CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS. 8.1 Random Sampling. 8.2 Some Important Statistics
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1 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 o the statistics computed from the sample data so that we could ifer somethig about the parameters ad obtai more iformatio about the populatio. Thus we must make sure that the samples must be good represetatives of the populatio ad pay attetio o the samplig bias ad variability to esure the validity of statistical iferece. To elimiate ay possibility of bias i the samplig procedure, it is desirable to choose a radom sample i the sese that the observatios are made idepedetly ad at radom. Radom Sample Let X 1,X 2,...,X be idepedet radom variables, each havig the same probability distributio f (x). Defie X 1,X 2,...,X to be a radom sample of size from the populatio f (x) ad write its joit probability distributio as f (x 1,x 2,...,x ) = f (x 1 ) f (x 2 ) f (x ). 8.2 Some Importat Statistics It is importat to measure the ceter ad the variability of the populatio. For the purpose of the iferece, we study the followig measures regardig to the ceter ad the variability Locatio Measures of a Sample The most commoly used statistics for measurig the ceter of a set of data, arraged i order of magitude, are the sample mea, sample media, ad sample mode. Let X 1,X 2,...,X represet radom variables. Bias Ay samplig procedure that produces ifereces that cosistetly overestimate or cosistetly uderestimate some characteristic of the populatio is said to be biased. Sample Mea To calculate the average, or mea, add all values, the divide by the umber of idividuals. X = X 1 + X X = 1 X i where X is the special symbol of the sample mea ad x = 1 x i deotes its value, or the realizatio of X.
2 38 Chapter 8. Fudametal Samplig Distributios ad Data Descriptios NOTE. The mea is the balace poit. It is the ceter of mass. EXAMPLE 8.1. The weights of a group of studets (i lbs) are give below: Fid the mea. If aother studet jois i the group ad his weight is 250 lbs, what would be the ew mea? Sample Media The umber such that half of the observatios are smaller ad half are larger, i.e., the midpoit of a distributio. { if is odd x = x (+1)/2 1 2 ( x/2 + x /2+1 ) if is eve EXAMPLE 8.2. The weights of a group of studets (i lbs) are give below: Fid the media. If aother studet jois i the group ad his weight is 250 lbs, what would be the ew media? Sample Mode The mode of a data set is the value that occurs most frequetly. The cases are uimodal, bimodal, multimodal ad o mode. The mode is/are the value(s) whose frequecies are the largest (the peaks). EXAMPLE 8.3. The weights of three group of studets (i lbs) are give below: (a) 135,105,118,163,172,183,122,150 (b) 135,105,118,163,172,183,122,135 (c) 135,135,118,118,122,118,122,135 Fid the mode for each group Variability Measures of a Sample The most commoly used statistics for measurig the ceter of a set of data, arraged i order of magitude, are the sample variace, sample stadard deviatio, ad sample rage. Let X 1,X 2,...,X represet radom variables. Sample Variace The sample variace S 2 is used to describe the variatio aroud the mea. We use s 2 = 1 1 (x i x) 2 ] = [ x 1 2 ( x)2 1 = (x2 ) ( x) 2 ( 1) to deote the realizatio or the computed values of S 2. Sample Stadard Deviatio The sample stadard deviatio is the squared root of the sample variace. ad s = S = S (x i x) 2 NOTE. Properties of Stadard Deviatio s measures spread about the mea ad should be used oly whe the mea is the measure of ceter. s = 0 oly whe all observatios have the same value ad there is o spread. Otherwise, s > 0. s gets larger, as the observatios become more spread out about their mea. s has the same uits of measuremet as the origial observatios. NOTE. The stadard deviatio of a populatio is defied by σ = 2, where N is the populatio (x µ) N size ad µ is populatio mea. Be careful with the deomiator iside square-root is N, istead of N 1. EXAMPLE 8.4. Calculate the sample variace ad the sample stadard deviatio of the followig set of data: Sample Rage The sample rage R of a data set is defied as R = X max X mi EXAMPLE 8.5. Refer to Example 8.4. Fid the sample rage. STAT-3611 Lecture Notes 2015 Fall X. Li
3 Sectio 8.4. Samplig Distributio of Meas ad the Cetral Limit Theorem Samplig Distributios Samplig Distributio I geeral, the samplig distributio of a give statistic is the distributio of the values take by the statistic i all possible samples of the same size form the same populatio. I other words, if we repeatedly collect samples of the same sample size from the populatio, compute the statistics (mea, stadard deviatio, proportio), ad the draw a histogram of those statistics, the distributio of that histogram teds to have is called the sample distributio of that statistics (mea, stadard deviatio, proportio). NOTE. The statistical applets are good tools to study the samplig distributio. Check out the Rice Uiversity Applets at sim/samplig_dist/idex.html. 8.4 Samplig Distributio of Meas ad the Cetral Limit Theorem Samplig Distributio of Sample Meas from a Normal Populatio Mea ad Stadard Deviatio of a Sample Mea Theorem. Let X be the sample mea of a radom sample of size draw from a populatio havig mea µ ad stadard deviatio σ, the the mea of X is µ X = µ ad the stadard deviatio of X is σ X = σ EXAMPLE 8.6. Prove the above theorem. The variatio of X is much smaller tha that of the populatio. The stadard deviatio of X decreases as the sample size icreases. The above results do NOT require ay assumptios o the shape of the populatio. However, a radom sample is a must. EXAMPLE 8.7. The mea ad stadard deviatio of the stregth of a packagig material are 55 kg ad 6 kg, respectively. A quality maager takes a radom sample of specimes of this material ad tests their stregth. If the maager wats to reduce the stadard deviatio of X to 1.5 kg, how may specimes should be tested? EXAMPLE 8.8. A soft-drik machie is regulated so that the amout of drik dispesed averages 240 milliliters with a stadard deviatio of 15 milliliters. Periodically, the machie is checked by takig a sample of 40 driks ad computig the average cotet. If the mea of the 40 driks is a value withi the iterval µ X ± 2σ X, the machie is thought to be operatig satisfactorily; otherwise, adjustmets are made. The compay official foud the mea of 40 driks to be x = 236 milliliters ad cocluded that the machie eeded o adjustmet. Was this a reasoable decisio? Samplig Distributio of Sample Meas from a Normal Populatio Theorem. Let X = 1 X i be the sample mea of a radom sample of size draw from a ormal populatio havig mea µ ad stadard deviatio σ, the X follows a exact ormal distributio with mea µ ad stadard deviatio σ/. That is, X i N(µ, σ) = X N ( µ, σ/ ). EXAMPLE 8.9. Prove the above theorem. NOTE. Oe of the essetial assumptios is a radom sample. The distributio of X has the EXACTLY ormal distributio if the radom sample is from a ormal populatio. EXAMPLE The cotets of bottles of beer vary accordig to a ormal distributio with mea µ = 341 ml ad stadard deviatio σ = 3 ml. (a) What is the probability that the cotet of a radomly selected bottle is less tha 339 ml? NOTE. The sample mea X is a ubiased estimator of the populatio mea µ ad is less variable tha a sigle observatio. (b) What is the probability that the average cotet of the bottles i a 12-pack of beer is less tha 339 ml? X. Li 2015 Fall STAT-3611 Lecture Notes
4 40 Chapter 8. Fudametal Samplig Distributios ad Data Descriptios EXAMPLE A patiet is classified as havig gestatioal diabetes if the glucose level is above 140 milligrams per deciliter (mg/dl) oe hour after a sugary drik is igested. Sheila s measured glucose level oe hour after igestig the sugary drik varies accordig to the ormal distributio with µ = 125 mg/dl ad σ = 10 mg/dl. (a) If a sigle glucose measuremet is made, what is the probability that Sheila is diagosed as havig gestatioal diabetes? (b) If measuremets are made o three separate days ad the mea result is compared with the criterio 140 mg/dl, what is the probability that Sheila is diagosed as havig gestatioal diabetes? (c) What is the level L such that there is probability oly 5% that the mea glucose level of three test results fall above L for Sheila s glucose level distributio The Cetral Limit Theorem (CLT) approximate probability statemet cocerig the sample mea, without kowledge of the shape of the populatio distributio. Agai, oe of the essetial assumptios is a radom sample. The distributio of X has the approximately ormal distributio if the radom sample is from a populatio other tha ormal. How large a sample size? Usually, it would safe to apply the CLT if 30. It also depeds o the populatio distributio, however. More observatios are required if the populatio distributio is far from ormal. EXAMPLE The time a family physicia speds seeig a patiet follows some right-skewed distributio with a mea of 15 miutes ad a stadard deviatio of 11.6 miutes. (a) Ca you calculate the probability that the doctor speds less tha 12 miutes with the ext patiet she sees? If so, do it. If ot, explai why. (b) What is the probability that the doctor speds a average time betwee 13 ad 18 miutes with her 30 patiets of the day? (c) Oe