Statistics for Economists
Statistics for Economists ECON 413
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This 3 page Class Notes was uploaded by Nakia Spencer on Tuesday October 27, 2015. The Class Notes belongs to ECON 413 at University of Wisconsin - Milwaukee taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/230291/econ-413-university-of-wisconsin-milwaukee in Economcs at University of Wisconsin - Milwaukee.
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Date Created: 10/27/15
University of Wisconsin Milwaukee Department of Economics Ozlem Eren Fall 2004 ECON 413 C0unting Rules Multiplication Principle If one experiment has m outcomes and another experiment has n outcomes then there are mgtltn possible outcomes for the two experiments Question A class has 15 boys and 20 girls We want to pick 1 boy and 1 girl from this class In how many different ways can we do this Extended Multiplication Principle If there are p experiments and the first has n1 possible outcomes the second has n2 and the p th has np possible outcomes then there are a total ofn1gtltnzgtlt gtltnp possible outcomes for the p experiments Question You are eating at a restaurant and the waiter informs you that you have a two choices for appetizers soup or juice b three for the main course a meat sh or vegetable dish and c two for dessert ice cream or cake How many possible choices do you have for a complete meal Permutations A permutation is an ordered a1rangement of objects Consider the set A a1 azan which has n elements Suppose we choose or take sample of r elements from this set and list them in order How many ways can this be done The answer depends on whether duplication is allowed or not Duplication is equivalent to sampling with replacement ie you pick an object from the set and you put it back into the set when you make the next selection No duplication means sampling without replacement ie you pick an object from the set but you don t put it back when you make the next selection Proposition For a set of size n and a sample of size r there are nr different ordered samples with replacement and ngtltn lgtltn 2gtltgtltn r 1 different ordered samples without replacement Corollary The number of orderings of n elements is ngtltn lgtlt gtltl n Question How many ways can five children be lined up Question Suppose we choose five children from ten children How many different lines are possible Question In some states license plates have six characters three letters followed by three numbers How many different license plates are possible University of Wisconsin Milwaukee Department of Economics Ozlem Eren Fall 2004 Proposition The number of distinct sequences of n objects of which n1 are of one kind n are ofa second kind and nk are ofa k th kind is 71 n Kn1n2nk nln2nk k where Zn 71 11 Question How many different arrangements can be obtained from the letters of the word moon using all the letters Answer the same question for the word lollipop Combinations If r objects are taken from a set of n objects without replacement and disregarding order how many different samples are possible Proposition The number of unordered samples of r objects from n objects without replacement is n n Kr n rr n We read Jas the number of combinations of n objects taken r at a time r n Note that the number Jappears as a coefficient in the binomial expansion of X y r n xyquot l L yw V J Because of this it is called a binomial coef cient Question How many different ways can you invest in four out of ten promising intemet stocks Question Suppose in a lottery you pick 6 numbers from the numbers 1 to 49 and you win if the numbers you pick match the winning numbers How many different selections can you make Instead suppose you have to pick 6 numbers from 1 to 53 How many different selections can you make now University of Wisconsin Milwaukee Department of Economics Ozlem Eren Fall 2004 Proposition The number of ways that n objects can be grouped into r classes with n in the ith class i 12 r and Zn n is 11 7 J n n1n2n nln2nr n The numbers i are called multinomial coef cients because they are n n 1 2 r coefficients in multinomial expansions n 7 n n2 n x1 x2 x Z 1 x2 xr n7 where Zn 71 11 Question In how many different ways can you diVide a group of 12 bridge players into 3 tables with 4 at each table
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