How is the President Doing? Sampling Distribution for the Mean. Now we move toward inference. Bush Approval Ratings, Week of July 7, 2003

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1 Samplig Distributio for the Mea Dr Tom Ilveto FREC How is the Presidet Doig? 2/1/2001 4/1/2001 Presidet Bush Approval Ratigs February 1, 2001 through October 6, /1/2001 8/1/ /1/ /1/2001 2/1/2002 4/1/2002 6/1/2002 8/1/ /1/ /1/2002 2/1/2003 4/1/2003 6/1/2003 8/1/ /1/2003 Bush Approval Ratigs, Week of July 7, 2003 Presidetial Poll Results Week of September 20-27, % 62% 60% 58% 56% 54% 52% 50% 1,006 adults Gallup 753 adults CBS Ipsos/Reid 764 adults ABC 1,006 adults Newsweek 1,017 adults 50% 48% 46% 44% 42% 40% 38% 36% 1,013 adults ICR LA Times 1,052 adults CNN ZOGBY 640 adults 1,213 adults Bush Gore VOTER.com 1,000 adults We epect variability from sample to sample we call it samplig error We epect variability from sample to sample we call it samplig error (Def1.19 p48) Now we move toward iferece Remember we oted that A parameter is a umerical descriptive measure of the populatio (Def3.15 p178) We use Greek terms to represet it It is hardly ever kow A sample statistic is a umerical descriptive measure from a sample (Def6.4 p311) Based o the observatios i the sample We wat the sample to be derived from a radom process Let s set up a small eperimet Toss a die three times Each time we toss the die three times we ote ad record the faces The calculate mea ad media We ca do this a umber of times 1

2 A Priori we have the followig epectatio X P() E() = 1(.1667) + 2(.1667) +3(.1667) + 4(.1667) + 5(.1667) + 6(.1667) E() = E(-:) 2 = (1-3.5) 2 (.1667) + (2-3.5) 2 (.1667) + (3-3.5) 2 (.1667) + (4-3.5) 2 (.1667) + (5-3.5) 2 (.1667) + (6-3.5) 2 (.1667) E(-:) 2 = F = As a eperimet Roll 3 times Roll 3 times Roll 3 times Roll 3 times Roll 3 times Mea Media Roll 3 times Results of 217 samples of size 3 Mea Media Mea Stadard Error Media Mode Stadard Deviatio Sample Variace Kurtosis Skewess Rage Miimum Maimum Sum Cout Note: I worked out all possible outcomes There are 6*6*6 = 216 differet combiatios of outcomes of rollig three die If I take the mea of each possible outcome Ad take the summary statistics (icludig the mea of the meas) I get the followig table (from Ecel) Samplig Distributio of Sample Mea for rollig 3 die Descriptives Mea 3.50 Stadard Error 0.07 Media 3.50 Mode 3.33 Stadard Deviatio 0.99 Sample Variace 0.98 Kurtosis Skewess 0.00 Rage 5.00 Miimum 1.00 Maimum 6.00 Sum 756 Cout 216 We said that the stadard deviatio for rollig a die was: F = Divide this figure by the Square root of 3 (sample Size), Samplig Distributio of Sample Mea for rollig 3 die Descriptives Mea 3.50 Stadard Error 0.07 Media 3.50 Mode 3.33 Stadard Deviatio 0.99 Sample Variace 0.98 Kurtosis Skewess 0.00 Rage 5.00 Miimum 1.00 Maimum 6.00 Sum 756 Cout 216 We said that the stadard deviatio for rollig a die was: F = Divide this figure by the Square root of 3 (sample Size), we get the Stadard Deviatio of, which is 0.99 This is also called the Stadard Error of (p317) 2