day, 35 patiets have a appoitmet to see the doctor. What is the probability that she will have to work overtime, beyod her 8-hour shift? Samplig Distributio of the Differece betwee Two Meas Theorem (Cetral Limit Theorem). If X is the mea of a radom sample of size take from a populatio with mea µ ad fiite variace σ 2, the as. Z = X µ σ/ (z;0,1) I other words, if a radom sample of size is selected from ay populatio with mea µ ad stadard deviatio σ, the X is approximately N ( µ, σ/ ), whe is sufficietly large. NOTE. The Cetral Limit Theorem is importat because, for reasoably large sample size, it allows us to make a Suppose that we have two populatios, the first with mea µ 1 ad stadard deviatio σ 1, ad the secod with mea µ 2 ad stadard deviatio σ 2. We take a radom sample of size 1 from the first populatio ad measure some variable X 1, ad take a idepedet radom sample of size 2 from the secod populatio ad measure the value of the some variable X 2. By the Cetral Limit Theorem, we kow that, if 1 ad 2 are sufficietly large, ad X 1 N(µ 1, σ 1 / 1 ), X 2 N(µ 2, σ 2 / 2 ). It ca also be show that X 1 X 2 N µ 1 µ 2, σ1 2 + σ STAT-3611 Lecture Notes 2015 Fall X. Li
5 Sectio 8.6. t-distributio 41 Theorem. If idepedet samples of size 1 ad 2 are draw at radom from two populatios, discrete or cotiuous, with meas µ 1 ad µ 2 ad variaces σ 2 1 ad σ 2 2, respectively, the the samplig distributio of the differeces of meas, X 1 X 2, is approximately ormally distributed with mea ad variace give by So, µ X1 X 2 = µ 1 µ 2 ad σ 2 X 1 X 2 = σ σ Z = ( X 1 X 2 ) (µ1 µ 2 ) σ 2 1 / 1 + σ 2 2 / 2 N(0, 1) NOTE. If both samples are from the ormal populatios, the samplig distributio of X 1 X 2 will be exactly ormal, istead of approximate ormal. EXAMPLE We take a radom sample of five 10- year-old boys ad four 10-year-old girls ad measure their heights. Suppose that we kow that heights X 1 of 10-year old boys follow a ormal distributio with mea 55.7 iches ad stadard deviatio 2.9 iches, ad that heights X 2 of 10-year old girls follow a ormal distributio with mea 54.1 iches ad stadard deviatio 2.6 iches. What is the probability that the mea height of the girls i the sample is smaller tha the mea height for the boys i the sample? EXAMPLE A research o bulimia amog college wome studies the coectio betwee childhood sexual abuse ad a measure of family cohesio (the higher the score, the greater the cohesio). Assume that sexually abused studets have a average family cohesio scale of 2.8 ad a stadard deviatio of 2.1, while o-abused studets have the average scale of 4.8 ad a stadard deviatio of 3.2. What is the probability that a radom sample of 49 o-abused studets will have a average family cohesio scale that is at least 0.5 scores higher tha the average scale of a radoms sample of 36 sexually abused studets? What ca you coclude? 8.5 Samplig Distributio of S 2 Distributio of ( 1)S 2 /σ 2 If S 2 is the variace of a radom sample of size take from a ormal populatio havig the variace σ 2, the the statistic χ 2 ( 1)S2 = σ 2 = ( Xi X ) 2 has a chi-squared distributio with ν = 1 degrees of freedom. σ 2 NOTE (Degrees of Freedom). There are degrees of freedom, or idepedet pieces of iformatio, i the radom sample from the ormal distributio. Whe the data (the values i the sample) are used to compute the mea (i.e., whe µ is replaced by x), a degree of freedom is lost i the estimatio of µ. Hece, there are the remaiig ( 1) degrees of freedom i the iformatio used to estimate σ 2. Let χ 2 α(ν) be the χ 2 value above which we fid a area of α uder the curve of the chi-squared distributio with ν degrees of freedom. That is, P ( χ 2 (ν) > χ 2 α(ν) ) = α. We use table A.5. to fid these critical values of the chi-squared distributio with ν degrees of freedom. EXAMPLE Fid the critical values (a) χ (4) (b) χ (22) EXAMPLE Fid k such that P ( χ 2 (12) < k ) = EXAMPLE Use Table A.5. to give the best estimate to each of the followig probabilities. (a) P ( χ 2 (5) 3 ) (b) P ( χ 2 (8) > 3.33 ) (c) P ( χ 2 (10) 6.66 ) (d) P ( χ 2 (25) > 99.9 ) 8.6 t-distributio We have leared that Z = X µ σ/ (exactly or approximately) follows the stadard ormal distributio, where the data are from a radom sample of size from the populatio with mea µ ad stadard deviatio σ. Ad, it is very likely that both µ ad σ are ukow parameters. I practice, it suffices that the distributio is symmetric ad sigle-peaked uless the sample is very small. Sice most of the simple work i statistical iferece focus o the ukow populatio mea µ, we will eed deal with the ukow σ especially whe is ot large. It is quite ituitive ad atural to estimate the ukow populatio stadard deviatio σ usig the sample stadard deviatio S. as a aa- We have aother statistic T = X µ S/ log sample versio of Z = X µ σ/. X. Li 2015 Fall STAT-3611 Lecture Notes
