Introduction There are two really interesting things to do in statistics.

Transcription

1 ECON 497 Lecture Notes E Page 1 of 1 Metropolita State Uiversity ECON 497: Research ad Forecastig Lecture Notes E: Samplig Distributios Itroductio There are two really iterestig thigs to do i statistics. The first is to be able to predict what a sample draw from a populatio is likely to look like, give what you kow about the populatio ad the size of the sample. The secod is to be able to predict what a populatio is likely to look like, give what you kow about a sample which has bee draw from it. I this chapter, we'll ivestigate the first of these. While there are some situatios i which the first thig is very compellig, the primary value of the aswers we'll fid is i turig the questio aroud ad approachig the secod of these. Samplig... Samplig is as much a art as it is a sciece. Drawig a truly radom sample from a populatio ca be very tricky ad has oly recetly bee practiced at a level earig perfectio. Populatios ca be regarded as fiite or ifiite. A fiite populatio ca be listed (the members of a class or the citizes of the U.S.) while a ifiite populatio caot be listed (all possible customers at McDoald's i a day). I the case that a populatio is fiite, the tet How to Coduct Your Ow Survey by Priscilla Salat ad Do Dillma offers the followig three steps i samplig (p. 58) 1. Idetify the target populatio as precisely as possible ad i a way that makes sese i terms of the purpose of the study. It is importat to be specific eough that everyoe ivolved i the research kows who is eligible for the survey ad who is ot. 2. Fid or put together a list of the target populatio, the list from which the sample will evetually be draw. 3. Select the sample. Samplig methods rage from simple to etremely comple. For may surveys of small populatios ad small areas, ucomplicated desigs like simple radom samplig ad systematic samplig are adequate. I the case that a populatio is ifiite (or at least impossible to list) selectig, for eample, every teth or hudredth idividual for samplig might be sufficiet.

2 ECON 497 Lecture Notes E Page 2 of 2 The importace of good samplig techique caot be over emphasized. If a samplig techique makes it more likely that some members of a populatio will be selected tha others, this ca seriously skew the resultig sample's characteristics ad the implicatios of the sample. The samplig distributio of the sample mea Ay time you select a sample, you are likely to get differet members of a populatio. The resultig sample mea ( ) will vary from sample to sample ad, thus, is a radom variable. As such, this radom variable has a distributio we ca discuss. The sample mea is a radom variable with a epected value equal to the populatio mea (µ) ad a stadard deviatio equal to the populatio stadard deviatio divided by the square root of the sample size. Put aother way: µ E() µ where µ is the populatio mea is the populatio stadard deviatio is the sample size

3 ECON 497 Lecture Notes E Page 3 of 3 EX: I.Q. tests geerally have a mea of 100 ad a stadard deviatio of 20. If such a test is admiistered to a populatio ad you take a radom sample of that populatio, what will the stadard deviatio of the sample mea score be if the sample size is 25, 49, 100, 196 ad 400? I each case, µ E() µ 100. The differece is i the stadard deviatio of the sample mea,. Because the populatio stadard deviatio 20, the stadard deviatio of the sample mea will be: I a graph, if these are ormally distributed, they look like: Distributios of IQ Sample Meas Aother way to describe the distributio of the sample meas is to list the probabilities that a sample mea is betwee, say, 98 ad 102, for each sample size. To calculate these, covert the limits (98 ad 102) to stadard ormal values by subtractig the populatio mea (100) ad dividig by the stadard deviatio of the sample mea:

4 ECON 497 Lecture Notes E Page 4 of 4 P (98 < < 102) Cetral Limit Theorem The cetral limit theorem says that for a sample draw from ay populatio (eve populatios with really weird distributios), if you draw eough observatios for a sample the the mea of that sample will approimately follow a ormal distributio. I practice, this usually meas drawig at least 30 observatios. So, if you take a sample from a populatio, the mea of that sample will be a ormally distributed radom variable with epected value equal to the mea of the populatio ad a stadard deviatio equal to the stadard deviatio of the populatio divided by the square root of the sample size. EX: * Populatio mea µ 200 * Populatio S.D. 50 * Sample size 100 The stadard deviatio of -bar is 50 * So, the sample mea,, is distributed ormally with mea 200 ad stadard deviatio 5. That is, ~ N(200,5). What is the probability that the sample mea will be withi +/- 5 of the populatio mea? Fid P(195 < < 205) Covert these to stadard ormal radom variable terms... * ( )/5-1 * ( )/5 +1 So, this is the same as... Fid P(-1 < z < 1)

5 ECON 497 Lecture Notes E Page 5 of 5 The samplig distributio of the populatio proportio If the variable of iterest is the percetage of the populatio satisfyig some requiremet (that they are left haded, voted i the last electio or drive Cadillacs, for eample) the tetbook calls this populatio proportio p. The proportio of the sample satisfyig the characteristic is p. It's ot that differet from. The sample proportio p is a radom variable with epected value p (the populatio proportio) ad stadard deviatio p p(1 p) pq q 1 p EX: Imagie that amog the world populatio, 10% of the people are left haded. Fid the distributio of the sample proportio which is left haded i a sample of size 81. What is the probability that, i a radom sample of size 81, more tha 12% of the people are left haded? That is Fid P ( p > 0.12) Because the sample size 81 is fairly large, we will assume that the sample proportio will be ormally distributed with mea µ ad stadard deviatio p pq p So, we ca covert the questio to oe ivolvig a stadard ormal variable: P(p > 0.12) P z > 0.12 µ p p P z > P z ( > 0.6) EX: Cosider the eample of a presidetial electio i the U.S. These electios are very epesive methods of determiig the proportio of the populatio that wats to vote for each cadidate. We could probably get the same wier most of the time by simply selectig a smaller radom sample of the populatio ad just askig them. How well would this work? Let's say that 51% of the populatio wat the democratic cadidate to wi. Let p be the proportio of the populatio that wat the democratic cadidate ad p0.51. If we take a sample of size 30, what is the probability that p, the portio of the sample favorig the democratic cadidate, will be greater tha or equal to 0.50 (allowig the democrats to wi)?

