Sampling. Inferential statistics draws probabilistic conclusions about populations on the basis of sample statistics

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1 Samling Inferential statistics draws robabilistic conclusions about oulations on the basis of samle statistics Probability models assume that every observation in the oulation is equally likely to be observed This ideal goal is hard (imossible?) to achieve Simle random samling (the ideal) Two roerties Every observation has an equal chance of being samled Observations are samled indeendently of each other If these roerties are met Every samle of size N has equal chance of being selected Stratified random samling Divide oulation according to roerties of interest and samle within each grou E.g., olitical ollsters divide the electorate according to arty affiliation, race, education, income levels, geograhic areas, urban-suburban-rural, likelihood of voting, Perhas simle random samle within grous, but even that often is difficult and other methods may be used Cluster samling Similar to above, but division occurs in convenient rather than theoretically motivated ways E.g., by hysical location Systematic samling Introduce randomness at some oint (e.g., location of samling); then samle every k th observation 1

2 Convenience samling Take the samle when and where one can Whether or not a convenience samles is bad deends on whether or not it is biased with resect to variables of interest Political olling by going to households in only one location, or by asking listeners to call in, will yield a biased samle Studying normal visual ercetion with Psychology 100 articiants who have normal vision (robably) will not The robability models used in inferential statistics generally assume that samles are selected in unbiased ways I.e., that all samles of size N are equally likely to occur If the nature of the samle bias is known, statistical corrections sometimes are ossible Outline of Probability Toics Some common errors in thinking about uncertainty What does robability measure or reresent? Reresenting the sace of events over which robabilities are defined Laws of robability Random variables numerical outcomes of exeriments Probability distributions alication of robability laws to random variables 2

3 Intuitions about Probability Linda is 45 years old, married with no children, outsoken, and very bright. As a student, she was deely concerned with issues of discrimination and social justice, and also articiated in antinuclear demonstrations. Rank the statements from most (1) to least (8) robable. A. Linda is an elementary school teacher B. Linda works in a bookstore and takes yoga classes C. Linda is active in the feminist movement D. Linda is a sychiatric social worker E. Linda belongs to the League of Women Voters F. Linda is a bank teller G. Linda is an insurance saleserson H. Linda is a bank teller and active in the feminist movement A UMCP undergraduate selected at random has taken Calculus 1-3, Intro. Lab Physics, Intro. Lab Biology lus other courses. Which category is this erson s major more likely in, or are all equally likely? humanities-fine arts, natural sciences, social sciences Which of the following sequences of tosses of a fair coin is more likely, or are they equally likely? HTHTHT HHHTTT TTTTTT HHTHTT 3

4 What Does Probability Reresent? It reresents long-run relative frequency Called objective robability (OP) The robability of an event is its exected longrun relative frequency Note: Probability is not defined in terms of equally-likely events (as many authors define it) That would be circular Problems Fine as a definition, but not easy to work with If a coin yields 492 heads in 1,000 tosses, do we say P(H)π.50? Should we toss it 2,000 times to be sure? 5,000 times? Does not aly to unique, non-reeatable events It reresents coherent ersonal oinion Called subjective or Bayesian robability (SP) Coherent: The oinions are internally consistent Two eole may have different subjective robability distributions about a universe of events Both are accetable if they are coherent» P(H)=.489, P(T)=.511 or P(H)=.500, P(T)=.500 As information accumulates, the two distributions will converge to the relative frequencies Problem Not clear what robabilities should be based on in the absence of information These are called rior robabilities Probabilities udated by information are osterior robabilities 4

5 Probability as a Measure The hilosohical differences between the objective and subjective aroach are great However, given enough observations, the objective and subjective aroaches converge The best way to think about robability is as a measure Just as area is a measure over surfaces Probability is a measure defined over elementary events (outcomes) and event classes Considering robability in this way will make the rules intuitive The measure reresents either SP or OP Deicting Outcomes and Event Classes Two methods are useful Venn diagrams and event trees The rovide different ways to concetualize relations among events A concrete examle first A single die has outcomes 1, 2, 3, 4, 5, 6 Some events are Even (outcomes 2, 4, 6) Less than 4 (outcomes 1, 2, 3) Even and less than 4 (outcome 2) Divisible by 3 (outcomes 3, 6) 5

6 Venn Diagram Divisible by 3 Even Less than 4 Venn diagrams are ways to reresent event saces Outcomes are not necessarily exlicitly shown Illustrate which events have outcomes in common Such events are said to intersect We talk about the intersection of events Outcomes are in the intersection of event A and B if they are in both A and B (roll of die is even and divisible by 3) If 2 events have no outcomes in common, their intersection is null. Such events are mutually exclusive The union of 2 events refers to one or the other 6

7 C A E B F H G I D E AB (A and B) F BC G CD H BCD I A or D The comlement of an event consists of all those outcomes not in the event A ~ is the comlement of A Venn Diagram for the outcomes of an exeriment consisting of tossing 3 coins Next figure 7

8 A Venn Diagram for H 1 Tossing 3 Coins T 1 HHT THT H 2 HHH THH H 3 HTH TTH T 2 HTT TTT Tree Diagram An alternative way to lay out event saces The method to use deends on the roblem Trees are often useful when the full exeriment consists of the union of distinct event-outcomes Tossing 3 coins Next figure 8