3 Stem ad Leaf of 3-die of a Priori Samplig Distributio Stem is the oes; leaf is the first decimal place. Compare the Mea ad Media of our Sample Distributio Mea Media Mea Stadard Error Media Mode Stadard Deviatio Sample Variace Kurtosis Skewess Rage Miimum Maimum Sum Cout The mea has Miimum Variace Samplig Distributio Sample statistics are radom variables They have a probability distributio based o repeatig the samplig eperimet may times. We may get differet sample statistics each time Repeatig the eperimet may times results i a samplig distributio The samplig distributio of a sample statistic calculated from a sample of measuremets is the probability distributio of the statistic (Def6.5 p311) What do we wat to see? If 0 is a good estimator of : We would epect the values of 0 to cluster aroud : We would t wat to cluster to be at a poit above or below : (ot be biased) Ad we might say our estimator is good if the cluster of the 0 s aroud : is tighter tha the samplig distributio of some other possible estimator (miimum variace) I asked a past class to help costruct a samplig distributio for the mea Results of the Eperimet based o repeated samples from a populatio with a mea of 75 ad a stadard deviatio of 10 It was repeated samples of size = 50 From ~ N (75, 10) The mea of the meas should equal the populatio parameter : The stadard deviatio of this ew distributio should be related to F But we might epect it to be smaller σ = σ 10/(50).5 = 1.41 AKA the Stadard Error (p317) Samplig Distributio for the Mea Stem = 10's place Leaf = oe s digit mea X = = 55 Std Dev = 1.35 Media = 74.8 The eercise resulted i a Std Dev = 1.35 This is close to the epected Std Error of 10/(50).5 =

4 Samplig theory ad samplig distributios help make ifereces to a populatio Let's use the eample of the mea to set up our discussio of a samplig distributio. Suppose we are lookig at a variable, e.g., blood pressure. We thik of the populatio (let's use the Populatio of adult males i Delaware age 18 to 85 i 2002). We believe there is a average blood pressure of this populatio, desigated as µ. We wat to take a sample to estimate µ. Ifereces from a sample Our sample estimator to populatio mea is: = The variace of our sample estimate is give as: 2 ( X ) 2 s = 1 where is equal to the sample size s 2 is a ubiased estimator of populatio variace σ 2 Ifereces from a sample The stadard deviatio represets the average deviatio aroud the sample mea. But we oly took oe sample out of a ifiite umber of possible samples. A reasoable questio would be what is the deviatio aroud our estimator (i.e., the sample mea). I.e., what if we took a whole buch of samples, ad recorded the mea of each sample Ifereces from a sample If we could take a ifiite umber of samples, each sample would most likely yield a differet sample mea. Yet, each oe would be epressed as a reasoable estimate of the true populatio mea. So, if we were able to take repeated samples, each of sample size, what would be the stadard deviatio of the sample estimates? Ifereces from a sample Samplig theory specifies the variace of the samplig distributio of a mea as: 2 σ Var( ) Var = = σ σ = This is called the Stadard Error of the mea (p317) Ifereces from a sample The stadard error of the mea is the stadard deviatio of a samplig distributio of meas with populatio parameters equal to µ ad F 2. If we do't kow F 2 we use the ubiased sample estimate of s 2 to estimate the samplig variace of the mea. 4

5 Here s Our Strategy We use the theoretical samplig distributio to make ifereces from our sample to the populatio. The samplig distributio of a estimator is based o repeated samples of size. We may ever actually take repeated samples But we could thik of this happeig Ad our observed sample as oe of may possible samples, of size, we could have draw from the populatio Here s our strategy We epect that the stadard deviatio of samplig distributio of the estimator (i this case the mea) will be smaller tha that of the populatio or the samples themselves. We epect some variability across samples, but ot as much as we would fid i the populatio. Thus the samplig error is smaller tha the stadard deviatio for the populatio. The stadard error depeds upo: The size of (as gets larger the SE gets smaller) The variace of the populatio variable itself. We ca thik of this as homogeeity. The larger the sample size, ad the more homogeeous the populatio, the smaller the stadard error for our estimator. Properties of the Samplig Distributio of (p317) If is the mea of a radom sample of size from a populatio with mea µ ad stadard deviatio σ, the: The samplig distributio of has a mea equal to the populatio mea µ. If we use µ deote the mea of, the µ = µ Properties of the Samplig Distributio of (cot.) Ad the samplig distributio of has a stadard deviatio equal to the stadard deviatio of the populatio stadard deviatio σ, divided by the square root of the sample size. If we use σ deote the stadard deviatio of, the: σ = σ Properties of the Samplig Distributio of (cot.) Ad the samplig distributio of has a stadard deviatio equal to the stadard deviatio of the populatio stadard deviatio σ, divided by the square root of the sample size. If we use do t kow σ, the we use the sample stadard deviatio to estimate it s = s 5

