Sampling Distributions and Estimation
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1 Cotets 40 Samplig Distributios ad Estimatio 40.1 Samplig Distributios 40. Iterval Estimatio for the Variace 13 Learig outcomes You will lear about the distributios which are created whe a populatio is sampled. For example, every sample will have a mea value; this gives rise to a distributio of mea values. We shall look at the behaviour of this distributio. We shall also look at the problem of estimatig the true value of a populatio mea (for example) from a give sample.
2 Samplig Distributios 40.1 Itroductio Whe you are dealig with large populatios, for example populatios created by the maufacturig processes, it is impossible, or very difficult ideed, to deal with the whole populatio ad kow the parameters of that populatio. Items such as car compoets, electroic compoets, aircraft compoets or ordiary everyday items such as light bulbs, cycle tyres ad cutlery effectively form ifiite populatios. Hece we have to deal with samples take from a populatio ad estimate those populatio parameters that we eed. This Workbook will show you how to calculate sigle umber estimates of parameters - called poit estimates - ad iterval estimates of parameters - called iterval estimates or cofidece itervals. I the latter case you will be able to calculate a rage of values ad state the cofidece that the true value of the parameter you are estimatig lies i the rage you have foud. Prerequisites Before startig this Sectio you should... Learig Outcomes O completio you should be able to... uderstad ad be able to calculate meas ad variaces be familiar with the results ad cocepts met i the study of probability be familiar with the ormal distributio uderstad what is meat by the terms sample ad samplig distributio explai the importace of samplig i the applicatio of statistics explai the terms poit estimate ad the term iterval estimate calculate poit estimates of meas ad variaces fid iterval estimates of populatio parameters for give levels of cofidece HELM (008): Workbook 40: Samplig Distributios ad Estimatio
3 1. Samplig Why sample? Cosiderig samples from a distributio eables us to obtai iformatio about a populatio where we caot, for reasos of practicality, ecoomy, or both, ispect the whole of the populatio. For example, it is impossible to check the complete output of some maufacturig processes. Items such as electric light bulbs, uts, bolts, sprigs ad light emittig diodes (LEDs) are produced i their millios ad the sheer cost of checkig every item as well as the time implicatios of such a checkig process reder it impossible. I additio, testig is sometimes destructive - oe would ot wish to destroy the whole productio of a give compoet! Populatios ad samples If we choose items from a populatio, we say that the size of the sample is. If we take may samples, the meas of these samples will themselves have a distributio which may be differet from the populatio from which the samples were chose. Much of the practical applicatio of samplig theory is based o the relatioship betwee the paret populatio from which samples are draw ad the summary statistics (mea ad variace) of the offsprig populatio of sample meas. Not surprisigly, i the case of a ormal paret populatio, the distributio of the populatio ad the distributio of the sample meas are closely related. What is surprisig is that eve i the case of a o-ormal paret populatio, the offsprig populatio of sample meas is usually (but ot always) ormally distributed provided that the samples take are large eough. I practice the term large is usually take to mea about 30 or more. The behaviour of the distributio of sample meas is based o the followig result from mathematical statistics. The cetral limit theorem I what follows, we shall assume that the members of a sample are chose at radom from a populatio. This implies that the members of the sample are idepedet. We have already met the Cetral Limit Theorem. Here we will cosider it i more detail ad illustrate some of the properties resultig from it. Much of the theory (ad hece the practice) of samplig is based o the Cetral Limit Theorem. While we will ot be lookig at the proof of the theorem (it will be illustrated where practical) it is ecessary that we uderstad what the theorem says ad what it eables us to do. Essetially, the Cetral Limit Theorem says that if we take large samples of size with mea X from a populatio which has a mea µ ad stadard deviatio σ the the distributio of sample meas X is ormally distributed with mea µ ad stadard deviatio σ. That is, the samplig distributio of the mea X follows the distributio ( ) σ X N µ, Strictly speakig we require σ <, ad it is importat to ote that o claim is made about the way i which the origial distributio behaves, ad it eed ot be ormal. This is why the Cetral Limit Theorem is so fudametal to statistical practice. Oe implicatio is that a radom variable which takes the form of a sum of may compoets which are radom but ot ecessarily ormal will itself be ormal provided that the sum is ot domiated by a small umber of compoets. This explais why may biological variables, such as huma heights, are ormally distributed. HELM (008): Sectio 40.1: Samplig Distributios 3
4 I the case where the origial distributio is ormal, the relatioship ( betwee ) the origial distributio X N(µ, σ) ad the distributio of sample meas X σ N µ, is show below. X N ( ) σ μ, μ Figure 1 X N(μ, σ) The distributios of X ad X have the same mea µ but X has the smaller stadard deviatio σ The theorem says that we must take large samples. If we take small samples, the theorem oly holds if the origial populatio is ormally distributed. Stadard error of the mea You will meet this term ofte if you read statistical texts. It is the ame give to the stadard deviatio of the populatio of sample meas. The ame stems from the fact that there is some ucertaity i the process of predictig the origial populatio mea from the mea of a sample or samples. Key Poit 1 For a sample of idepedet observatios from a populatio with variace σ, the stadard error of the mea is σ σ. Remember that this quatity is simply the stadard deviatio of the distributio of sample meas. 4 HELM (008): Workbook 40: Samplig Distributios ad Estimatio
5 Fiite populatios Whe we sample without replacemet from a populatio which is ot ifiitely large, the observatios are ot idepedet. This meas that we eed to make a adjustmet i the stadard error of the mea. I this case the stadard error of the sample mea is give by the related but more complicated formula σ,n σ N N 1 where σ,n is the stadard error of the sample mea, N is the populatio size ad is the sample size. Note that, i cases where the size of the populatio N is large i compariso to the sample size, the quatity N N 1 1 so that the stadard error of the mea is approximately σ/. Illustratio - a distributio of sample meas It is possible to illustrate some of the above results by settig up a small populatio of umbers ad lookig at the properties of small samples draw from it. Notice that the settig up of a small populatio, ( say ) of size 5, ad takig samples of size eables us to deal with the totality of samples, 5 there are 5! 10 distict samples possible, whereas if we take a populatio of 100 ad!3! ( ) 100 draw samples of size 10, there are 100! 51, 930, 98, 370, 000 possible distict samples 10 10!90! ad from a practical poit of view, we could ot possibly list them all let aloe work with them! Suppose we take a populatio cosistig of the five umbers 1,, 3, 4 ad 5 ad draw samples of size to work with. The complete set of possible samples is: (1, ), (1, 3), (1, 4), (1, 5), (, 3), (, 4), (, 5), (3, 4), (3, 5), (4, 5) For the paret populatio, sice we kow that the mea µ 3, the we ca calculate the stadard deviatio by (1 3) + ( 3) σ + (3 3) + (4 3) + (5 3) For the populatio of sample meas, 1.5,,.5, 3,.5, 3, 3.5, 3.5, 4, 4.5 their mea ad stadard deviatio are give by the calculatios: ad (1.5 3) + ( 3) + + (4 3) + (4.5 3) We ca immediately coclude that the mea of the populatio of sample meas is the same as the populatio mea µ. HELM (008): Sectio 40.1: Samplig Distributios 5
6 Usig the results give above the value of σ,n should be give by the formula σ,n σ N N 1 with σ 1.414, N 5 ad. Usig these umbers gives: σ,5 σ N N as predicted. N Note that i this case the correctio factor ad is sigificat. If we take samples N 1 of size 10 from a populatio of 100, the factor becomes N N ad for samples of size 10 take from a populatio of 1000, the factor becomes N N Thus as N N 1 1, its effect o the value of σ reduces to isigificace. Task Two-cetimetre umber 10 woodscrews are maufactured i their millios but packed i boxes of 00 to be sold to the public or trade. If the legth of the screws is kow to be ormally distributed with a mea of cm ad variace 0.05 cm, fid the mea ad stadard deviatio of the sample mea of 00 boxed screws. What is the probability that the sample mea legth of the screws i a box of 00 is greater tha.0 cm? Your solutio 6 HELM (008): Workbook 40: Samplig Distributios ad Estimatio