9 H T H T H T H T H T H T H T HHH HHT HTH HTT THH THT TTH TTT Probability Laws Probabilities are numbers, 0 (E) 1, that reresent the relative likelihoods of events and conform to 3 laws These laws are easily understood by thinking of them as relative areas of events reresented in Venn diagrams or as relative likelihoods of outcomes reresented in a tree The articular reresentation to use deends on the roblem at hand First law 0 (E) 1 and (sure event) = 1 9

10 Second law For 2 mutually exclusive events, E 1 and E 2, ( E or E ) = ( E ) ( ) E2 Examle: An urn has 40 red, 30 green, 20 blue, 10 yellow balls. Drawing one ball at random from the urn, what is the robability that it is red or blue? (Use tree or Venn-tye diagram to see this) Corollary for n mutually exclusive events n ( E or E or Lor E ) ( ) = 1 2 n E 1 i Corollary for event E and its comlement ( E) = 1 ( E) E Corollary for n mutually exclusive and exhaustive events n 1 ( E ) = 1 i 40/100 30/100 Red Green 20/100 Blue 10/100 Yellow 40/100 30/100 20/100 10/100 10

11 Third law For any 2 events, A and B, A and B = B A A ( ) ( ) ( ) ( A and B) ( A B) ( B) = P(A B) is read robability of A given B It is the robability that an outcome is in A once you know it is in B We will often write (AB) instead of (A and B) Examle: If 10 balls of each color in revious e.g. have a 0 on them and the rest have a 1, what is the robability in a single random draw of obtaining a red ball with a 1 on it? (A tree diagram is easiest here) of obtaining a ball with a 1 on it? Corollary A = A B B + ( ) ( ) ( ) ( A B ~ ) ( B ~ ) 40/100 30/100 Red Green 20/100 Blue 10/100 Yellow 10/40 30/40 10/30 20/30 10/20 10/20 10/10 0/ R0 R1 G0 G1 B0 B1 Y0 Y1 ( R1) ( R) ( 1 R) = ( R1 ) = ( )( 30 40) = =. 30 () 1 = ( R1) + ( G1) + ( B1) ( Y1) + () 1 = =

12 0 1 R ( R )=.40 G ( G )=.30 B ( B )=.20 Y 10 ( Y )=.10 ( 0 )=.40 ( 1 )=.60 Definition Two events, A and B, are indeendent if and only if A B = A B ~ = A ( ) ( ) ( ) When the above is true, it is also the case that ( B A) = ( B ) ( B) A ~ = Corollary to Law 3 If 2 events, A and B, are indeendent, then ( AB) = ( B) ( A) Questions Are color and number indeendent in revious e.g. What is the difference between indeendence and mutually exclusive 12

13 From Laws 2 and 3 For any 2 events, A and B ( A or B) = ( A) + ( B) ( AB) ( A or B) = ( A) + ( B) ( A B) ( B) If A and B are indeendent ( A or B) = ( A) + ( B) ( A) ( B) A B If (A)=.6, (B)=.3, (AB)=.1, what is (A or B) what is AB ~ ) what is ( A ~ B ~ ) What is (B A) A B 13

14 A samle consists of 100 men and 200 women Of the men, 40 are Democrats, 50 are Reublican, 10 are indeendent Of the women, 110 are Democrats, 80 are Reublican, 10 are indeendent What is the robability that a erson randomly samled from the grou is Democrat Reublican Indeendent Male or Reublican Male given Democrat 100/ /300 Male Female 40/100 10/100 50/ /200 10/200 80/200 Dem Ind Re Dem Ind Re ( D ) = ( MD) + ( FD) = ( ) + ( ) =. 50 ( R) = ( MR) + ( FR) = ( ) + ( ). 43 ( M or R) + ( M) + ( R) ( MR) =

15 For the last question, what is (M D)? From the third law, we know MD ( M D) = D ( M D) = ( ) ( ) ( D M) ( M) ( D) This equation is known as Bayes Rule How do we find (D)? ( D) = ( MD) + ( FD) ( D) = ( D M) ( M) ( D F) ( F) + Answer the question using the equations, a tree diagram, or a Venn-tye diagram.13 M D =..50 ( ) 27 Bayes Rule simultaneously is a simle consequence of the robability laws, very imortant, and very controversial Imortant because it underlies imortant uses of robability in diagnosis, ublic olicy, medicine, and numerous other areas The robability of disease given a ositive test outcome The robability of a hurricane given certain weather atterns In cases such as these, all the robabilities are estimated on the basis of relative frequency data Controversial when used to test scientific hyotheses Prior robabilities of hyotheses often are subjective Bayesian statistics is becoming increasingly accetable these days 15

16 Examle An eidemic of disease X has broken out The best estimates are that 5% of the adult oulation is affected A screening test for X has a true ositive rate of.98 and a false ositive rate of.30 ~ P ( + X ) =. 98 P ( + X ) =. 30 What is the robability that someone who tests ositive for X really has X? ( ) ( + X) ( X) X + = + = + X X + + X ~ X ~ ( + ) ( ) ( ) ( ) ( ) ( ) ( X + ) = = =

Mathematics. ( : Focus on free Education) (Chapter 16) (Probability) (Class XI) Exercise 16.2

Mathematics. ( : Focus on free Education) (Chapter 16) (Probability) (Class XI) Exercise 16.2 ( : Focus on free Education) Exercise 16.2 Question 1: A die is rolled. Let E be the event die shows 4 and F be the event die shows even number. Are E and F mutually exclusive? Answer 1: When a die is