6 We use two theorems to help us make ifereces I the case of the mea, we use two theorems cocerig the ormal distributio that help us make ifereces Oe depeds upo the variable beig ormally distributed The other does ot - Cetral Limit Theorem For variables that are distributed ormally If repeated samples of a variable Y of size are draw from a ormal distributio, with mea µ ad variace F 2, the samplig distributio will be ormal with mea µ ad variace F 2 /. For variables that are distributed ormally What this meas: If we could repeatedly take radom samples of size from a ormal distributio. Ad the take the mea of each sample We would epect the mea of the sample meas to equal µ Ad the variace of the sample meas would equal F 2 / Cetral Limit Theorem If repeated sample of Y of size are draw from ay populatio (regardless of its distributio as ormal or otherwise) havig a mea µ ad variace F 2, the samplig distributio of the sample meas approaches ormality, with µ ad variace F 2 /. As log as the sample size is sufficietly large (p317) Cetral Limit Theorem The Cetral Limit Theorem is a very powerful theorem. It relaes the assumptio of the distributio of the populatio variable Samplig distributio of for differet populatio ad differet sample sizes (page 319) Note: this is based o the otio that our samples are draw o a probability basis. That is, each elemet of the populatio has a equal chace of beig selected 6

7 Ifereces from a Sample Compariso of the Characteristics of the Populatio, Sample, ad the Samplig Distributio for the Mea Referred to as How it is viewed Mea Variace Stadard Deviatio Populatio Parameters Assumed real : = 3X/N F 2 = 3(X- :) 2 /N Our Sample Sample Statistics s = (s 2 ).5 Note: N ad X are for the Populatio, ad are for the sample F Observed 0 = 3/ s 2 = 3(- 0) 2 /(-1) Samplig Distributio Statistics Theoretical : = 30/N F 2 0 = F2 / F 0 = F// So how do we use this iformatio? We draw a radom sample We thik of our sample as oe of may possible samples of size from a populatio with parameters : ad F. If the variable is distributed ormally, we ca use iformatio about the samplig distributio of the mea to make ifereces from the sample to the populatio. Eve if the variable is ot distributed ormally, if our sample size () is large eough, we ca assume the samplig distributio of sample mea is distributed ormally (Cetral Limit Theorem p317) Rare Evet Approach Most ifereces will be made usig a rare evet approach We will take a sample Ad compare it to a hypothesized populatio Ad see how close or far away our sample estimate is from the samplig distributio Automobile Batteries The maufacturer claims that life of his automobile batteries is 54 moths o average, with a stadard deviatio of 6 moths. We are ivolved i a cosumer group ad we decide to take a sample of 100 batteries ad test the claim. We select 100 batteries at radom Test them over time ad record the legth of the battery life Our Sample Data The mea battery life for our sample is: Mea = 52 moths Std Dev = 4.5 moths So what do we do et? Our batteries did t last as log o average as the maufacturer said, but it is just a sample. How ca we test to see if the claim is bogus? Our Testig Strategy If the world works as the maufacturer says Ad I would have take repeated samples of size 100 The samplig distributio would be a ormal distributio Ad have a mea equal to the populatio mea for battery life, i.e., 54 moths Ad a stadard deviatio of F divided by the square root of 6 σ = =