7 Aswer Sice the populatio is very large ideed, we are effectively samplig from a ifiite populatio. The mea ad stadard deviatio are give by 0.05 µ cm ad σ cm 00 Sice the paret populatio is ormally distributed the meas of samples of 00 will be ormally distributed as well. Hece P(sample mea legth >.0) P(z >.0 ) P(z > 1.5) Statistical estimatio Whe we are dealig with large populatios (the productio of items such as LEDs, light bulbs, pisto rigs etc.) it is extremely ulikely that we will be able to calculate populatio parameters such as the mea ad variace directly from the full populatio. We have to use processes which eable us to estimate these quatities. There are two basic methods used called poit estimatio ad iterval estimatio. The essetial differece is that poit estimatio gives sigle umbers which, i the sese defied below, are best estimates of populatio parameters, while iterval estimates give a rage of values together with a figure called the cofidece that the true value of a parameter lies withi the calculated rage. Such rages are usually called cofidece itervals. Statistically, the word estimate implies a defied procedure for fidig populatio parameters. I statistics, the word estimate does ot mea a guess, somethig which is rough-ad-ready. What the word does mea is that a agreed precise process has bee (or will be) used to fid required values ad that these values are best values i some sese. Ofte this meas that the procedure used, which is called the estimator, is: (a) cosistet i the sese that the differece betwee the true value ad the estimate approaches zero as the sample size used to do the calculatio icreases; (b) ubiased i the sese that the expected value of the estimator is equal to the true value; (c) efficiet i the sese that the variace of the estimator is small. Expectatio is covered i Workbooks 37 ad 38. You should ote that it is ot always possible to fid a best estimator. You might have to decide (for example) betwee oe which is cosistet, biased ad efficiet ad oe which is cosistet, ubiased ad iefficiet whe what you really wat is oe which is cosistet, ubiased ad efficiet. HELM (008): Sectio 40.1: Samplig Distributios 7
8 Poit estimatio We will look at the poit estimatio of the mea ad variace of a populatio ad use the followig otatio. Notatio Estimatig the mea This is straightforward. ˆµ x Populatio Sample Estimator Size N Mea µ or E(x) x ˆµ for µ Variace σ or V(x) s ˆσ for σ is a sesible estimate sice the differece betwee the populatio mea ad the sample mea disappears with icreasig sample size. We ca show that this estimator is ubiased. Symbolically we have: ˆµ x 1 + x + x so that E(ˆµ) E(x 1) + E(x ) + + E(x ) E(X) + E(X) + + E(X) E(X) µ Note that the expected value of x 1 is E(X), i.e. E(x 1 ) E(X). Similarly for x 1, x,, x. Estimatig the variace (x µ) This is a little more difficult. The true variace of the populatio is σ which suggests N (x µ) the estimator, calculated from a sample, should be ˆσ. However, we do ot kow the true value of µ, but we do have the estimator ˆµ x. Replacig µ by the estimator ˆµ x gives (x x) ˆσ This ca be writte i the form (x x) ˆσ x ( x) Hece E(ˆσ ) E( x ) E{( X) } E(X ) E{( X) } 8 HELM (008): Workbook 40: Samplig Distributios ad Estimatio
9 We already have the importat result E(x) E( x) ad V( x) V(x) Usig the result E(x) E( x) gives us E(ˆσ ) E(x ) E{( x) } E(x ) {E(x)} E{( x) } + {E( x)} E(x ) {E(x)} (E{( x) } {E( x)} ) V(x) V( x) σ σ 1 σ This result is biased, for a ubiased estimator the result should be σ ot 1 σ. Fortuately, the remedy is simple, we just multiply by the so-called Bessel s correctio, amely ad obtai the result ˆσ (x x) (x x) There are two poits to ote here. Firstly (ad rather obviously) you should ot take samples of size 1 sice the variace caot be estimated from such samples. Secodly, you should check the operatio of ay had calculators (ad spreadsheets!) that you use to fid out exactly what you are calculatig whe you press the butto for stadard deviatio. You might fid that you are calculatig either (x µ) σ (x x) or ˆσ N 1 It is just as well to kow which, as the first formula assumes that you are calculatig the variace of a populatio while the secod assumes that you are estimatig the variace of a populatio from a radom sample of size take from that populatio. From ow o we will assume that we divide by 1 i the sample variace ad we