#### Transcript Sampling Distribution Proportion

Sampling Distribution of a Sample Proportion Lecture 28 Sections 8.1 – 8.2 Wed, Mar 7, 2007 Sampling Distributions Sampling Distribution of a Statistic The Sample Proportion Let p be the population proportion. Then p is a fixed value (for a given population). Let p^ (“p-hat”) be the sample proportion. Then p^ is a random variable; it takes on a new value every time a sample is collected. The sampling distribution of p^ is the probability distribution of all the possible values of p^. Example Suppose that this class is 3/4 freshmen. Suppose that we take a sample of 1 student. Find the sampling distribution of p^. Example 3/4 F P(F) = 3/4 N P(N) = 1/4 1/4 Example Let X be the number of freshmen in the sample. The probability distribution of X is x 0 1 P(x) 1/4 3/4 Example Let p^ be the proportion of freshmen in the sample. (p^ = X/n.) The sampling distribution of p^ is x 0 1 P(p^ = x) 1/4 3/4 Example Now we take a sample of 2 student, sampling with replacement. Find the sampling distribution of p^. Example 3/4 3/4 F P(FF) = 9/16 N P(FN) = 3/16 F P(NF) = 3/16 N P(NN) = 1/16 1/4 1/4 3/4 N F 1/4 Example Let X be the number of freshmen in the sample. The probability distribution of X is x 0 1 2 P(x) 1/16 6/16 9/16 Example Let p^ be the proportion of freshmen in the sample. (p^ = X/n.) The sampling distribution of p^ is x 0 1/2 1 P(p^ = x) 1/16 6/16 9/16 Samples of Size n = 3 If we sample 3 people (with replacement) from a population that is 3/4 freshmen, then the proportion of freshmen in the sample has the following distribution. x 0 P(p^ = x) 1/64 = .02 1/3 2/3 1 9/64 = .14 27/64 = .42 27/64 = .42 Samples of Size n = 4 If we sample 4 people (with replacement) from a population that is 3/4 freshmen, then the proportion of freshmen in the sample has the following distribution. x P(p^ = x) 0 1/256 = .004 1/4 12/256 = .05 2/4 54/256 = .21 3/4 108/256 = .42 1 81/256 = .32 The pdf when n = 1 0 1 The pdf when n = 2 0 1/2 1 The pdf when n = 3 0 1 The pdf when n = 4 0 1/4 2/4 3/4 1 The pdf when n = 8 0 1/4 2/4 3/4 1 The pdf when n = 16 0 1/4 2/4 3/4 1 The pdf when n = 48 0 1/4 2/4 3/4 1 Observations and Conclusions Observation: The values of p^ are clustered around p. Conclusion: p^ is probably close to p. Observations and Conclusions Observation: As the sample size increases, the clustering becomes 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 p. Observations and Conclusions Observation: The distribution of p^ appears to be approximately normal. Conclusion: We can use the normal distribution to calculate just how close to p we can expect p^ to be. One More Observation However, we must know the values of and for the distribution of p^. That is, we have to quantify the sampling distribution of p^. The Central Limit Theorem for Proportions It turns out that the sampling distribution of p^ is approximately normal with the following parameters. Mean of pˆ p p 1 p Variance of pˆ n Standard deviation of pˆ p 1 p n The Central Limit Theorem for Proportions The approximation to the normal distribution is excellent if np 5 and n1 p 5. Why Surveys Work Suppose that we are trying to estimate the proportion of the population who own a cell phone. Suppose the true proportion is 75%. If we survey a random sample of 1000 people, how likely is it that our error will be no greater than 5%? Why Surveys Work First, describe the sampling distribution of p^ if the sample size is n = 1000 and p = 0.75. np = 750 5 and n(1 – p) = 250 5, so p^ is approximately normal. Check: Why Surveys Work Then find the parameters p^ and p^. pˆ p 0.75. pˆ p1 p n 0.750.25 0.01369. 1000 Why Surveys Work Now find the probability that p^ is between 0.70 and 0.80. normalcdf(.70, .80, .75, .01369) = 0.9997. It is virtually certain that our estimate with be within 5% of 75%. Why Surveys Work What if we had surveyed only 200 people? Surveys What range of percentages contains 95% of the sample proportions? Surveys Suppose that Candidate X has 48% of the vote and Candidate Y has 52% of the vote. What is the probability that a survey of 100 people will indicate that Candidate X is ahead? Surveys What is the probability that a survey of 2000 people will indicate that Candidate X is ahead? Quality Control A company will accept a shipment of components if they are convinced that no more than 5% of them are defective. H0: 5% of the parts are defective. H1: More than 5% of the parts are defective. Quality Control They will take a random sample of 100 parts and test them. If no more than 10 of them are defective, they will accept the shipment. What is ? What is ?