Class Note for STAT 528 at OSU 56
Class Note for STAT 528 at OSU 56
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Date Created: 02/06/15
Stat 528 Autumn 2008 Elly Kaizar The sampling distribution of counts and proportions Reading Section 51 o The Bernoulli distribution 0 Mean and stdev of a Bernoulli RV o The binomial distribution 0 Binomial probabilities 0 Mean and stdev of a Binomial RV 0 Mean and stdev of a sample proportion o The normal approximation 0 The continuity correction The sampling distribution for counts and proportions 0 Suppose we ask a random sample of size n a yes no question 0 Let the RV X denote the number of individuals answering yes a count 0 The proportion of individuals answering yes is the sample proportion 13 Xn o What is the sampling distribution of The Bernoulli trial and distribution 0 Consider a chance experiment with two possible outcomes 1 X 1 denotes a success a yes i suppose this occurs with probability p 2 X 0 denotes a failure a no i suppose this occurs with probability 1 p 0 Let X denote the random variable RV of a Bernoulli trial We say X follows a Bernoulli distribution with parameter p or that X has a Bernltpgt distribution 0 The probability distribution for X can be written as 1 1 k0 p kl PX k ux p 0 130 1 The binomial trial o The setup 1 There are a xed number of n observations in the sample 2 The n observations are all independent 3 There are only two possible outcomes for each observa tion success 1 or failure 4 The probability of success p is constant for each obser vation 0 Then the number count of successes X has a binomial distribution With parameters 71 and p 0 We say X N 30113 0 Note X is the sum of n independent Bernoulli trials each With constant success probability p Examples Consider the following situations For each comment on the ap propriateness of the binomial model for the RV X o A coin is ipped 1000 times X is the number of tails 0 You draw 5 cards from a well shuf ed deck X is the number of diamonds among the drawn cards 0 You observe the sex of the next 50 children born at a local hospital X is the number of girls among them More Examples o A couple decides to continue having children until their rst girl is born X is the number of children the couple have 0 You want to know what percentage of married people believe that mothers of young children should not be employed out side the home You plan to interview 50 people and for the sake of convenience you decide to interview both the husband and the wife in 25 married couples X is the number among the 50 persons who think mothers should not be employed Binomial probabilities 0 Suppose X has a BM p distribution Then the probability distribution of X is M kgt Zpklt1 29W 0 Remember the RV X counts the number of successes in n independent Bernoulli trials 1 By the independence of the trials we can multiply the probabilities of success or failure 2 The probability of k successes is pk 3 The probability of n k failures is 1 p k 4 But there may be many ways to get k successes and n k failures Picking and choosing o For n gt O and 0 g k g n the binomial coef cient is and is equal to the number of ways that one can select k items of one type out a collection of 71 items 0 kl means k factorial and is de ned by 1 n O n Mn 1 71 gt1 Binomial probabilities cont 0 Table C lists the binomial probabilities 0 Each entry lists PX k k O1n for various com binations of parameters 71 and p Only 0 g p g 05 is listed swap success and failure to calculate for p gt 05 Values of n tabulated are pretty small see later 0 Alternatively binomial probabilities can be obtained from Minitab Use the menu sequence Calc Probability Dis tributions Binomial to get the dialog box To nd PX k click on the Probability dot To nd PX g k click on the Cumulative proba bility dot Is the binomial distribution valid for a SRS o No Recall our discussion of sampling from a nite popula tion Sampling Without replacement produces dependent draws from the population It is like drawing cards from a deck 0 When the population is much larger the books rule is at least 10 times as large than the sample then the count of successes X in a SR8 of size n has approximately a B n p distribution 0 Note The sample proportion I Xn does not have a binomial distribution ln this setting it is approximately a binomial proportion 10 Marriage example According to government data 25 of employed women have never been married 1 If 10 employed women are selected at random what is the probability that exactly two have never been married 2 What is the probability that 2 or fewer have never been mar ried 3 What is the probability that at least 8 have been married 11 The mean and stdev of a binomial RV and sample proportion 0 Let X be a binomial RV with parameters 71 and p Then uX Hp 0 np1 p and 0X xnp1 p X 2211 K Where Y is Bernp and the Ys are inde pendent of one another 0 Let 1 X n denote the sample proportion of successes in a SR8 of size n drawn from a large population having popula tion proportion p of successes Then 39w l l R k E and 05 0 Thus I is an unbiased estimator of p 12 Poverty level example According to government data 21 of American children under the age of six live in households With incomes less that the o icial poverty level A study of learning in early childhood chooses a SR8 of 300 children 0 What is the mean number of children in the sample Who come from poverty level households What is the standard deviation of this number 0 What is the probability that at least 80 of the children in the sample live in poverty 13 Normal approximation for counts and proportions 0 Draw a SR8 of size n from a large population having a pop ulation proportion p of success 0 Verify that the approximation is reasonable Make sure that the population contains far more than 71 individuals Make sure that neither up nor 711 p is too small Many rules of thumb exist Most commonly make sure that up 2 5 and M1 p Z 5 The book recommends up 2 10 and n1 p Z 10 0 Then the number count of successes is the sample X is approximately a normal RV with mean and stdev equal to the mean and stdev of the sampling distribution of X 0 Also the sample proportion of successes I Xn is approximately a normal RV with mean and stdev equal to the mean and stdev of the sampling distribution of 14 The continuity correction o The binomial RV X is a discrete random variable We are approximating its distribution using the dis tribution of a continuous normal RV The continuity correction is used to make the approxima tion better Think about approximating a histogram with a smooth curve that is in turn approximated by a histogram 15 16 78 79 80 81 82 0 000 0 001 1 0 002 0 003 0 004 0 005 0 005 50 100 150 200 250 300 78 79 80 81 82 000 001 002 003 004 005 1 1 1 1 1 0 000 0 001 0 002 0 003 0 004 0 005 0 005 1 1 1 1 1 1 Example n 300 p 021 Generic Examples Suppose X is BM p We approximate this with a Nnp nplt1 13 distribution and 1 We approximate P X as using P X 2 We approximate PX Z 33 by HXZ 3 We approximate PX gt as by HXZ 17
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