Sampling Distributions and Confidence Intervals for Proportions Statistics and parameters Sampling distribution of a proportion Confidence intervals for proportions Margin of error and critical values Assumptions and conditions Choosing the sample size Statistical Experiment Data are fixed Experiment is random Many possible samples Survey Every group of 1000 people is equally likely Experiment happen Each assignment to control or treatment could Statistics and Parameters Statistics calculated x s Sample proportion p Parameters unknown Population proportion p Inference Use statistic to describe the parameter Parameter is the goal Statistic is the tool Sampling variability Each time we take a random sample from a population we are likely to get a different set of individuals and calculate a different statistic This is called sampling variability If we take a lot of random samples of the same size from a given population the variation from sample to sample the sampling distribution will follow a predictable pattern Sampling Distribution Distribution of the statistic obtained from repeated samples or repeated trials of an experiment using the same number of observations Changes with parameter Random Variable Imagine the data are random variables Statistic function of data Statistic is also random Survey Simple Random Sample 78 agree with statement Repeat survey 77 agree 73 agree Sample is random Sample proportions The proportion of successes can be more informative than the count In statistical sampling the sample proportion of successes is used to estimate the proportion p of successes in a population p For any SRS of size n the sample proportion of successes is successes in the sample count of p n X n In an SRS of 50 students in an undergrad class 10 are Hispanic 10 50 0 2 proportion of Hispanics in sample p The 30 subjects in an SRS are asked to taste an unmarked brand of coffee and rate it would buy or would not buy Eighteen subjects rated the coffee would buy 18 30 0 6 proportion of would buy p If the sample size is much smaller than the size of a population with proportion p of successes then the mean and standard deviation of p are p p p p p 1 n Because the mean is p we say that the sample proportion in an SRS is an unbiased estimator of the population proportion p The variability decreases as the sample size increases So larger samples usually give closer estimates of the population proportion p Sampling distribution of the sample proportion The sampling distribution of is never exactly normal But as the sample size increases the sampling distribution of becomes approximately normal p p The normal approximation is most accurate for a large fixed n when p is close to 0 5 and least accurate when p is near 0 or near 1 Rules for Sample Proportion 1 A population with a fixed proportion p 2 Random Sample Independent Equal chance 3 Sample size is large np 9 n 1 p 9 Describes a binomial experiment with normal approximation then is approximately Normal with mean and standard deviation How does this help What is p p 1 p so plug in p 1 2 worst case Or Example Suppose that a soda bottler claims that only 5 of the soda cans are underfilled A quality control technician randomly samples 200 cans of soda What is the probability that more than 10 of the cans are underfilled n 200 zP zP 24 3 pP 10 10 05 95 05 200 1 9994 0006 S underfilled can p P S 05 q 95 np 10 nq 190 OK to use the normal approximation n 200S underfilled canp P S 05q 95np 10 nq 190 Confidence intervals for proportions According to the empirical rule 95 between 2 and 2 500 people in Washington surveyed 120 agree with recent changes to bankruptcy laws Sample proportion 120 500 0 24 Estimate SE p Can we say 24 of people in Washington agree with the recent changes to bankruptcy laws No It is probably true that 24 of people in Washington agree with the recent changes to bankruptcy laws No We are 95 confident that between 24 2 1 9 and 24 2 1 9 of people in Washington agree with the recent changes to bankruptcy laws Yes 95 Confidence What does the term 95 confidence really mean 95 of samples of this size will produce confidence intervals that capture the true proportion 95 Confidence cont What does the term 95 confidence really mean We expect 5 of our samples to produce intervals that fail to capture the true proportion Margin of Error A 95 CI for a population proportion p can be written as p 2 SE p Half of the width of the CI is called the margin of error so the CI can be written as estimate ME Critical Values Confidence level 95 we ve been using 2 for the empirical rule but if we use software table we get z 1 96 so this is the critical value and 95 CI is 96 1 p SE p Confidence level 90 z 1 645 so 90 CI is 1 p 645 SE p 90 Confidence Interval Critical Values Confidence Level Critical Value z 90 95 98 99 99 9 1 645 1 96 2 326 2 576 3 29 Confidence Intervals Point estimate Critical value x Standard error Generally 95 Confidence Interval Example Give a 95 Confidence Interval for p Supposed that 43 of 1006 people surveyed in Washington are in favor of a financial reform Standard Error Margin of error Upper limit Lower limit BLS Household survey 60 000 people surveyed 6 8 unemployed 90 confidence interval Assumptions and Conditions 1 Independence Assumption Are sample observations independent of each other 1 Randomization Condition Was the sample randomly generated 2 10 Condition If sampling is done without replacement then the sample size n must be no larger than 10 of the population 2 Success Failure Condition The sample size must be large enough so that both np and n 1 p are at least 10 Questions Let s think about the 95 confidence interval we just computed for the proportion of people in Washington in favor of a financial reform 1 If we wanted to be 99 confident would our confidence interval need to be wider or narrower 2 Our margin of error was about 3 If we wanted to reduce it to 2 would our level of confidence be higher or lower 3 If we had polled more people would the interval s margin of error have been larger or smaller Sample size for a desired margin of error You may need to choose a sample size large enough to achieve a specified margin of error However because the sampling distribution p of is a function of the population proportion p this process requires that you guess a likely value for p p pNp p 1 np 2 n z m p 1 p The margin of error will be less than or equal to m if p is chosen to be 0 5 Remember though that sample
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