Lecture09 - Introduction Inference for Proportions...

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Inference for Proportions Inference for a Single Proportion Section 8.1 © 2009 W.H. Freeman and Company Introduction ! Many studies collect data on categorical variables , such as voting preferences, occupation of a person, the make of a car, etc. ! The parameters of interest in these settings are population proportions , labeled p . ! The statistic used to estimate a population proportion is the sample proportion, labeled The population parameter, p ! If our data are nominal we can count the number of occurrences of each outcome to describe the population. ! From counts we can calculate proportions. ! We studied the binomial distribution using the count of the # of successes, X. ! We now deal with sample proportions because we want to estimate the probability of success, p in a population. Sampling Distribution of a Proportion Draw a random sample of size n from a population having population proportion p of successes. Let X be the count of successes in the sample and = X/n. When n is large, the sampling distribution of is approximately normal:
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Sampling distribution of a sample proportion The sampling distribution of a sample proportion is approximately normal when the sample size is large enough and when p is closer to .5 Rule of Thumb: np and n(1 – p) > 10 Conditions for inference on p Assumptions: 1. The data used for the estimate are an SRS from the population studied. 2. The sample size n is large enough that the sampling distribution can be approximated with a normal distribution. How large a sample size is required depends on the value of p and the test conducted. (Otherwise, rely on the binomial distribution) 3. If the number of successes and the number of failures are both at least 15, we can construct the confidence interval. Standard Deviation Vs. Standard Error ! In terms of confidence intervals, we see a problem with the standard deviation of the sample proportion, . ! The standard deviation above depends on the unknown population parameter, p . ! In our calculations, we estimate p with in the calculation of the standard error… Large-sample confidence interval for p Use this method when the number of successes and the number of failures are both at least 15. C Z * ! Z * m m Confidence intervals contain the population proportion p in C % of samples. For an SRS of size n drawn from a large population and with sample proportion calculated from the data, an approximate level C confidence interval for p is: C is the area under the standard normal curve between ! z * and z *.
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Example: Public Opinion Polls Public opinion polls occur on a daily basis. Polling companies are clear about their sampling techniques and provide the reader this information on their websites. One such company, Rasmussen Reports, provides comprehensive coverage of public opinion and is covering the presidential election of 2008.
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