Math 111, section 08.x supplement: The Central Limit Theorem notes by Tim Pilachowski

Transcription

1 Math 111, sectio 08.x supplemet: The Cetral Limit Theorem otes by Tim Pilachowski Table of Cotets 1. Backgroud 1. Theoretical 3. Practical 3 4. The Cetral Limit Theorem 4 5. Homework Exercises 7 1. Backgroud Gatherig Data. Iformatio is beig collected ad aalyzed all the time by various groups for a vast variety of purposes. For example, maufacturers test products for reliability ad safety, ews orgaizatios wat to kow which cadidate is ahead i a political campaig, advertisers wat to kow whether their wares appeal to potetial customers, ad the govermet eeds to kow the rate of iflatio ad the percet of uemploymet to predict expeditures. The items icluded i the aalysis ad the people actually aswerig a survey are the sample, while the whole group of thigs or people we wat iformatio about is the populatio. How well does a sample reflect the status or opiios of the populatio? The aswer depeds o may variables, some easier to cotrol tha others. Sources of Bias. Oe source of bias is carelessly prepared questios. For example, if a pollster asks, Do you thik our overworked, uderpaid teachers deserve a raise? the aswers received are likely to reflect the slated way i which the questio was asked! Aother dager is a poorly selected sample. If the perso coductig the survey stads outside a shoppig ceter at pm o a weekday, the sample may ot be represetative of the etire populatio because people at work or people who shop elsewhere have o chace of beig selected. Radom Samplig. To elimiate bias due to poor sample selectio, statisticias isist o radom samplig, i which every member of the populatio is equally likely to be icluded i the sample. For a small populatio, simple radom samplig ca be achieved by puttig everyoe s ame o a slip of paper, placig all the slips i a large bi, mixig well, ad selectig the sample. Techically, each slip should be replaced before the ext slip is draw, so the probability does t chage for subsequet draws. That is, the evets should be idepedet, ad the probability ot coditioal. Of course, the slips should be mixed before each draw, as well. I practice, most populatios beig tested are large eough that the coditioal probabilities do t sigificatly alter the results. Represetative Samples. It is importat to ote that sayig the sample is free of bias does ot guaratee that it will perfectly represet the populatio. A radom sample may, by chace variatio, be very differet from the populatio. However, the mathematics uderlyig statistics allows us to specify how much a radom sample is likely to vary ad how ofte it will be extremely o-represetative. Bias, o the other had, affects the results i ukow ways that caot be quatified or specified i ay such maer. page 1.

2 Example 1 Suppose that a club has 30 members, ad five are to be radomly selected to represet the club at a regioal coferece. Names of female members are i italics. No-italicized ames belog to male members. Abby Greg Mark Sam Zelda Be Hele Norris Tom Art Charlie Iva Oscar Ursula Bob David Jamal Patricia Vic Curt Eric Kevi Queti Walt Da Frak Larry Rob Yvette Eva Suppose you were to select te differet radom samples of five club members from this populatio of 30. Eve though exactly 1/5 of the populatio is wome, it is very likely your samples would ot all iclude exactly 1/5 female. Some samples might have o wome, two wome, or maybe eve all five picks would be wome. This does't mea the samplig process was biased it simply illustrates samplig variability. Example Similarly, if you flip a coi te times, you would ot be surprised if it came up heads oly four times or six times rather tha exactly 1/, or five, of the times. If, o the other had, you flipped a coi te times ad got te heads, you probably would be surprised. We kow ituitively that it is ulikely we will get such a extremely urepresetative sample. It is ot impossible, just very rare. We may eve decide the coi must be weighted i some way so that heads are more likely to appear. Samplig variability is also affected by the umber of observatios we iclude. If we flip a coi te times ad get oly 3 heads, or 30%, we may ot be very surprised. If we flip the same coi 1000 times ad oly get 300 heads, however, this is more surprisig. Agai, we may begi to woder whether the coi is fair. Over a larger umber of trials, the percet of heads should be closer to the 50% heads we expect.. Theoretical If you were able to costruct every possible sample of a specified size from the same populatio, you would create what statisticias call a samplig distributio. The Cetral Limit Theorem tells us, i short, that a samplig distributio (based o the theoretical possibility of creatig every possible sample of size ) is close to a ormal distributio. Furthermore, the Cetral Limit Theorem tells us that the peak of a samplig distributio is the populatio mea (symbolized by µ). The techical statistical term for this true mea is the populatio parameter or just parameter. I cotrast, the mea calculated from a give sample (symbolized by x ) is a sample statistic or just statistic. The mea of the samplig distributio is the mea of the populatio [ E ( x) = µ ]. Calculated sample meas (statistics) will vary while they are most likely to occur ear the true populatio mea, they may also be far away from the populatio mea, depedig o the particular characteristics of the particular sample selected. Also, we kow some facts about ormal distributios that will be helpful i determiig how likely it is that a sample statistic occurs ear the populatio parameter. Oe characteristic of ormal probability distributios: 68% of the outcomes are withi 1 stadard deviatio of the mea 95% of the outcomes are withi stadard deviatios of the mea 99.7% of the outcomes are withi 3 stadard deviatios of the mea. page.

