S301 Week 4 Lecture & Textbook Notes
S301 Week 4 Lecture & Textbook Notes STAT-S301
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This 5 page Class Notes was uploaded by Lauren Detweiler on Tuesday February 10, 2015. The Class Notes belongs to STAT-S301 at Indiana University taught by Hannah Bolte in Spring2015. Since its upload, it has received 140 views. For similar materials see Business Statistics in Statistics at Indiana University.
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Date Created: 02/10/15
S301 Textbook Week 4 Ch 7 amp 8182 pgs 152166 176184 192193 Required concepts not covered in lecture I Law of Large Numbers LLN a The relative frequency of an outcome converges to a number the probability of the outcome as the number of observed outcomes increases b In other words as you increase the number of observations the probability will be closer to the predicted probability i ie if you keep tossing a coin the proportion of tosses that are heads will eventually be close to 12 ii Often misunderstood because we forget that it only applies in the long run II Essential Rules a Rule 1 The probability of an outcome in the sample space is 1 i PS 1 ii Remember sample space is the collection of all possible outcomes b Rule 2 For any event A the probability of A is between 0 and 1 i 0 S PA S 1 ii Makes sense because probability of an event can never be negative and also cannot exceed the sample space 1 c Rule 3 The probability of a union of disjoint events is the sum of the probabilities If A and B are disjoint events then PA or B PA PB d Rule 4 Complement rule The probability of one event is one minus the probability of its complement PA 1 PAC Rule 5 Addition rule For two events A and B the probability that one or the other occurs is the sum of the probabilities minus the probability of their intersection PA or B PA PB PA and B III Vocabulary a Disjoz39nt Events Events that have no outcomes in common mutually exclusive events b Mutually Exclusive Events Events are mutually exclusive if no element of one is in the other e Independent Events Events that do not in uence each other the probabilities of independent events multiply i PA and B PA gtlt PB IV Boole s Inequality PA1rA3 or orAg S p p pk a b Probability of a union S sum of the probabilities of the events C Boole s Inequality is most useful when the events have small probabilities d Best practices 1 ii iii Make sure your sample space includes all the probabilities Include all of the pieces when describing an event Check that the probabilities assigned to all possible outcomes add up to 1 Only add probabilities of disjoint events Be clear about independence when you use probabilities Only multiply probabilities of independent events e 176 of the textbook a Focuses on household income and education in the US b Spend some time thinking about how to interpret this table iv v vi V Table 81 pag VI Multiplication Rule for Dependent Events a Covered in lecture b Best practices 1 ii iii iv v vi Think conditionally Presume events are dependent and use the multiplication rule Use tables to organize probabilities Use probability trees for sequences of conditional probabilities Check that you have included all the events Use Bayes Rule to reverse the order of conditioning Required concepts covered in lecture Chap 5 amp 6 I III IV V Equation of a Line a In high school algebra we use y mx b b In statistics we use 3 a bx c To nd the trendline you need to nd a and b It turns out b r x sysx d The slope is the correlation times the ratio of the variables standard deviations i a 37 b x a e The intercept is what s left over when you subtract the average x value scaled by b from the average y value f NOTE that i 7 andx i 9 Know Your Symbols a We have several versions of most variables i Memorize what the symbols indicate and S301 will be a lot less confusing b a or 37 i The variable s AVERAGE is denoted by drawing a line or bar over it ii We say xbar and ybar c y i The variable s EXPECTATION is denoted by giving it a hat ii We say yhat d x or y or xi or yi i The variable s actual observed value is denoted by letting it go topperless ii Sometimes it has a subscript ito indicate it s an INDIVIDUAL observation but so can expectations wearing hats iii We say y or ysubi Standardizing the Regression Equation a b Starts as y mx b Rewritten as zy rzx Computing the Trendline and Making Predictions a Using Excel to nd a and b bbrx ay bxf c See Week 3 Correlation ExampleXlsx 9 Correlation InClass B d Why this is important i Once we know the regression equation we can use it to predict outcomes given various inputs ii See Week 3 Correlation Examplexlsx 9 Correlation B Complete Tips and Pitfalls a Don t use correlations if the data are categorical b Don t treat association and correlation as causation c Don t assume that correlation of 0 means the two variables aren t associated Chap 7 amp 8182 I II III IV VI VII VIII IX Important ideas a Experiment any actionprocess observation that has random consequencesoutcomesrealizations b Sample Space the collection of all possible outcomes denoted as S c Event an event is any collection of outcomes denoted by A B or A1 A2 Example a Experiment On Friends who will Rachel end up with b Sample Space S Joey Ross Paolo Gavin etc i EventA Ross ii Event B Joey iii Event C Someone other than Ross or Joey c Each event is a subset of S It is some collection of the possible outcomes Law of Large Numbers LLN a If you repeat the same experiment over and over forever independently and under the same conditions then the number of times an event occurs divided by the number of experiments performed will get closer and closer to the probability of the event Rule 3 a Suppose two events can t happen at the same time b Addition rule for disjoint events PA U B PA PB Mutually Exclusive or Disjoint Events a Two events are mutually exclusive or disjoint if the occurrence of one means that the other CANNOT occur i It s impossible for both to happen at the same time ii They exclude each other b ie PRoss PJoey PChandler PRoss or Joey or Chandler c Question Can independent events be mutually exclusive No Exhaustive Events a Events are exhaustive if one of the must occur b ie PRoss PJoey PChandler 1 Rule 5 a Suppose two events happen at the same time b PM U B PA PB PA n B Multiplication Rule for Independent Events a PM n B PA x PB Probability XI XII XIII a Classical When all possible outcomes are equally likely the probability of event E is the number of outcomes in which E occurs divided by the total number of possible outcomes b Relative The probability of an event is the proportion ie fraction percentage of time it is expected to happen when the process is repeated over and over forever independently and under the same conditions i Subjective Your own guess based on intuition or feelings c In practice many probabilities are based on statistical models i May include classical frequency and subjective elements ii Whether one should believe the model depends on its quality Contingency Tables a Be able to understand them See Table 81 in the book Independence a Two events are independent if the conditional probability of the second does not change whether or not the rst occurs b If the conditional probability of the second changes depending on whether or not the rst event occurs the two events are dependent i This is not the same as mutually exclusive The Multiplication Rule Mathematical De nition a Remember definition of conditional probability PAB PAnB PB b Leads to the multiplication rule for dependent events PA n B PAB x PB Probability Trees and Conditioning a Probability tree a way of listing all possible combinations of events that can occur b To get the probability of one combination of events multiply along the branches of the tree 0 To get the probability of a more complicated event you may have to multiply several branches and add the results
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