The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Chap 5 Lecture 14 Sampling Distributions for Counts and Proportions 10 20 09 Lecture 14 1 Statistic Its Sampling Distribution A statistic is any numeric measure calculated from data It is a random variable its value varies from sample to sample count proportion Ex number proportion of free throws made by a Tar Heel player who shoots 20 free throws in a practice sample mean Ex average SAT score of a group of 10 students randomly selected from STAT 155 The probability distribution of a statistic is called its sampling distribution It depends on the population distribution and the sample size 10 20 09 Lecture 14 2 Binomial Experiment n trials with n fixed in advance Each trial has two possible outcomes success S and failure F The probability of success p remains the same from one trial to the next The trials are independent i e the outcome of each trial does not affect outcomes of other trials 10 20 09 Lecture 14 3 Example The experiment randomly draw n balls with replacement from an urn containing 10 red balls and 20 black balls Let S represent drawing a red ball and F represent drawing a black ball Then this is a binomial experiment with p 1 3 Q Would it still be a binomial experiment if the balls were drawn without replacement No 10 20 09 Lecture 14 4 Binomial Distribution 10 20 09 Lecture 14 5 Do they follow binomial distributions approximately X number of stocks on the NY stock exchange whose prices increase today X number of games the Tar Heel will win next season A couple decides to have children until they have a girl X number of boys the couple will have Answer NO in all 3 cases Why 10 20 09 Lecture 14 6 Binomial Distribution If X B n p then X np 2X np 1 p P X x depends on n and p which can be calculated using software or Table C for some n and p or a Binomial Formula page 329 a simple argument given in class 10 20 09 Lecture 14 7 Binomial Table for n 20 and certain values of p Table C Page T 6 10 20 09 Lecture 14 8 Credit Card Example Records show that 5 of the customers in a shoe store make their payments using a credit card This morning 8 customers purchased shoes 1 Use the binomial table to answer the following questions Find the probability that exactly 6 customers did not use a credit card 2 X number of customers who did not use a credit card Then X B 8 0 95 which is not on the table Y number of customers who did use a credit card Then Y B 8 0 05 which is on the table P X 6 P Y 2 0515 What is the probability that at least 3 customers used a credit card See the board 10 20 09 Lecture 14 9 Credit Card Example continued 3 What is the expected number of customers who used a credit card Y np 8 05 0 4 4 What is the standard deviation of the number of customers who used a credit card 2Y np 1 p 8 05 95 0 38 The standard deviation is 0 38 0 62 Y 10 20 09 Lecture 14 10 Parking Example bad impact Sarah drives to work everyday but does not own a parking permit She decides to take her chances and risk getting a parking ticket each day Suppose A parking permit for a week 5 days cost 30 A parking fine costs 50 The probability of getting a parking ticket each day is 0 1 Her chances of getting a ticket each day is independent of other days She can get only 1 ticket per day What is her probability of getting at least 1 parking ticket in one week 5 days What is the expected number of parking tickets that Sarah will get per week Is she better off paying the parking permit in the long run 10 20 09 Lecture 14 11 Sample Proportion If X B n p the sample proportion is defined as X of successes p n sample size mean variance of a sample proportion p p 10 20 09 p p 1 p n Lecture 14 12 Example Clinton s vote 43 of the population voted for Clinton in 1992 Suppose we survey a sample of size 2300 and see if they voted for Clinton or not in 1992 We are interested in the sampling distribution of the sample proportion p for samples of size 2300 What s the mean and variance of p 10 20 09 Lecture 14 13 Count Proportion of Success A Tar Heel basketball player is a 95 free throw shooter Suppose he will shoot 5 free throws during each practice X number of free throws he makes in a practice p proportion of free throws made during practice P X 3 P p 0 6 Why 10 20 09 Lecture 14 14 10 20 09 Lecture 14 15 Normal Approximation for Counts and Proportions Let X B n p and p X n If n is large then X is approx N np np 1 p p is approx N p p 1 p n Rule of Thumb np 10 n 1 p 10 10 20 09 Lecture 14 16 Switches Inspection A quality engineer selects an SRS of 100 switches from a large shipment for detailed inspection Unknown to the engineer 10 of the switches in the shipment fail to meet the specifications Software tells us that the actual probability that no more than 9 of the switches in the sample fail inspection is P X 9 0 4513 What will the normal approximation say 10 20 09 Lecture 14 17 10 20 09 Lecture 14 18 Switches Inspection The normal approximation to the probability of no more than 9 bad switches is the area to the left of X 9 under the normal curve X np 100 1 10 X np 1 p 100 1 9 3 Using Table A we have X 10 9 10 P X 9 P P Z 33 3707 3 3 The approximation 3707 to the binomial probability of 4513 is not very accurate In this case np 10 10 20 09 Lecture 14 19 10 20 09 Lecture 14 20 Continuity Correction The normal approximation is more accurate if we consider X 9 to extend from 8 5 to 9 5 X 10 to extend from 9 5 to 10 5 and so on Example Cont X 10 9 5 10 P X 9 P X 9 5 P 3 3 P Z 17 4325 10 20 09 Lecture 14 21 Continuity Correction P X 8 replaced by P X 8 5 P X 14 replaced by P X 13 5 P X 8 P X 7 replaced by P X 7 5 For large n the effects of the continuity correction factor is very small and will be omitted 10 20 09 …
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