6 ECON 497 Lecture Notes E Page 6 of 6 So, we're tryig to fid P ( p > 0.50). This is a ormal radom variable, ad to fid the probability, we eed to covert it to a stadard ormal radom variable. To do this, we eed the mea ( µ ) ad the stadard deviatio p p(1 p) p Covertig 0.50 to a comparable stadard ormal radom variable limit yields P(p > 0.50) P z > P(z > 0.109) So, a radom sample of 30 voters would yield the correct wier about 54% of the time if 51% of the populatio preferred oe cadidate. A radom sample of 100 voters will give the correct wier or about 58% of the time. A radom sample of 10,000 voters will give the correct wier or almost 98% of the time, eve whe the support for the most popular cadidate is at oly 51%. As a result, we ca coclude that if we just asked a radom sample of 10,000 people who they wated for presidet, we would get the correct wier (the oe with 51% support) almost 98% of the time. This would elimiate much of the epese of atioal electios without sigificatly affectig the results. Here's graph showig the probability of selectig the correct cadidate whe the populatio proportio supportig her (or him) is 51% versus 49% for the other:

7 ECON 497 Lecture Notes E Page 7 of 7 1 Probability of the more popular cadidate wiig whe p How High Ca I Make Your Grades? Whe I teach a class, there is a assumptio about what the mea grade should be ad if I deviate too much from that mea grade I get heat from admiistrators. Basically, though, the grade for ay class is the mea grade for the sample of people i the class. The larger the size of the class, the smaller the stadard deviatio of the sample mea ad the closer the average grade i the class eeds to be to the accepted stadard mea. For eample, if I was teachig a class with oe perso i it, I could probably get away with givig that perso a 4.0 because it is ot icoceivable that there would be oe perso whose work is of such a high quality. However, i a class of 40 people, givig a average grade of 4.0 would be difficult to justify because it is ulikely that i a radom sample of 40 people, all 40 would be sufficietly taleted ad hard workig to deserve so high a grade. EX: A survey showed that a family speds a average of $ per day while o vacatio with a stadard deviatio of $ For the followig questios, assume a sample size of 40 families. µ ad ad 40. A. Fid the samplig distributio of the sample mea,. is distributed ormally with mea ad stadard deviatio 85.00/40 1/

8 ECON 497 Lecture Notes E Page 8 of 8 B. What is the probability that the simple radom sample of 40 families will provide a sample mea that is withi $20 of the populatio mea? P( < < ) ( )/ / ( )/ / P( < z < 1.488) C. What is the probability that the simple radom sample of 40 families will provide a sample mea that is withi $10 of the populatio mea? P( < < ) ( )/ / ( )/ / P(-0.74 < z < 0.74)

9 ECON 497 Lecture Notes E Page 9 of 9 EX: Amog adults i the U.S., 17% voted for George Bush i Assume a sample of 800 adults is take. p0.17, 800. A. What is the samplig distributio of the sample proportio of people that voted for George Bush? p is ormally distributed with mea 0.17 ad stadard deviatio (0.17*0.83/800)1/ B. What is the probability that the sample proportio is withi two percetage poits of the populatio proportio? P(0.15 < p < 0.19) ( )/ / ( )/ / P(-1.51 < z < 1.51) C. If the sample size is doubled to 1600, what is the probability that the sample proportio is withi two percetage poits of the populatio mea? p is ormally distributed with mea 0.17 ad stadard deviatio (0.17*0.83/1600) 1/ P(0.15 < p < 0.19) ( )/ / ( )/ / P(-2.13 < z < 2.13) EX: A productio ru is ot acceptable for shipmet to customers if a sample of 100 items cotais 5% or more defective items. If a productio ru has a populatio proportio defective of p0.10, what is the probability that p-bar will be at least 0.05? p0.10, 100 Fid P( p > 0.05) p is distributed ormally with mea 0.10 ad stadard deviatio (0.10*0.90/100) 1/ ( )/ / P(z > -1.67) So oly about 4.75% of the samples will be acceptable ad 95.25% will be rejected

10 ECON 497 Lecture Notes E Page 10 of 10 EX: Studies show that 10% of the U.S. populatio is fuctioally illiterate. Four hudred people are selected at radom to participate i a project. If more tha 12% of the people selected are illiterate the the project will fail. Fid the probability that this project will fail. p is distributed ormally with mea 0.10 ad stadard deviatio [(0.10)(0.90)/400] 1/ P( p > 0.12) P(Z > ( )/0.015) P(Z > 0.02/0.015) P(Z > 1.33)