually a family of distributios. However, ulike the ormal, we caot trasform each t distributio ito a sigle t equivalet of the N(0,1) distributio (i.e. stadard ormal distributio). It is, however, the case that each member of the family of t distributios is symmetric about zero. So every t distributio is cetered at zero. Therefore, the oly thig that ca chage from t distributio to t distributio is the shape of the curve. With the ormal distributio we ca chage both the locatio (the mea) ad the shape (the variace or stadard deviatio). The t distributio is a sigle parameter distributio. The parameter is called degrees of freedom ad is usually deoted by df. This is i cotrast to the ormal distributio which is a two parameter distributio with parameters ad. μ Page -1-

4 If the t distributio is based o a sample of size, the it is said to have degrees of freedom. So sample size is what distiguishes the various t distributios. Sample size determies the shape of the distributio. 1 I order to calculate t we ot oly have to estimate the populatio mea μ by the x 2 s 2 sample mea, but we also have to estimate the populatio variace by (the sample variace) or estimate the populatio stadard deviatio by s (the sample stadard deviatio). This meas that we should expect more variability with the t distributio tha with the ormal distributio (i.e. we should expect that the t distributio will have larger [or fatter] tails tha the ormal distributio). Below we have a t distributio with 3 degrees of freedom. The graph shows that the poit cuts off the upper 2.5 percet of the area uder the curve or the lower 97.5 percet of the curve. For the ormal distributio with mea = 0 ad SD = 1, the poit 1.96 would cut off these same percetages (i.e. you have to go further out o the t distributio tha o the ormal distributio to cut off oly 2.5%)..4 The desity fuctio for a t distributio with 3 degrees of freedom f(t).3.2 tde(3,3.182) = (3.182, 0.019) t I am goig to itroduce three Stata fuctios that are useful i workig with the t-distributio. The curve above is the desity for the t-distributio with 3 degrees of Page -2-

5 freedom. Notice i the graph above the elemets o the x-axis are represeted by t ad the elemets o the y-axis are represeted by f(t). So the poits o the curve are (t, f(t)). Stata fuctio tde(df, t): This fuctio allows us to graph the t distributio. We ca use it to fid the poits (t, f(t)) where t plays the role of x ad f(t) plays the role of y. Let t = The to fid f(3.182) we will use the Stata fuctio tde(df, t). The degrees of freedom (df) idicate which t distributio we are usig ad t (here 3.182) gives us the poit o the x-axis. So above we have the t distributio associated with df = 3 or sample size = 4 (i.e. df = - 1). Notice that Stata gives the fuctio as tde(, t). Stata s is ot the sample size but the degrees of freedom. I will try to cosistetly use df to avoid cofusio.. di tde(3,3.182) The poit (3.182, 0.019) is o the curve. How do we get a graph of the t desity fuctio usig tde(df,t). clear *the dofile is tdist3df.do *We kow that the t distributio is cetered at zero *but we will eed some idea of what maximum ad miimum *values we should use for t *I checked Table 5 o page 831 of Roser. The table *idicates that for the t distributio with 3 df *99% of the area uder the curve is to the left of 4.5 *ad 99.5% of the area uder the curve is to the left of 5.8. *So if we graph from -5 to 5 we should be i good shape * *I decidig how to create the values for t that we will eed *to get a graph, we have to use small icremets so that we'll have *a smooth curve. I have decided to use To get from -5 to -4 *will take 4 of the 0.25's. So to get from -5 to 5 we will eed 40 *observatios ad we eed to take oe more to get the edpoit * set obs 41 * ge t = sum(0.25) ge foft = tde(3,t) twoway (coected foft t, msymbol(oe)), ytitle(f(t))/* */xtitle(t) xlabel(-5(1)5)/* */ title(the desity fuctio for a t distributio, size(medium))/* */ subtitle(with 3 degrees of freedom, size(medium)) But what is more useful to us is to be able to take a particular value of t ad fid the Page -3-