8 Our Testig Strategy We wat to look at our sample as beig part of the theoretical Samplig Distributio. That is, ~ N ( µ, σ ) I this case, ~ N (54, 0.6) Ad see how likely it is that our sample came from that distributio I other words, how likely is it to get a sample mea of 52 from a radom sample of 100 batteries whe the true populatio mea is 54 moths How do I do this? I hypothesize that the true mea is 54 I calculate a z-score based o my sample value (52.0) ad the hypothesized mea ad stadard error (of the samplig distributio) I look up the probability of fidig a z- score equal to or less tha our calculated value The Test Calculate my z-score (52 54) z = = Look up the value i the stadard ormal table Ad p =.0010 after that poit Draw it out! z = correspods to a probability of.4990 up to that poit This is really a rare evet give the claim of the maufacturer that the batteries really last 54 moths o average 52 moths 54 moths What is our sample size was 30? The stadard error of the samplig distributio would chage σ z = = 6 = (52 54) = Ad p =.0336 after that poit Draw it out! z = correspods to a probability of.4664 up to that poit 52 moths 54 moths While this is still a relatively rare evet, I m ot as cofidet that I ca refute the claim of the maufacturer 8

9 Eample Problem The average live weight of a farmer s steers prior to slaughter i past years was 380 pouds. This year, 50 radomly chose steers were fed o a ew diet. Test to see if the mea weight for the sample o the ew diet is larger tha 380. Sample statistics =50 Mea = 390 s = 35.2 NOTE: s is a ubiased estimator of σ. We use s whe σ is ukow Strategy for the Problem Preted that our sample really does come from a populatio with a mea = 380 Compare our sample to the theoretical samplig distributio for = 50 with : = 380 F = 35.2/(50).5 = 4.98 Strategy for the Problem Net we see how likely it would be that our sample would come from a populatio so described, i.e., we see where our sample would lie i the samplig distributio if the populatio mea really was 380. The book calls this a rare-evet approach (p215-16) This compariso is doe usig a z-score test statistic Differece of our sample from the populatio parameter, divided by the stadard deviatio of the samplig statistic Steer Problem Calculate Test Statistic z = ( )/[35.2/(50).5 ] z = 10/4.98 = 2.01 Steer Problem Look up the Test Statistic i the book z= 2.01 relates to a probability of.4778 o the right side of the curve Which meas we would epect to fid a value out there or further oly at =.022 Or 2.2% of the time Steer Problem : coclusio While it is possible that a sample with a mea of 390 could come from a populatio with a mea of 380 The probability is very small,.022 If I was bettig o this oe I would have good evidece to suggest that the ew diet resulted i icreased weight I.e., that the sample did ot come from a populatio with : = 380 9

10 I could also poit a Boud of Error or Cofidece Iterval o my sample estimate I specify a level of cofidece that I wat 95% C.I. Fid the z value associated with this level I take the 95% level Divide it by 2 for each side.475 Ad search for the z-value associated with this probability level 1.96 Cofidece Iterval I would calculate the cofidece iterval as 390 " 1.96[35.2/(50).5 ] 390 " 1.96(4.98) 390 " to What does the Cofidece Iterval Mea? I repeated samples 95% of the samples draw Would have a cofidece iterval that cotais the true populatio parameter It does ot mea that the populatio parameter or the true value is betwee those two umbers I calculated for my sample we do t kow for sure. We just kow that i repeated samples, 95% of the samples would geerate a cofidece iterval that cotais : Summary We ca make ifereces from a sample to a populatio if: The sample is draw radomly We use a estimator to geerate sample statistics We kow somethig about the Samplig Distributio of our estimator - This helps us geerate the Stadard Error of our estimator Summary We ca use a iterval estimate via a Cofidece Iterval Or a poit estimate i a Hypothesis Test To make ifereces to the populatio parameter. 10