3 What does this mea for radom samplig? It tells us that 68% of the time, a radom sample will give us a result a statistic withi 1 stadard deviatio of the true parameter. We would expect that 95% of the time, a radom sample will give a statistic withi stadard deviatios of the populatio parameter, ad 99.7% of the time, a radom sample will give a statistic withi 3 stadard deviatios of the populatio parameter. I a begiig level statistics course, you would be itroduced to cofidece itervals ad hypothesis tests, each of which makes use of the percets listed above. 3. Practical Costructig every possible sample of a specified size from the populatio is a theoretical costruct. I practical terms, it s ever goig to happe. Time ad cost would prevet researchers from eve tryig. Likewise, a ifiite umber of trials is a impossibility. However, the Cetral Limit Theorem also helps us i this regard. A sample of size = 1 is ot likely to be represetative of the etire populatio. Neither is a sample of size =. It makes sese ituitively that a larger sample size would be more likely to be represetative of the populatio. (See Example above). The Cetral Limit Theorem ot oly verifies this ituitio, but also tells us that, as sample size icreases, the shape of the samplig distributio grows closer to the shape of a ormal distributio. The followig series of graphics illustrate this. They were draw by a appelet foud o the website I each case, the graph o the left illustrates a samplig distributio for a set of samples of size = 1, ad gives a rough idea of the shape of the populatio distributio. For the middle graph the sample size has bee icreased to 10. I the graph o the right, the sample size is = 30. Notice that, o matter what the shape of the populatio distributio, as the sample size icreases, the shape of the samplig distributio becomes very much like the shape of a ormal distributio. Close eough, i fact, that we ca use the ormal distributio table to determie probabilities. page 3.

4 4. The Cetral Limit Theorem Give a populatio with mea µ ad stadard deviatio : 1) As the sample size icreases, or as the umber of trials approaches ifiite, the shape of a samplig distributio becomes icreasigly like a ormal distributio. ) The mea of samplig distributio, E ( x), = µ. 3) The stadard deviatio of samplig distributio, called the stadard error, = page 4.. For statistics, a sample size of 30 is usually large eough to use the ormal distributio probability table for hypothesis tests ad cofidece itervals. For the homework exercises below, you ll use the ormal distributio table to fid various probabilities ivolvig sample meas ( x ). Example 3 The populatio of fish i a particular pod is kow to have a mea legth µ = 15 cm with stadard deviatio = 5. a) You catch oe fish from the pod. What is its expected legth? The legths of the fish i the pod exhibit some variability we kow this because the stadard deviatio does ot equal 0. So, while we caot say with absolute certaity that the fish we catch will be ay specific legth, our best (educated) guess is that we expect the fish to be 15 cm i legth. (This is why the mea is also called expected value.) b) You catch oe fish from the pod. What is the probability that its legth is less tha 14 cm? We caot aswer this questio because we do ot kow the shape of the probability distributio. Specifically, sice it is ot explicitly stated that the distributio is ormal, we caot use the ormal distributio table. c) You catch fifty fish from the pod ad measure their legths. What are the expected value of the sample mea ( x ) ad stadard error for the samplig distributio? While the legth of ay sigle fish from the pod is ot likely to be represetative of the whole populatio, a sample of size 50 is large eough for the Cetral Limit Theorem to apply, ad we ca say how likely it is that this sample will be represetative of the populatio. By the Cetral Limit Theorem, for sample size = 50, 5 expected value E ( x) = µ = 15 ad stadard error = = d) You catch fifty fish from the pod ad measure their legths. What is the probability that the average legth of those fish (i.e. sample mea x ) is less tha 14 cm? x µ z = = 1.41, P( x < 14) = P(Z < 1.41) = e) You catch fifty fish from the pod ad measure their legths. What is the probability that the average legth of those fish (i.e. sample mea x ) is betwee 14 cm ad 16.5 cm? x µ z = =.1, P( x < 16.5) = P(Z <.1) = P(14 < x < 16.5) = P( 1.41 < Z <.1) = P( < Z <.1) P( < Z < 1.41) = =