6 area that goes with that cutoff. Or give a area, to be able to fid the cutoff. This is because area is syoymous with probability. Stata fuctio ttail(df, t): This fuctio will allow us to fid the area to the right of the cutoff give that we kow the degrees of freedom ad the value of the cutoff. Let df = 3 ad t = di ttail(3,3.182) So 2.5% is the area to the right of Notice that this meas the area to the left of is I have bee workig with oly the upper tail for the t distributio. But sice the t distributio is symmetric, we kow that the area to the left of is What happes if I put i 3 degrees of freedom ad the poit 1.96?. di ttail(3,1.96) This idicates we have more area to the right of 1.96 usig the t distributio with 3 degrees of freedom tha we would have with the N(0,1) distributio where the area would be The graph below makes this same poit. Graph that shows t 1,1 (a/2) > z 1 (a/2) N(0,1) t distributio with 3 df Page -4-

7 Stata fuctio ivttail(df,p): ivttail(df,p) provides the iverse of ttail(). If ttail(df,t) = p, the ivttail(df,p) = t, where t is the cutoff ad p is the area to the right of t. If I use ivttail ad eter the degrees of freedom (3) ad the area i the upper tail (0.025), I get back the value that cuts off that area (3.182).. di ivttail(3,0.025) Now cosider the graph below: I the graph below, the dotted lie represets a N(0,1) curve. Notice that its tails fit uder the tails of all of the t distributios. This meas for ay poit particular poit the tail of the distributio, the area to the right of the area to the right of t 0 o the t distributios. t 0 t 0 o the ormal curve will be less tha Notice that the smaller the degrees of freedom, the fatter the tails. Also otice that as the umber of degrees of freedom get larger, the t distributio begis to look more like the ormal distributio. i Page -5-

8 Usig the fuctio ttail, I ca give you the idea that as the sample size gets very large, the t distributio approaches the stadard ormal distributio (i.e. the stadard ormal distributio is the limit for the t distributio as gets large). For the stadard ormal distributio [i.e. N(0,1)], the umber 1.96 cuts off 2.5% of the upper tail of the distributio.. di ttail(10, 1.96) Sample size = di ttail(20, 1.96) Sample size = di ttail(100, 1.96) Sample size = di ttail(200, 1.96) Sample size = di ttail(1000, 1.96) Sample size = di ttail(10000, 1.96) Sample size = 10, So it is clear that the larger the sample size (ad hece the degrees of freedom), the closer the area comes to Let me remid you that the whole reaso we care about the t distributio (at least at this poit) is that we eed it to get the cofidece iterval for the mea of the populatio whe either or is kow. I should also poit out that it is much more commo μ for us ot to kow tha it is for us to kow. X ~ N( μ, 2 ) X ~ N( μ, 2 ) Whe is kow ad, the where is the sample size, the 1 - α cofidece iterval about the sample mea is: x z x + z, 1 ( α/ 2) 1 ( α/ 2) Page -6-

9 Remember that we were able to derive this cofidece iterval because we kew that X ~ N( μ, 2 ) implies that Z X μ = ( / ) is distributed N(0,1). So z < X μ ( / ) < 1 ( α/ 2) 1 ( α/ 2) z X μ t = ( S / ) But whe is ot kow we have which follows the t distributio with 1 degrees of freedom where is the sample size. So we should expect the cofidece iterval to look somethig like x s t x t s, + somethig somethig Remember we ca t covert all of the t distributios to a t equivalet of N(0,1). So we have to idicate ot oly the α-level (or area), but we also have to specify which member of the family of t distributios we are usig. If the sample size is, the the degrees of freedom for the t distributio are - 1, so the t equivalet to is. z 1 ( α 2) t 11, ( α 2) So the geeral form for the cofidece iterval usig the t distributio is s x t x t s, + 11, ( α/ 2) 11, ( α/ 2) Page -7-

