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1 Stat 40, sectio 5.4 The Cetral Limit Theorem otes by Tim Pilachowski If you have t doe it yet, go to the Stat 40 page ad dowload the hadout 5.4 supplemet Cetral Limit Theorem. The homework (both practice ad had-i) homework for sectio 5.4 will be from that supplemet. From sectio 5.3: Radom variables,,, form a (simple) radom sample of size if they meet two (importat) requiremets:. The i s are idepedet radom variables.. Every i has the same probability distributio. Data is collected from the sample, i.e., the radom variables,,, each receive values x, x,, x. These values are used to calculate sample statistics. The sample statistics we ll be most iterested i are:. The sample total T0 = The sample mea = T0. (The calculated sample mea is symbolized by x.) 3. The sample variace S = ( i ) i =. (The calculated sample variace is symbolized by s.) Note that T0, ad S are themselves radom variables. Sectios 5.3 ad 5.4 focus o what the probability distributios of these radom variables look like, ad what they ca tell us about the overarchig populatio s distributio. Theory Probability models exist i a theoretical world where everythig is kow. If you costructed every possible sample of a specified size from a give populatio (Example i the supplemet), or were able to toss a coi a ifiite umber of times (Example i the supplemet), you would create what statisticias call a samplig distributio. I the Examples below, we ll use a hypothetical populatio Ψ cosistig of the umbers 0, 0, 30, 40 ad 50. The parameter ad statistic we ll cosider first is the mea. Example A-: Calculate the mea ad stadard deviatio of a populatio Ψ which cosists of elemets from the set {0, 0, 30, 40, 50} with probabilities give i the table below P( = x) Example A-: Costruct a histogram for populatio Ψ. We would get the same histogram if we were to cosider all possible samples of size = that could be take from the populatio Ψ ad calculated each sample s expected value (mea).

2 Example A-3: Costruct all possible samples of size = that ca be made from the elemets of Ψ, desigate the probability of each beig picked, ad calculate each sample s expected value (mea). sample P x sample P x sample P x sample P x 0, 0 0, 0 30, 0 50, 0 0, 0 0, 0 30, 0 50, 0 0, 30 0, 30 30, 30 50, 30 0, 50 0, 50 30, 50 50, 50 Example A-4: Draw the histogram for the samplig distributio from Example A P = x ( ) Example A-5: Calculate mea ad stadard deviatio of the samplig distributio of Ψ for sample size =. Mea of samplig distributio = E ( ) = 0 ( 0.6) + 5( 0.6) + 0( 0.0) + 5( 0.08) + 30( 0.0) + 35( 0.08) + 40( 0.08) + 45( 0.0) + 50( 0. 04) = Variace of samplig distributio = ( ) V = ( 0 4) ( 0.6) + ( 5 4) ( 0.6) + ( 0 4) ( 0.0) + ( 5 4) ( 0.08) + ( 30 4) ( 0.0) + ( 35 4) ( 0.08) + ( 40 4) ( 0.08) + ( 45 4) ( 0.0) + ( 50 4) ( 0.04) = Stadard deviatio of samplig distributio ( ) = σ = Theory: = σ stadard error = (Example A-6 is foud o the ext page.)

3 Example A-6: Draw the histogram ad calculate the mea ad stadard deviatio of the samplig distributio of Ψ for sample size = 4. Mea of samplig distributio = E ( ) = Variace of samplig distributio = Var ( ) = Stadard deviatio of samplig distributio ( ) = σ = = σ stadard error = The histogram for this samplig distributio (sample size = 4) looks like this. Example A-7: O your ow, explore the histogram for samplig distributios of Ψ for various sample sizes. (That is, coduct a series of simulatio experimets usig various values of.) The supplemet provides two sources, alog with illustratios: Notes followig Examples A: (Proofs for coclusios about samples of ay size are o the ext page.)

