274 CHAPTER 6 Normal Probability Distributions Let’s formally define sampling distribution, the main character in the preceding statistics story. DEFINITION The sampling distribution of a statistic (such as a sample proportion or sample mean) is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population. (The sampling distribution of a statistic is typically represented as a probability distribution in the format of a probability histogram, formula, or table.) Sampling Distribution of Sample Proportion The preceding general definition of a sampling distribution of a statistic can now be restated for the specific case of a sample proportion: We need to distinguish between a population proportion p and a sample proportion, and the following notation is common and will be used throughout the remainder of this book, so it’s very important. Notation for Proportions x = number of successes n = sample size N = population size pn = x>n denotes the sample proportion p = x>N denotes the population proportion HINT pn is pronounced “p-hat.” When symbols are used above a letter, as in x and pn, they represent statistics, not parameters. Behavior of Sample Proportions 1. The distribution of sample proportions tends to approximate a normal distribution. 2. Sample proportions target the value of the population proportion in the sense that the mean of all of the sample proportions pn is equal to the population proportion p; the expected value of the sample proportion is equal to the population proportion. DEFINITION The sampling distribution of the sample proportion is the distribution of sample proportions (or the distribution of the variable pn), with all samples having the same sample size n taken from the same population. (The sampling distribution of the sample proportion is typically represented as a probability distribution in the format of a probability histogram, formula, or table.)
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