10 So for a 95% cofidece iterval, α = 0.05 ad α/2 = which implies 1 - α/2 = So for = 16 ( i.e. - 1 = 15) ad α = 0.05, the cutoff for is t 11, ( α 2) t 15, was obtaied by usig Stata commad ivttail(15,0.025) So the cofidece iterval for the baselie heart rate i the cardiology example we used earlier (ifed.dta) is , = (67.25, 86.37) sice x = ad s = sum( heartlv0) if trtgrp == 1 Variable Obs Mea Std. Dev. Mi Max heartlv We used the stadard deviatio because the populatio stadard deviatio was ukow. But this is a lot of work. Surely Stata ca do this for us. We ca get the cofidece iterval usig the Stata ci commad.. ci heartlv0 if trtgrp == 1 Variable Obs Mea Std. Err. [95% Cof. Iterval] heartlv This is called the ormal cofidece iterval by Stata. This meas that the radom variable X is approximately ormal ot that it is usig z 1 ( α / 2) i the formula (it is, i fact, usig the t distributio). As I ve said before, i real life you almost ever kow the value of. So Stata does t eve bother to provide you with that cofidece iterval. The cofidece itervals derived usig the t-distributio are loger tha those usig the ormal distributio eve if = s because t > z 11, ( α/ 2) 1 ( α/ 2) Page -8-

11 All of the 95% cofidece itervals usig the ormal distributio have the same legth (give that the sample size stays fixed) although the locatios vary. The cofidece itervals based o the t distributio will have differet legths as well as differet locatios because ot oly will the sample meas vary from sample to sample, but the sample stadard deviatios will also vary from sample to sample eve whe the sample size remais the same. But it is still the case eve with the t distributio that we have cofidece that 95% of the 95% cofidece itervals will cover the true mea of the populatio. Now we have bee behavig like the oly type of cofidece iterval that exists is the 95% cofidece iterval but that is ot the case. It is the case that the default for the cofidece iterval i Stata is the 95% CI (i.e. if you do ot specify a level, Stata assumes 95%), but you ca request other levels. Below I have first requested the 95% cofidece iterval so that you ca see it is exactly what we got above whe we did t specify a level.. ci heartlv0 if trtgrp == 1,level(95) Variable Obs Mea Std. Err. [95% Cof. Iterval] heartlv If we request a 90% cofidece iterval, this meas that α = 0.10 so that α/2 = Notice that the 90% cofidece iterval below is cetered about the mea ad that it i fits iside the 95% CI.. ci heartlv0 if trtgrp == 1,level(90) Variable Obs Mea Std. Err. [90% Cof. Iterval] heartlv % 90% Baselie Heart Rate beats/mi Page -9-

12 The more cofidet you are, the loger the cofidece iterval.. ci heartlv0 if trtgrp == 1,level(99) Variable Obs Mea Std. Err. [99% Cof. Iterval] heartlv Before we go o to the chi square distributio let me give you a little iformatio about the ma who came up with the t distributio. W. S. Gosset was a famous chemist ad statisticia. He worked for the Guiess Brewery i Dubli durig the early 1900's. He iveted the t-test to hadle small samples for quality cotrol i brewig. Gosset discovered the form of the t distributio by a combiatio of mathematical ad empirical work with radom umbers, a early applicatio of the Mote-Carlo method. It is said that he published uder the ame Studet because he did t wat other breweries to lear of the potetial applicatio of his work for a brewery. This is take from the web ad from the book Biostatistics by Forthofer ad Lee, published i Brief Overview To date our cofidece itervals have looked like x z x + z, 1 ( α/ 2 ) 1 ( α/ 2 ) Cofidece iterval 1 or x s t x t s, + 11, ( α/ 2) 11, ( α/ 2) Cofidece iterval 2 Both of these cofidece itervals are itervals about the mea of the populatio of iterest. Page -10-

