Sampling Distribution of a Sample Proportion Lecture 26 Section 8 4 Robb T Koether Hampden Sydney College Mon Mar 5 2012 Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 1 35 Outline 1 Computing the Sampling Distribution of p 2 The Central Limit Theorem for Proportions 3 Applications 4 Assignment Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 2 35 Outline 1 Computing the Sampling Distribution of p 2 The Central Limit Theorem for Proportions 3 Applications 4 Assignment Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 3 35 Sampling Distributions Definition Sampling Distribution of a Statistic The sampling distribution of a statistic is the distribution of values of that statistic over all possible samples of a given size n from the population We may sample with or without replacement For our purposes it will be simpler to sample with replacement Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 4 35 The Sample Proportion We will work out the sampling distribution for p for sample sizes of 1 2 and 3 Then I will show you the sampling distribution for p for sample sizes of 4 5 and 10 Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 5 35 Example Suppose that 45 of all people approve of President Obama s performance Suppose that we select one person at random We may diagram the 2 possibilities Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 6 35 Example Y 0 45 0 55 N Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 7 35 Example Now we take a sample of 2 people sampling with replacement Find the sampling distribution of p Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 8 35 Example 0 45 Y YY 2 0 45 2 0 2025 N YN 1 0 45 0 55 0 2475 Y NY 1 0 55 0 45 0 2475 N NN 0 0 55 2 0 3025 Y 0 45 0 55 0 45 0 55 N 0 55 Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 9 35 Example Let x be the number of people out of 2 who strong disapprove of President Obama s performance The probability distribution of p p 0 1 2 1 is P p 0 2025 0 4950 0 3025 Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 10 35 Example Now we take a sample of 3 people sampling with replacement Find the sampling distribution of p Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 11 35 Example 0 45 0 45 Y YYY 3 0 45 3 0 0911 0 55 N YYN 2 0 45 2 0 55 0 1114 0 45 Y YNY 2 0 45 0 55 0 45 0 1114 0 55 N YNN 1 0 45 0 55 2 0 1361 0 45 Y NYY 2 0 55 0 45 2 0 1114 0 55 N NYN 1 0 55 0 45 0 55 0 1361 0 45 Y NNY 1 0 55 2 0 45 0 1114 0 55 N NNN 0 0 55 3 0 1664 Y Y 0 45 0 55 0 45 N Y 0 55 N 0 55 N Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 12 35 Example Let p be the sample proportion of people who strong disapprove of President Obama s performance The sampling distribution of p is p 0 1 3 2 3 1 P p 0 1664 0 4084 0 3341 0 0911 Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 13 35 Samples of Size n 4 If we sample 4 people then the sampling distribution of the sample proportion is P p p 0 0 0915 1 4 0 2995 2 4 0 3675 3 4 0 2005 1 0 0410 Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 14 35 Samples of Size n 5 If we sample 5 people then the sampling distribution of the sample proportion is p P p 0 0 0503 1 5 0 2059 2 5 0 3369 3 5 0 2757 4 5 0 1128 1 0 0185 Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 15 35 Samples of Size n 6 If we sample 6 people then the sampling distribution of the sample proportion is p P p 0 0 0277 1 6 0 1359 2 6 0 2780 3 6 0 3032 4 6 0 1861 5 6 0 0609 1 0 0083 Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 16 35 Samples of Size n 10 If we sample 10 people then the sampling distribution of the sample proportion is p 0 00 0 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 1 00 P p 0 0025 0 0207 0 0763 0 1665 0 2384 0 2340 0 1596 0 0746 0 0229 0 0042 0 0003 Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 17 35 Outline 1 Computing the Sampling Distribution of p 2 The Central Limit Theorem for Proportions 3 Applications 4 Assignment Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 18 35 The Central Limit Theorem for Proportions Theorem The Central Limit Theorem for Proportions For any population and any sample size the sampling distribution of p has the following mean and standard deviation p p r p p 1 p n Furthermore the sampling distribution of p is approximately normal provided n is large enough Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 19 35 The Central Limit Theorem for Proportions The Sample Size n is large enough if np 5 and n 1 p 5 If n is small then we have to work out the distribution by hand Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 20 35 Outline 1 Computing the Sampling Distribution of p 2 The Central Limit Theorem for Proportions 3 Applications 4 Assignment Robb T Koether Hampden Sydney College Sampling Distribution of a Sample Proportion Mon Mar 5 2012 21 35 Applications Suppose that 60 of all high school students own a cell phone If we survey 3 high school students how likely is it that we will find that at least 2 of them own a cell phone …
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