4 I Example A- we foud E() ad σ (). I Examples A-5 ad A-6, we foud that ( ) E( ) σ ( ) ( ) σ =. Will the same be true for ay sample size? E = ad 4.3 Example D revisited. Distributio of ACT scores is approximately ormal. I 00 mea score for the ACT =.6, with a stadard deviatio of 4.3. What is the probability that a sigle studet chose at radom has a ACT score betwee ad 4? aswer: (source: Usefuless of High School Average ad ACT Scores i Makig College Admissio Decisios, retrieved from Theory: samplig distributio for a ormally distributed populatio

5 4.3 Example D a ew questio. Distributio of ACT scores is approximately ormal. I 00 mea score for the ACT =.6, with a stadard deviatio of 4.3. a) If a radom sample of 50 studets who took the ACT is selected, what is the shape of the resultig samplig distributio? b) What are E ( ) ad σ? c) What is the probability that the sample mea is betwee ad 4? aswers: ormal;.6, 0.608; 0.88 This is fie for a populatio kow to be ormally distributed, but ca we make ay statemets about samplig distributios from a populatio which may ot (or defiitely do ot) have a ormal probability distributio? Recall the various versios of Example A doe earlier. Populatio Ψ did ot have a ormal distributio, ad was ot eve symmetric. However, the shape of the samplig distributio took o a shape close to that of a ormal distributio as icreased. Eter the Cetral Limit Theorem: Give a populatio with mea µ ad stadard deviatio σ : ) 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 = the mea of the populatio, E ( ) µ =. σ 3) The stadard deviatio of samplig distributio = stadard error, σ =. The proof of () i the Cetral Limit Theorem requires momet geeratig fuctios which we do ot have yet, ad which we may (but probably wo t) get to before the ed of the semester. 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 Lecture examples, ad for homework exercises from the hadout, we ll use the ormal distributio table to fid various probabilities for sample statistics. The Cetral Limit Theorem tells us, i short, that a samplig distributio is ofte close to a ormal distributio. 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 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 statistics, the Cetral Limit Theorem is the justificatio for costructig cofidece itervals ad coductig hypothesis tests.

6 Example B. A populatio has mea µ = 50 ad stadard deviatio σ =. For a radom sample of size 47, calculate a) the expected value of the sample mea ad b) the stadard error. Fid the followig probabilities: 45 < < 53 P > 54. c) P ( ) ad d) ( ) aswers: 50,, , WARNING: Example B is ot like those we did i sectio 4.3! That is, it does ot fid probabilities for sigle µ values of for a ormally-distributed populatio, which uses Z =. (I fact, because we do ot kow σ the probability distributio, we caot specify probabilities for idividual subjects.) Rather, Example B looks at oe sample ad fids probabilities ivolvig the mea of that sample,, cosidered as part of all the hypothetical samples which could have bee costructed (the samplig µ µ distributio). Therefore, for example B we used Z = =. σ σ Also, i some homework exercises, you ll eed to use the skills from previous sectios to determie µ ad σ.

7 Example C. A radom variable has probability desity fuctio ( x) ( x) 6x 0 x f =. 0 otherwise a) What are the expected value ad stadard deviatio for a sigle radomly-chose value of? b) You radomly select a sample of size = 00. What is the expected value for the sample mea, E ( ) is the stadard error, σ, for the samplig distributio?? What c) What is the probability that a sigle radomly-chose subject from this populatio will exhibit a value of at least 0.55? d) You radomly select a sample of size = 00. What is the probability that the sample mea will be at least 0.55? e) You radomly select a sample of size = 00. There is a 5% probability that the sample mea will be below what value? aswers:, ; 5, ; 0.455; 0.05; Go to the supplemet for more examples worked out. There are examples similar to each of the homework exercises.

Sampling Distributions & Estimators

Sampling Distributions & Estimators API-209 TF Sessio 2 Teddy Svoroos September 18, 2015 Samplig Distributios & Estimators I. Estimators The Importace of Samplig Radomly Three Properties of Estimators 1. Ubiased 2. Cosistet 3. Efficiet I