will simply write s for s 1. Iterval estimatio We will look at the process of fidig a iterval estimatio of the mea ad variace of a populatio ad use the otatio used above. Iterval estimatio for the mea This iterval is commoly called the Cofidece Iterval for the Mea. Firstly, we kow that while the sample mea x x 1 + x + + x is a good estimator of the populatio mea µ. We also kow that the calculated mea x of a sample of size is ulikely to be exactly equal to µ. We will ow costruct a iterval aroud x i such a way that we ca quatify the cofidece that the iterval actually cotais the populatio mea µ. Secodly, we kow that for sufficietly large samples take from a large populatio, x follows a ormal distributio with mea µ ad stadard deviatio σ. HELM (008): Sectio 40.1: Samplig Distributios 9
10 Thirdly, lookig at the followig extract from the ormal probability tables, Z X µ σ we ca see that 47.5% 95% of the values i the stadard ormal distributio lie betwee ±1.96 stadard deviatio either side of the mea. So before we see the data we may say that P (µ 1.96 σ x µ σ ) 0.95 After we see the data we say with 95% cofidece that µ 1.96 σ x µ σ which leads to x 1.96 σ µ x σ This iterval is called a 95% cofidece iterval for the mea µ. Note that while the 95% level is very commoly used, there is othig sacrosact about this level. If we go through the same argumet but demad that we eed to be 99% certai that µ lies withi the cofidece iterval developed, we obtai the iterval x.58 σ µ x +.58 σ sice a ispectio of the stadard ormal tables reveals that 99% of the values i a stadard ormal distributio lie withi.58 stadard deviatios of the mea. The above argumet assumes that we kow the populatio variace. I practice this is ofte ot the case ad we have to estimate the populatio variace from a sample. From the work we have see above, we kow that the best estimate of the populatio variace from a sample of size is give by the formula (x x) ˆσ 1 It follows that if we do ot kow the populatio variace, we must use the estimate ˆσ i place of σ. Our 95% ad 99% cofidece itervals (for large samples) become where x 1.96 ˆσ µ x ˆσ ad x.58 ˆσ µ x +.58 ˆσ ˆσ (x x) 1 Whe we do ot kow the populatio variace, we eed to estimate it. Hece we eed to gauge the cofidece we ca have i the estimate. I small samples, whe we eed to estimate the variace, the values 1.96 ad.58 eed to be replaced by values from the Studet s t-distributio. See HELM (008): Workbook 40: Samplig Distributios ad Estimatio
11 Example 1 After 1000 hours of use the weight loss, i gm, due to wear i certai rollers i machies, is ormally distributed with mea µ ad variace σ. Fifty idepedet observatios are take. (This may be regarded as a large sample.) If observatio i is y i, the y i 497. ad yi i1 Estimate µ ad σ ad give a 95% cofidece iterval for µ. i1 Solutio We estimate µ usig the sample mea: ȳ yi We estimate σ usig the sample variace: s 1 (yi ȳ) 1 { y i 1 [ ] } yi { } gm gm s The estimated stadard error of the mea is gm 50 s The 95% cofidece iterval for µ is ȳ ± That is < µ < Exercises 1. The voltages of sixty omially 10 volt cells are measured. Assumig these to be idepedet observatios from a ormal distributio with mea µ ad variace σ, estimate µ ad σ. Regardig this as a large sample, fid a 99% cofidece iterval for µ. The data are: The atural logarithms of the times i miutes take to complete a certai task are ormally distributed with mea µ ad variace σ. Sevety-five idepedet observatios are take. (This may be regarded as a large sample.) If the atural logarithm of the time for observatio i is y i, the y i ad y i Estimate µ ad σ ad give a 95% cofidece iterval for µ. Use your cofidece iterval to fid a 95% cofidece iterval for the media time to complete the task. HELM (008): Sectio 40.1: Samplig Distributios 11
12 Aswers 1. y i 611.0, y i ad 60. We estimate µ usig the sample mea: ȳ yi V We estimate σ usig the sample variace: s 1 (yi ȳ) 1 { y i 1 [ ] } yi { } The estimated stadard error of the mea is s V The 99% cofidece iterval for µ is ȳ ±.58 s /. That is < µ < We estimate µ usig the sample mea: ȳ yi We estimate σ usig the sample variace: s 1 (yi ȳ) 1 { y i 1 [ ] } yi { } The estimated stadard error of the mea is s The 95% cofidece iterval for µ is ȳ ± 1.96 s /. That is < µ <.005 The 95% cofidece iterval for the media time, i miutes, to complete the task is e < M < e.005 That is 6.93 < M < HELM (008): Workbook 40: Samplig Distributios ad Estimatio
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