Math 140 Introductory Statistics

Transcription

1 6. Probability Distributio from Data Math Itroductory Statistics Professor Silvia Ferádez Chapter 6 Based o the book Statistics i Actio by A. Watkis, R. Scheaffer, ad G. Cobb. We have three ways of specifyig a populatio:. List of all (idividual) uits. Frequecy Table (p. 68). Relative Frequecy or Proportio Table (p. ) How ca we calculate mea ad SD o each? List of all uits Frequecy Tables Number Type Value populatio mea Type Dime Value Frequecy f. f populatio mea + + f 6 8 Dime - 6 SD ( ) Sum f SD ( ) f Dime Total cois Sum cets 6.6.6

2 Relative Frequecy or Proportio Table Summary of Mea/SD Type Value Proportio of cois P().. P(). populatio mea P ( ) List of all uits Frequecy Table Relative Frequecy (or Proportio) Table Dime Sum..... SD ( ) P ( ) (.) + (.) + 6 (.) ( ) f ( ) f P () ( ) P ( ) Eample (page ) Number of Motor Vehicles (per household), Proportio of households, P() The umber of motor vehicles per household. The represets or more, but the proportio of households with more tha four vehicles is very small. [Source: U.S. Cesus Bureau, America Commuity Survey,, factfider.cesus.gov.] How to Sample We have three ways of specifyig a populatio:. List of all (idividual) uits. Frequecy Table (p. 68). Relative Frequecy or Proportio Table (p. ) How ca we get (or simulate) a sample from these distributios?

3 How to Sample Ways to geerate radom umbers List of all Uits ad Frequecy Tables.. Make a umbered list of all uits. If usig a frequecy table remember to repeat each value accordig to its multiplicity.. Draw as may umbers at radom as eeded. Relative Frequecy (or Proportio) Tables. Assig a umber to each proportio i the table, usig as may digits as the decimal precisio of the proportios.. Select as may of these umbers at radom as eeded.. Use the assigatio to realize the characteristics of the values i the sample. By usig rad or radit i your calculator (Preferred way) By usig a strig or a table of radom digits (page 88) By writig umbers i slips of paper, miig, ad drawig at radom. Eample (page ) Eample (part ) Number of Motor Vehicles (per household), Proportio of households, P() Number of Motor Vehicles (per household), Proportio of households, P().88. Radom Numbers Represetig this Category , The umber of motor vehicles per household. The represets or more, but the proportio of households with more tha four vehicles is very small. [Source: U.S. Cesus Bureau, America Commuity Survey,, factfider.cesus.gov.] Note: If you are usig a radom-digit table the you should use as the equivalet of

4 Eample (part ) Samplig with or without replacemet Number of Motor Vehicles (per household), Proportio of households, P() Radom Numbers Represeti g this Category Suppose we have the followig strig of radom digits (already separated by groups of ) The correspodig umber of vehicles per household i this sample are: Radom Number 8 Vehicles i household Without replacemet. Usual choice i practice. Do ot retur slips of paper to the bo. Disregard repetitios whe usig tables of radom digits or radom umbers from a calculator. Note: The two possibilities yield almost the same results as log as the sample size is small compared to the populatio size. With replacemet Seldom used i practice. However it is much easier to do calculatios i this case. (see.,.) Do retur slips of paper to bo. Allow repetitios of umbers whe usig tables of radom digits or radom umbers from a calculator.. Geeratig Samplig Distributios Eample Samplig Distributios. Distributio of summary statistics obtaied from takig repeated radom samples. Steps for geeratig a samplig distributio: I. Take a radom sample of a fied size from a populatio. II. Compute a Summary Statistic for this sample. III. Repeat steps I ad II may times. IV. Display the distributio of the Summary Statistic. Westvaco case. Radomly select three workers from the group of with ages above, ad calculate the mea age of the three selected Note: A way to remember these steps is, Radom Sample, Summary Statistic, Repetitio, Distributio.