13 The major differece betwee them is the distributio that we use to obtai the cofidece level. For Cofidece Iterval 1 the distributio used is the ormal distributio. For Cofidece Iterval 2 the distributio is the t distributio. So i order to obtai a cofidece iterval, we eed a kow probability distributio that is associated with the samplig distributio of the parameter (of the origial populatio) of iterest. The parameter for each of the two cofidece itervals above is the mea of the populatio. We use Cofidece Iterval 1 whe we kow the stadard deviatio of the populatio ad Cofidece Iterval 2 whe we kow oly the sample estimate (s) for the populatio stadard deviatio. For quality cotrol purposes the parameter that teds to be of iterest is the variace or stadard deviatio rather tha the mea. It happes that either the ormal or the t distributio will work here. We will have to fid a distributio that is associated with the samplig distributio of the variace (i.e. the distributio that you get whe you take repeated radom samples from the origial populatio ad get the variace, rather tha the mea for each sample). It turs out that the chi square distributio is the oe that works. Remember that the radom variable associated with the samplig distributio for both Cofidece itervals 1 ad 2 above was X. This was because we were lookig for a cofidece iterval about the populatio mea ad cosequetly the samplig distributio was the distributio of the meas of all samples from the populatio of size. We will fid that the patter for the cofidece iterval for the variace is differet from Cofidece Itervals 1 ad 2 above. This is because we will be lookig for a cofidece iterval about the variace (or stadard deviatio) ad the samplig distributio will be the distributio of variaces of all samples of size from the origial populatio. We called the radom variable associated with the samplig distributio of meas. X seemed a reasoable choice sice the otatio X is similar to the otatio of the sample mea x. Note that we used a upper case letter for the radom variable ad a lower case for the sample mea. Followig the same patter we will use a upper case S 2 for the radom variable associated with the distributio of the variaces of the radom samples ad a lower case s 2 for the sample variace. Notice that I have also used differet fots for the radom variable ad the sample estimate so they will be easier to distiguish. X Page -11-

14 X μ / I additio, we kow that the distributio of, for large eough, is approximately N(0,1), so we deoted the ratio as Z. X μ S / I a similar fashio we kow that follows the t distributio with 1 degrees of freedom, so we deoted the ratio as t. ( 1) S 2 2 We will use the fact that is distributed chi square with - 1 degrees of freedom. Roser s example 6.39 o page 196 is ice example of a problem ivolvig the variace rather tha the mea of the populatio. Cofidece iterval for the variace ad stadard deviatio We have already discovered that the variace is useful i describig data, ad ow we ll cosider the use of the variace as a way of describig how reliable a process is. Example: For may years the Ceters for Disease Cotrol has ru a iterlaboratory program to maitai cotrol over measuremets of lead i blood. Approximately 100 laboratories participate each moth. Each laboratory is to ascertai the blood lead level i a sample that has bee provided by CDC (all of the samples are supposed to have the same blood lead level). The histogram below gives the results for April, We ca see that there is cosiderable variatio from lab to lab. It is this variatio that is of cocer. Page -12-

15 Idividual measuremets of blood lead cocetratios by approximately 100 differet laboratories 20 Frequecy Blood lead cocetratio i μg/dl The data i the histogram above is my guess at readig the values from a graph preseted i a 1980 issue of Sciece. I order to get the cofidece iterval for or 2, we will eed to itroduce a ew distributio, amely the chi square distributio (for those of you who have studied such thigs, the chi square is a gamma distributio). Page -13-

16 Properties of the chi square distributio 1) The chi square distributio is a cotiuous distributio like both the ormal ad t distributios. 2) Ulike the ormal ad t distributios, the chi square distributio is ot symmetric about its mea. The chi square distributio is skewed with the tail to the right. 3) Like the t distributio, the chi square distributio is a family of distributios whose shapes deped o degrees of freedom. That is, you ca select a uique member of the family by specifyig just the degrees of freedom. So the chi square distributio is a sigle parameter distributio. 4) Because the chi square distributio is cotiuous, probability will be associated with area as it is with the ormal ad t distributios. Defiitio of chi square distributio: G = If X 2 where for each ad the are idepedet (i.e. the i = 1 i i, X ~ N( 01, ) X' s i X s are idepedet ad idetically distributed which is abbreviated as iid), the G is said to follow a chi-square distributio with degrees of freedom (df). So G is the sum of the squares of idetically distributed idepedet distributios. 2 χ χ The distributio is deoted by ( is the Greek letter chi). The mea of the chi square distributio with degrees of freedom is ad the variace is 2. X 1 If = 1 so that we have oly oe radom variable, say, ad that radom variable is X 1 2 distributed N(0, 1), is distributed chi square with oe degree of freedom. The graphs below are a attempt to show you how this works. A lack of ormality i the distributios formig G has a greater impact o the validity of the cofidece itervals derived here tha it does for the cofidece itervals usig t ad z distributios to get a cofidece iterval about the mea. So we eed to be careful whe we use the cofidece iterval for the variace. Page -14-

