Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment Sampling Distribution of a Sample Proportion Lecture 25 Sections 8 1 8 2 Robb T Koether Hampden Sydney College Fri Oct 10 2008 Outline Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 1 Experiment Results 2 Computing the Sampling Distribution of p 3 PDFs for n 1 2 3 30 4 Observations 5 The Central Limit Theorem for Proportions 6 Why Surveys Work 7 Assignment Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment Experiment Results Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment The Results of the Experiment In our experiment we collected a total of 100 samples each of size 5 The distribution of sample proportions we observed was Sample Proportion 0 0 0 2 0 4 0 6 0 8 1 0 Frequency 0 4 22 32 32 10 Homework Review Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment The Results of the Experiment To describe this distribution we could describe its shape and give the mean and standard deviation Keep in mind we are working with a sample of values of p That is a sample of samples This is not the entire population of all samples of size 5 Homework Review Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 The Results of the Experiment The shape is approximately normal To find the mean and standard deviation we may use the TI 83 Enter the distinct values of p in list L1 Enter the frequencies in list L2 Use 1 Var Stats as before but enter both lists Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment 1 Var Stats L1 L2 We get 0 644 for the mean and 0 2061 for the standard deviation Homework Review Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment The Results of the Experiment Theory says that the shape is approximately normal and that p p and r p p 1 p n We calculate p 0 66 and r r p 1 p 0 66 0 34 0 2118 p n 5 Our experimental results were very close to what the theory predicts Sampling Distributions Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment 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 large populations the difference is insignificant For our purposes it will be easier to sample with replacement allowing repetitions The Sample Proportion Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment The letter p represents the population proportion The symbol p p hat represents the sample proportion So p is a random variable The sampling distribution of p is the probability distribution of all the possible values of p Example Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment Suppose that 2 3 of all males wash their hands after using a public restroom Suppose that we take a sample of 1 male Find the sampling distribution of p Example Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results W W 1 2 3 N N 0 1 3 2 3 Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment 1 3 Example Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment Let x be the number of males in the sample who wash The probability distribution of x is x P x 0 1 3 0 3333 1 2 3 0 6667 Example Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment Let p be the sample proportion of males who wash p 1x The sampling distribution of p is P p p 0 1 3 0 3333 1 2 3 0 6667 Example Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment Now we take a sample of 2 males sampling with replacement Find the sampling distribution of p Example Sampling Distribution of a Sample Proportion 2 3 W WW 2 4 9 N WN 1 2 9 W NW 1 2 9 N NN 0 1 9 Robb T Koether W Experiment Results 2 3 1 3 Computing the Sampling Distribution of p PDFs for n 1 2 3 30 2 3 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment 1 3 N 1 3 Example Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment Let x be the number of males in the sample who wash The probability distribution of x is x P x 0 1 9 0 1111 1 4 9 0 4444 2 4 9 0 4444 Example Sampling Distribution of a Sample Proportion Robb T Koether Experiment Results Computing the Sampling Distribution of p PDFs for n 1 2 3 30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment Let p be the sample proportion of males who wash p 2x The
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