5 Shape, Ceter, ad Spread Shape, Ceter, ad Spread A good descriptio of a samplig distributio is the trio shape, ceter, ad spread. Recall the rectagles activity.. (See displays. ad.8) Sample mea of the areas of rectagles Populatio of Rectagle Areas Shape: Irregular Sample mea of the areas of rectagles Shape: Normal with a hit of skew to the right. Populatio of Rectagle Areas Ceter Mea. Spread Stadard Deviatio. Ceter Mea. Spread Stadard Deviatio SE. Shape, Ceter, ad Spread Notatio Sample mea of the areas of rectagles Shape: Normal with a hit of skew to the right. Ceter Mea. Spread Stadard Deviatio SE. Notes. The stadard deviatio of the samplig distributio is ofte called the Stadard Error (SE) Most sample distributios are early ormal, we ll see more about this later. Values that are i the middle % of a radom distributio are called Reasoably Likely. Values that are i the outer % of a radom distributio are called Rare Evets. Mea Stadard Deviatio Size Populatio N Sample s Samplig Distributio

6 Properties of The Samplig Distributio of The Sample Mea The mea of the samplig distributio of equals the mea of the populatio : The stadard deviatio of the samplig distributio of, also called the stadard error of the mea, equals the stadard deviatio of the populatio divided by the square root of the sample size : The Shape of the samplig distributio will be approimately ormal if the populatio is approimately ormal; for other populatios, the samplig distributio becomes more ormal as icreases. This property is called the Cetral Limit Theorem. Eample Problems usually ivolve a combiatio of the three properties of the Samplig Distributio of the Sample Mea, together with what we leared about the ormal distributio. Eample: Average Number of Childre What is the probability that a radom sample of families i the Uited States will have a average of. childre or fewer? Eample Eample Eample: Average Number of Childre What is the probability that a radom sample of families i the Uited States will have a average of. childre or fewer? Number of Childre (per family), or more Proportio of families, P() Mea (of populatio).8 Stadard Deviatio

7 Eample Fid z-score of the value. mea z SD ormalcdf(,.6). So i a radom sample of families there is a.% probability that the mea umber of childre per family will be less tha. Eample Eample: Reasoably Likely Averages What average umbers of childre are reasoably likely i a radom sample of families? Recall that the values that are i the middle % of a radom distributio are called Reasoably Likely. Eample Eample: Reasoably Likely Averages What average umbers of childre are reasoably likely i a radom sample of families? Recall that the values that are i the middle % of a radom distributio are called Reasoably Likely. Note that by calculatig the z-scores of.% ad.% we fid that the Reasoably Likely values are those values withi.6 stadard deviatios from the mea. That is, betwee.6 ad +.6 Fidig Probabilities for Sample Totals Sometimes situatios are stated i terms of the total umber i the sample rather tha the average umber: What is the probability that there are or fewer childre i a radom sample of families i the Uited States? You have the choice of two equivalet ways to do this problem. Method I: Fid the equivalet average umber of childre,, by dividig the total umber of childre,, by the sample size, :. The you ca use the same formulas ad procedure as i the previous eamples. Method II: Covert the formulas from the previous eamples to equivalet formulas for the sum, the proceed as i the et eample.

8 Samplig Distributio of the Sum of a Sample If a radom sample of size is selected with mea ad stadard deviatio, the the mea of the samplig distributio of the sum is sum the stadard error of the samplig distributio of the sum is sum the shape of the samplig distributio will be approimately ormal if the populatio is approimately ormal; for other populatios, the samplig distributio becomes more ormal as icreases. Eamples ad E: The Probability of or fewer Childre What is the probability that a radom sample of families i the Uited States will have a total of childre or fewer? E: Reasoably Likely Totals I a radom sample of families, what total umbers of childre are reasoably likely? Note: To get the sum formulas just multiply by 8