17 Stadard Normal Distributio = N(0, 1).5 f(x).25 Area is approximately Area is approximately x 1.75 Note that = 3.84 g(x).5.25 Area is approximately 0.05 where 0.05 = 2 X x Chi Square Distributio with 1 degree of freedom Note that the actual cutoff for for the N(0,1) distributio is ±1.96 ad that = I have used 2 for 1.96 ad 4 for 3.84 to make the graphs easier to follow. Page -15-

18 Stadard Normal Distributio = N(0, 1).5 Area of each side is approximately 0.34 f(x).25 We are cosiderig the middle 68% of this N(0,1) distributio as two 34% sectios x 1.75 g(x).5.25 Area is approximately 0.68 (0.68 = 2 X 0.34) x Chi Square Distributio with 1 degree of freedom Page -16-

19 Chi square distributios with various degrees of freedom.8 g(x).6.4 df = 1 df = 2.2 df = 5 df = 10 df = x What happes if the radom variables are idepedet, idetically ormally distributed, but ot N(0,1)? If X, X,X,...,X - N(μ, 2 ) ad the X's are idepedet, the Z i = Xi μ - N(0, 1) for = 1, 2,..., i ad sice the X i are idepedet, the Z i are also idepedet. [Notice it is Xot Xso we divide by ot.] This meas that χ 2 i = 1 2 Z i - because it is the sum of the squares of ormal idepedet radom variables with mea = 0 ad variace = 1. Page -17-

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

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

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

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

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

Stat 421-SP2012 Interval Estimation Section

Stat 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 information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

Economics Spring 2015

Economics 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 information

Chapter 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 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 information

Exam 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

Exam 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 information

Chapter 6. Sampling and Estimation

Chapter 6. Sampling and Estimation Samplig ad Estimatio - 34 Chapter 6. Samplig ad Estimatio 6.. Itroductio Frequetly the egieer is uable to completely characterize the etire populatio. She/he must be satisfied with examiig some subset

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

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

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

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

Big Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates.

Big 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 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

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

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

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

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

Estimation for Complete Data

Estimation for Complete Data Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of

More information

Read through these prior to coming to the test and follow them when you take your test.

Read through these prior to coming to the test and follow them when you take your test. Math 143 Sprig 2012 Test 2 Iformatio 1 Test 2 will be give i class o Thursday April 5. Material Covered The test is cummulative, but will emphasize the recet material (Chapters 6 8, 10 11, ad Sectios 12.1

More information

Power and Type II Error

Power and Type II Error Statistical Methods I (EXST 7005) Page 57 Power ad Type II Error Sice we do't actually kow the value of the true mea (or we would't be hypothesizig somethig else), we caot kow i practice the type II error

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

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

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 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS. 8.1 Random Sampling. 8.2 Some Important Statistics

CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS. 8.1 Random Sampling. 8.2 Some Important Statistics CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 8.1 Radom Samplig The basic idea of the statistical iferece is that we are allowed to draw ifereces or coclusios about a populatio based

More information

Homework 5 Solutions

Homework 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 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

A statistical method to determine sample size to estimate characteristic value of soil parameters

A statistical method to determine sample size to estimate characteristic value of soil parameters A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig

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

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

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

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

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 5

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 5 CS434a/54a: Patter Recogitio Prof. Olga Veksler Lecture 5 Today Itroductio to parameter estimatio Two methods for parameter estimatio Maimum Likelihood Estimatio Bayesia Estimatio Itroducto Bayesia Decisio

More information

NCSS Statistical Software. Tolerance Intervals

NCSS Statistical Software. Tolerance Intervals Chapter 585 Itroductio This procedure calculates oe-, ad two-, sided tolerace itervals based o either a distributio-free (oparametric) method or a method based o a ormality assumptio (parametric). A two-sided

More information

Resampling 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.

