Lesson_05_Notes - Sampling Distributions

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Sampling Distributions Introduction Learning objectives for this lesson Upon completion of this lesson, you should be able to: determine the standard error for the sample proportion determine the standard error for the sample mean apply the Central Limit Theorem properly to a set of continuous data Sampling Distributions of Sample Statistics Two common statistics are the sample proportion, , (read as “p-hat”) and sample mean, , (read as “x-bar”). Sample statistics are random variables and therefore vary from sample to sample. For instance, consider taking two random samples, each sample consisting of 5 students, from a class and calculating the mean height of the students in each sample. Would you expect both sample means to be exactly the same? As a result, sample statistics also have a distribution called the sampling distribution . These sampling distributions, similar to distributions discussed previuosly, have a mean and standard deviation. However, we refer to the standard deviation of a sampling distribution as the standard error . Thus, the standard error is simply the standard deviation of a sampling distribituion. Often times people will interchange these two terms. This is okay as long as you understand the distinction between the two: standard error refers to sampling distributions and standard deviation refes to probability distributions. Sampling Distributions for Sample Proportion, p-hat If numerous repetitions of samples are taken, the distribution of is said to approximate a normal curve distribution. Alternatively, this can be assumed if BOTH n *p and n *(1 - p) are at least 10. [ SPECIAL NOTE: Some textbooks use 15 instead of 10 believing that 10 is to liberal. We will use 10 for our discussions.] Using this, we can estimate the true population proportion, p, by and the true standard deviation of p by s.e.( ) = , where s.e.( ) is interpreted as the standard error of Probabilities about the number X of successes in a binomial situation are the same as probabilities about corresponding proportions. In general, if np >= 10 and n (1- p ) >= 10, the sampling distribution of is about normal with mean of p and standard error SE( ) = . Example. Suppose the proportion of all college students who have used marijuana in the past 6 months is p = .40. For a class of size N = 200, representative of all college students on use of marijuana, what is the chance that the proportion of students who have used mj in the past 6 months is less than .32 (or 32%)? Sampling Distributions http://onlinecourses.science.psu.edu/stat200/book/export/html/42 1 of 8 9/27/2010 11:48 PM
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Solution. The mean of the sample proportion is p and the standard error of is SE( ) = . For this marijuana example, we are given that p = .4. We then determine SE( ) = = = = 0.0346 So, the sample proportion is about normal with mean p = .40 and SE( ) = 0.0346. The z-score for .32 is z = (.32 - .40) / 0.0346 = -2.31. Then using
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This note was uploaded on 10/08/2010 for the course STAT 200 taught by Professor Barroso,joaor during the Fall '08 term at Pennsylvania State University, University Park.

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Lesson_05_Notes - Sampling Distributions

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