5 Example 4 A populatio exhibits the followig probability distributio: X P(X = x) a) What are the expected value ad stadard deviatio for a sigle radomly-chose subject from this populatio? E X = µ = = = Var ( ) ( ) ( ) ( ) ( ) ( ) 4 ( X ) = ( 10 4) ( 0.4) + ( 0 4) ( 0.) + ( 30 4) ( 0.) + ( 40 4) ( 0.0) + ( 50 4) ( 0.) = 4 = 4 = b) What is the probability that a sigle radomly-chose subject from this populatio will exhibit a value of at most 30? I cotrast to Example 3, we do kow the shape of this probability distributio, ad ca calculate a aswer directly from the probability distributio. P X 30 = P X = 10 + P X = 0 + P X = 30 = = 0. ( ) ( ) ( ) ( ) 8 c) You select a sample of 45 from this populatio. (sample size = 45). What are the expected value ad stadard error for the samplig distributio? By the Cetral Limit Theorem, for = 45, 4 14 expected value = µ = 4 ad stadard error = = d) You select a sample of 45 from this populatio. (sample size = 45). What is the probability that the sample mea is at most 30? x µ 30 4 z = =.69, P( x 30) = P(Z.69) = e) You select a sample of 45 from this populatio. (sample size = 45). What is the probability that the sample mea is at least 30? This is the complemet of the aswer to part d). P( x 30) = P(Z.69) = = Note the differece betwee the questios asked i parts a) ad b) as opposed to the questios asked i parts c), d) ad e). Parts a) ad b) asked about oe sigle subject, radomly chose from withi the populatio. The probabilities give i the probability distributio applied. Parts c), d) ad e) asked about 45 subjects, radomly chose from withi the populatio. The sample thus created ca be viewed as oe of the may possible samples of size = 45 that could have bee created from the populatio. That is, the sample we eded up with is part of the samplig distributio, ad we were able to make use of the Cetral Limit Theorem. The calculatios i parts a) ad b) ivolved x ad P(X = x), that is, they focused o oe specific value. The calculatios i parts c), d) ad e) ivolved x ad stadard error, that is, they focused o the mea of a sample of size. page 5.