6 42 Chapter 8. Fudametal Samplig Distributios ad Data Descriptios Studet t distributio Let X 1,X 2,...,X be idepedet radom variables that are all ormal with mea µ ad stadard deviatio σ. Let X = 1 X i ad S 2 = 1 1 ( Xi X ) 2. The the radom variable T = X µ S/ has a t-distributio with ν = 1 degrees of freedom. NOTE. Whe is very large, s is a very good estimate of σ, ad the correspodig t distributios are very close to the ormal distributio. The t distributios become wider for smaller sample sizes, reflectig the lack of precisio i estimatig σ from s. Because the symmetrically property, t 1 α = t α. We use table A.4. to fid these critical values of the t distributio with ν degrees of freedom. NOTE. The t table, as well as the χ 2 table, gives us the UPPER tail probabilities, while the z table gives the lower tail probabilities. EXAMPLE Fid the critical values. (a) t (5), t 0.05 (5), t 0.5 (5), t 0.85 (5), t (5) (b) t 0.10 (10), t 0.20 (20), t 0.30 (30), t 0.40 (40), t 0.60 (60) (c) t 0.90 (10), t 0.95 (15), t 0.99 (19) EXAMPLE Let T (ν) deote the Studet t-distributio with ν degrees of freedom. Fid k such that (a) P(T (8) > k) = (b) P(T (18) < k) = (c) P(T (28) k) = EXAMPLE Use Table A.4. to give the best estimate to each of the followig probabilities. (a) P(T (5) 1.11) Chapter 8 Fudametal Samplig Distributios ad Data Descriptios Importat Properties of the Studet t distributio The t distributio is differet for differet sample sizes, or differet degrees of freedom..1: Let X 1,X 2,...,X be idepedet radom variables that are all ormal with mea µ ad stadard deviatio σ. Let X = 1 X i ad S 2 = 1 The the radom variable T = of freedom. (X i X) 2. The t distributio 1 has the same geeral symmetric bell shape as the stadard ormal distributio, but has a t-distributio with v = 1 degrees X µ S/ it reflects the greater variability (with wider distributios) that is expected with small samples. The probability distributio of T was first published i 1908 i a paper writte by W. S. Gosset. At the time, Gosset was employed by a Irish brewery that prohibited publicatio of research by members of its staff. To circumvet this restrictio, he published his work secretly uder the ame Studet. Cosequetly, the distributio of T is usually called the Studet t-distributio or simply the t- distributio. I derivig the equatio of this distributio, Gosset assumed that the samples were selected from a ormal populatio. Although this would seem to be a very restrictive assumptio, it ca be show that oormal populatios possessig early bell-shaped distributios will still provide values of T that approximate the t-distributio very closely. the t-distributio Look Like? The t distributio has a mea of t = 0. The stadard deviatio of the t distributio varies with the sample size, but it is greater tha 1. As the sample size gets larger, the t distributio gets closer to the stadard ormal distributio. The distributio of T is similar to the distributio of Z i that they both are symmetric about a mea of zero. Both distributios are bell shaped, but the t- distributio is more variable, owig to the fact that the T -values deped o the fluctuatios of two quatities, X ad S 2, whereas the Z-values deped oly o the chages i X from sample to sample. The distributio of T differs from that of Z Let t α (ν) be the t value above which we fid a area of α uder the curve of the t distributio with ν degrees of freedom. That is, i that the variace of T depeds o the sample size ad is always greater tha 1. Oly whe the sample size will the two distributios become the same. I Figure 8.8, we show the relatioship betwee a stadard ormal distributio (v = ) ad t-distributios with 2 ad 5 degrees of freedom. The percetage poits of the t-distributio are give i Table A.4. v v 5 v P(T (ν) > t α (ν)) = α. t 1 α tα 0 tα t (b) P(T (8) < 2.22) (c) P(T (10) 3.33) (d) P(T (15) > 4.44) NOTE. Clearly, More geerally, T = = X µ σ/ [( 1)S 2 /σ 2 ] 1 Z χ 2 ( 1) 1 Theorem. Let Z be a stadard ormal radom variable ad V a chi-squared radom variable with ν degrees of freedom. If Z ad V are idepedet, the the distributio of the radom variable T, where T = Z V /ν is give by the desity fuctio h(t) = ( ) (ν+1)/2 Γ[(ν + 1)/2] Γ(ν/2) 1 + t2, < t <. πν ν This is kow as the t-distributio with ν degrees of freedom. t-distributio curves for v = 2, 5, Figure 8.9: Symmetry property (about 0) of the t-distributio. STAT-3611 Lecture Notes 2015 Fall X. Li
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