Resampling 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 information

1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable

More information

Understanding Samples

Understanding 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 information

Median and IQR The median is the value which divides the ordered data values in half.

Median and IQR The median is the value which divides the ordered data values in half. STA 666 Fall 2007 Web-based Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5-umber summary mea ad stadard deviatio Media

More information

Eco411 Lab: Central Limit Theorem, Normal Distribution, and Journey to Girl State

Eco411 Lab: Central Limit Theorem, Normal Distribution, and Journey to Girl State Eco411 Lab: Cetral Limit Theorem, Normal Distributio, ad Jourey to Girl State 1. Some studets may woder why the magic umber 1.96 or 2 (called critical values) is so importat i statistics. Where do they

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

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

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

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

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

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would

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

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

Chapter 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 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

Introducing Sample Proportions

Introducing Sample Proportions Itroducig Sample Proportios Probability ad statistics Aswers & Notes TI-Nspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,

More information

(6) Fundamental Sampling Distribution and Data Discription

(6) Fundamental Sampling Distribution and Data Discription 34 Stat Lecture Notes (6) Fudametal Samplig Distributio ad Data Discriptio ( Book*: Chapter 8,pg5) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye 8.1 Radom Samplig: Populatio:

More information

Direction: This test is worth 250 points. You are required to complete this test within 50 minutes.

Direction: 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 information

STAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)

STAT 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 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

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would

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

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

The 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.

The 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 information

CHAPTER I: Vector Spaces

CHAPTER I: Vector Spaces CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig

More information

BIOSTATS 640 Intermediate Biostatistics Frequently Asked Questions Topic 1 FAQ 1 Review of BIOSTATS 540 Introductory Biostatistics

BIOSTATS 640 Intermediate Biostatistics Frequently Asked Questions Topic 1 FAQ 1 Review of BIOSTATS 540 Introductory Biostatistics BIOTAT 640 Itermediate Biostatistics Frequetly Asked Questios Topic FAQ Review of BIOTAT 540 Itroductory Biostatistics. I m cofused about the jargo ad otatio, especially populatio versus sample. Could

More information

Lecture 24 Floods and flood frequency

Lecture 24 Floods and flood frequency Lecture 4 Floods ad flood frequecy Oe of the thigs we wat to kow most about rivers is what s the probability that a flood of size will happe this year? I 100 years? There are two ways to do this empirically,

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

PH 425 Quantum Measurement and Spin Winter SPINS Lab 1

PH 425 Quantum Measurement and Spin Winter SPINS Lab 1 PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured

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

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1. Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics

ACCESS 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 information

t distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference

t distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference EXST30 Backgroud material Page From the textbook The Statistical Sleuth Mea [0]: I your text the word mea deotes a populatio mea (µ) while the work average deotes a sample average ( ). Variace [0]: The

More information

BIOSTATISTICS. Lecture 5 Interval Estimations for Mean and Proportion. dr. Petr Nazarov

BIOSTATISTICS. 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 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

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

Computing Confidence Intervals for Sample Data

Computing Confidence Intervals for Sample Data Computig Cofidece Itervals for Sample Data Topics Use of Statistics Sources of errors Accuracy, precisio, resolutio A mathematical model of errors Cofidece itervals For meas For variaces For proportios

More information

Bayesian Methods: Introduction to Multi-parameter Models

Bayesian Methods: Introduction to Multi-parameter Models Bayesia Methods: Itroductio to Multi-parameter Models Parameter: θ = ( θ, θ) Give Likelihood p(y θ) ad prior p(θ ), the posterior p proportioal to p(y θ) x p(θ ) Margial posterior ( θ, θ y) is Iterested

