Sampling Distribution of a Sample Mean Lecture 29 Section 8 4 Fri Mar 9 2007 Introduction And now for something completely different Statz 4 Life Sampling Distribution of the Sample Mean Sampling Distribution of the Sample Mean With or Without Replacement If the sample size is small in relation to the population size 5 then it does not matter whether we sample with or without replacement The calculations are simpler if we sample with replacement In any case we are not going to worry about it Example Suppose a population consists of the numbers 6 12 18 Using samples of size n 1 2 or 3 find the sampling distribution of x Draw a tree diagram showing all possibilities The Tree Diagram n 1 n 1 6 mean 6 12 mean 12 18 mean 18 The Sampling Distribution n 1 The sampling distribution of x is The parameters are 12 2 24 x P x 6 1 3 12 1 3 18 1 3 The Sampling Distribution n 3 The shape of the distribution density 1 3 6 8 10 12 14 16 18 mean The Tree Diagram n mean 2 6 12 18 6 6 12 9 18 12 6 9 12 12 18 15 6 12 12 15 8 18 The Sampling Distribution n 2 The sampling distribution of x is The parameters are 12 2 12 x P x 6 1 9 9 2 9 12 3 9 15 2 9 18 1 9 The Sampling Distribution n 3 The shape of the distribution density 3 9 2 9 1 9 6 8 10 12 14 16 18 mean The Tree Diagram n 3 6 6 12 18 6 12 12 18 6 18 12 18 mean 6 12 18 6 12 18 6 12 18 6 12 18 6 12 18 8 10 12 10 12 14 8 10 12 6 12 18 6 12 18 6 12 18 12 14 16 10 12 14 12 14 16 6 12 18 14 16 18 6 8 10 10 12 14 The Sampling Distribution n 3 The sampling distribution of x is The parameters are 2 2 8 x P x 6 1 27 8 3 27 10 6 27 12 7 27 14 6 27 16 3 27 18 1 27 The Sampling Distribution n 3 The shape of the distribution density 9 27 6 27 3 27 6 8 10 12 14 16 18 mean Data From Excel Assignment 2 Student 1 40 30 20 10 0 65 68 71 74 77 80 83 86 89 Data From Excel Assignment 2 Student 2 40 30 20 10 0 65 68 71 74 77 80 83 86 89 Data From Excel Assignment 2 Student 3 40 30 20 10 0 65 68 71 74 77 80 83 86 89 Data From Excel Assignment 2 Student 4 40 30 20 10 0 65 68 71 74 77 80 83 86 89 Data From Excel Assignment 2 Student 5 40 30 20 10 0 65 68 71 74 77 80 83 86 89 Data From Excel Assignment 2 Student 6 40 30 20 10 0 65 68 71 74 77 80 83 86 89 Data From Excel Assignment 2 Student 7 40 30 20 10 0 65 68 71 74 77 80 83 86 89 Observations and Conclusions Observation The values of x are clustered around Conclusion x is likely to be close to Observations and Conclusions Observation As the sample size increases the clustering is tighter Conclusion Larger samples give more reliable estimates Conclusion For sample sizes that are large enough we can make very good estimates of the value of More Observations and Conclusions Observation The distribution of x appears to be approximately normal Conclusion We can use the normal distribution to calculate just how close to we can expect x to be One More Observation However we must know the values of and for the distribution of x That is we have to quantify the sampling distribution of x The Central Limit Theorem Begin with a population that has mean and standard deviation For sample size n the sampling distribution of the sample mean is approximately normal with Mean of x 2 Variance of x n Standard deviation of x n The Central Limit Theorem The approximation gets better and better as the sample size gets larger and larger That is the sampling distribution morphs from the distribution of the original population to the normal distribution The Central Limit Theorem For many populations the distribution is almost exactly normal when n 10 For almost all populations if n 30 then the distribution is almost exactly normal The Central Limit Theorem Also if the original population is exactly normal then the sampling distribution of the sample mean is exactly normal for any sample size This is all summarized on pages 536 537 Bag A vs Bag B Bag A 10 20 30 40 50 60 Bag B 10 20 30 40 50 60 Bag A vs Bag B Use the TI 83 to compute the mean and standard of each population Bag A and Bag B Bag A vs Bag B The Bag A population 23 16 14 53 The Bag B population 46 84 14 53 Bag A vs Bag B The hypotheses H0 The bag is Bag A H1 The bag is Bag B Suppose that we sample 10 vouchers one at a time with replacement If the average of the 10 vouchers is less than 35 we will accept H0 Bag A vs Bag B What is Find What is Find the sampling distribution of x if H0 is true the sampling distribution of x if H1 is true How reliable is this test
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