6 Example revisited a special case You flip a fair coi. Defie a radom variable Y = 0 for tails ad Y = 1 for heads. a) What is the expected value for a sigle toss? E(Y) = µ = 0(0.5) + 1(0.5) = 0.5 b) What are the variace ad stadard deviatio of Y for a sigle toss? Var(Y) = (0 0.5) (0.5) + (1 0.5) (0.5) = 0.5 = 0.5 = 0.5 c) You flip a fair coi forty times (sample size = 40). What are the expected value ad stadard error for the samplig distributio? By the Cetral Limit Theorem, for = 40, 0.5 expected value = µ = 0.5 ad stadard error = = d) You flip a fair coi forty times (sample size = 40). What is the probability that the average value for Y (i.e. the sample mea y ) is less tha 0.4? y µ z = = 1.6, P( y < 0.4) = P(Z < 1.6) = This Example is a special case, ot quite the same as Example 3 ad Example 4. Tossig a coi, for which there are two possible outcomes, is a biomial experimet. For this type of experimet, we have usually defied our radom variable as X = umber of heads (successes) i tosses of the coi. For Example we have defied our radom variable as a proportio, i.e. umber of heads X Y = = = p (probability of success). umber of tosses From the perspective of proportios, the aswers above would be iterpreted as follows. a) The expected value for a sigle toss = E(Y) = p = 0.5. traslatio: If we toss a sigle coi, the probability of heads is 50%. c) You flip a fair coi forty times (sample size = 40). Expected value of the samplig distributio = E ( y) = 0.5. traslatio: We would expect 50% of the tosses to be heads. importat ote for the stadard error: If we geeralized the mathematics used above, we would fid that the stadard error for the samplig pq distributio for a proportio will always equal, i this case ( 0.5)( 0.5) 1 = d) You flip a fair coi forty times (sample size = 40). The probability that the average value for Y (i.e. the sample mea y ) is less tha 0.4 = traslatio: If we toss a coi forty times, there is a 10.38% chace that less tha 40% of the tosses will be heads. importat ote for the z-score calculatio: y p I terms of proportios, the z-score formula ad calculatio would be z = = 1.6. page 6. pq For more o proportios ad the correspodig samplig distributio ad stadard error, see Appedix E, pages i your text.

7 5. Homework Exercises 1. I 010, the mea score for the three sectios of the SAT was µ = 1509, with a stadard deviatio = 339. (source: College Etrace Examiatio Board, College-Boud Seiors: Total Group Profile (Natioal) Report, through ) a) Oe studet takes the SAT. What is her expected score o the three parts? b) Oe hudred studets take the SAT. What is the expected value of their average score (i.e. expected value of the sample mea, x )? What is the stadard error for the samplig distributio? c) Oe hudred studets take the SAT. What is the probability that their average score is less tha 1450? d) Oe hudred studets take the SAT. What is the probability that their average score is betwee 1450 ad 1600? e) Oe hudred studets take the SAT. What is the probability that their average score is above 1600?. A experimet has the followig probability distributio: X P(X) a) What are the expected value ad stadard deviatio for a sigle radomly-chose value of X? b) You radomly select a sample of size = 50. What is the expected value for the sample mea, x? What is the stadard error for the samplig distributio? c) You radomly select a sample of size = 50. What is the probability that the sample mea is less tha.? d) You radomly select a sample of size = 50. What is the probability that the sample mea is betwee. ad.5? 3. You pay $1 to play a game i which you roll oe stadard six-sided die. You lose your dollar if the die is 1,, 3 or 4. You get your dollar back if the die is a 5, ad if the die is a 6 you get your dollar back plus $ more (total of $3). a) Calculate expected value ad stadard deviatio for a sigle toss. (Be sure to iclude the dollar you pay to play the game.) b) If you play the game 100 times, what are the expected value ad stadard error for the samplig distributio? c) If you play the game 100 times, what is the probability that your average outcome will be positive? (That is, you walk away with more moey tha what you had before the game.) d) If you play the game 100 times, your average wiigs have a 90% probability of beig below what value? 4. You flip a fair coi 100 times. Defie radom variable Y = proportio of heads. a) What is the expected proportio of heads for this experimet? b) What is the stadard error for the samplig distributio? c) What is the probability that the proportio of heads is betwee 0.45 ad 0.60? d) What is the probability that the proportio of heads is greater tha 0.60? 5. Lab results idicate that a certai drug is effective 75% of the time (success) ad ieffective 5% of the time (failure). As a trial, the drug is admiistered to a sample of 1000 patiets. a) What is the expected proportio of successes for this experimet? b) What is the stadard error for the samplig distributio? c) What is the probability that betwee 710 ad 770 of the patiets are helped by the drug? (Remember to thik i terms of proportio: i.e. 710 is what proportio of the 1000 tested? 770 is what proportio of the 1000 tested?) d) What is the probability that at least 770 of the patiets are helped by the drug? page 7.