More information

Introducing Sample Proportions

Introducing Sample Proportions Itroducig Sample Proportios Probability ad statistics Studet Activity TI-Nspire Ivestigatio Studet 60 mi 7 8 9 10 11 12 Itroductio A 2010 survey of attitudes to climate chage, coducted i Australia by the

More information

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22 CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first

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

1 Lesson 6: Measure of Variation

1 Lesson 6: Measure of Variation 1 Lesso 6: Measure of Variatio 1.1 The rage As we have see, there are several viable coteders for the best measure of the cetral tedecy of data. The mea, the mode ad the media each have certai advatages

More information

[ ] ( ) ( ) [ ] ( ) 1 [ ] [ ] Sums of Random Variables Y = a 1 X 1 + a 2 X 2 + +a n X n The expected value of Y is:

[ ] ( ) ( ) [ ] ( ) 1 [ ] [ ] Sums of Random Variables Y = a 1 X 1 + a 2 X 2 + +a n X n The expected value of Y is: PROBABILITY FUNCTIONS A radom variable X has a probabilit associated with each of its possible values. The probabilit is termed a discrete probabilit if X ca assume ol discrete values, or X = x, x, x 3,,

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

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

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso product-momet correlatio

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

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

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

Lecture 7: Properties of Random Samples

Lecture 7: Properties of Random Samples Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ

More information

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the

More information

4. Partial Sums and the Central Limit Theorem

4. Partial Sums and the Central Limit Theorem 1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems

More information

Lesson 10: Limits and Continuity

Lesson 10: Limits and Continuity www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

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

Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date:

Mathacle. 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 information

Stat 139 Homework 7 Solutions, Fall 2015

Stat 139 Homework 7 Solutions, Fall 2015 Stat 139 Homework 7 Solutios, Fall 2015 Problem 1. I class we leared that the classical simple liear regressio model assumes the followig distributio of resposes: Y i = β 0 + β 1 X i + ɛ i, i = 1,...,,

More information

Chapter 18 Summary Sampling Distribution Models

Chapter 18 Summary Sampling Distribution Models Uit 5 Itroductio to Iferece Chapter 18 Summary Samplig Distributio Models What have we leared? Sample proportios ad meas will vary from sample to sample that s samplig error (samplig variability). Samplig

More information

Basis for simulation techniques

Basis for simulation techniques Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios

More information

GG313 GEOLOGICAL DATA ANALYSIS

GG313 GEOLOGICAL DATA ANALYSIS GG313 GEOLOGICAL DATA ANALYSIS 1 Testig Hypothesis GG313 GEOLOGICAL DATA ANALYSIS LECTURE NOTES PAUL WESSEL SECTION TESTING OF HYPOTHESES Much of statistics is cocered with testig hypothesis agaist data

More information

Statisticians use the word population to refer the total number of (potential) observations under consideration

Statisticians use the word population to refer the total number of (potential) observations under consideration 6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space

More information

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. Comments:

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. Comments: Recall: STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Commets:. So far we have estimates of the parameters! 0 ad!, but have o idea how good these estimates are. Assumptio: E(Y x)! 0 +! x (liear coditioal

More information

Because it tests for differences between multiple pairs of means in one test, it is called an omnibus test.

Because it tests for differences between multiple pairs of means in one test, it is called an omnibus test. Math 308 Sprig 018 Classes 19 ad 0: Aalysis of Variace (ANOVA) Page 1 of 6 Itroductio ANOVA is a statistical procedure for determiig whether three or more sample meas were draw from populatios with equal

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

Lecture 8: Non-parametric Comparison of Location. GENOME 560, Spring 2016 Doug Fowler, GS

Lecture 8: Non-parametric Comparison of Location. GENOME 560, Spring 2016 Doug Fowler, GS Lecture 8: No-parametric Compariso of Locatio GENOME 560, Sprig 2016 Doug Fowler, GS (dfowler@uw.edu) 1 Review What do we mea by oparametric? What is a desirable locatio statistic for ordial data? What

More information