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Chapter 18 1 Sampling distributions • sample proportion ( pˆ ) fraction (or percentage) of measurements in a sample that are “successes”: € pˆ = (# successes in sample) / n • population proportion ( p ) € fraction (or percentage) of the measurements in the population that are “successes”; we also write q = 1 − p to represent the fraction of “failures” in the population € • sampling error/variability variability in values of a statistic when it is measured in different samples of the same size randomly selected from the same population • sampling distribution of a statistic distribution of all possible values of the statistic computed for every possible choice of (randomly selected) sample of a fixed size n chosen from the population Chapter 18 2 • sampling distribution model for pˆ A SRS of size n with sample proportion pˆ is selected from a population, chosen so that it is € € (1) small enough to be no more than 10% of the size of the population (the 10% Condition), but (2) large enough that it includes at least 10 successes and 10 failures (the Success/Failure Condition), then the normal model governs the sampling distribution of values of pˆ . The random variable represented by our measurements of pˆ has an expected (mean) value of E( pˆ ) = p, equal to the € population proportion, and a standard deviation of € pq € SD( pˆ ) = . n € € € That is, the sampling distribution model for pˆ is pq N p, . n € Moreover, this normal model is a better description of the sampling distribution when the sample size n € is true because of… is larger. This Chapter 18 3 the most fundamental theorem in all of statistical theory… • the Central Limit Theorem Regardless of the population from which we are sampling or the statistic we may be measuring, the distribution of the sampling statistic comes closer to being normally distributed the larger the sample size n becomes, and while the mean of the distribution does not depend on n, the standard deviation will decrease as n gets larger; therefore, the statistic we are sampling will become a better and better approximation of its true mean value for larger and larger n. Chapter 18 4 • sampling distribution model for y A SRS of size n with sample mean y is selected from a population with population mean µ and standard deviation σ. The sample is chosen so that € its measurements are independent of each other, a € condition that is difficult to check, but reasonable to assume, unless the sample is drawn without replacement, in which case it should be small enough to be no more than 10% of the size of the population (the 10% Condition). Then the normal model governs the sampling distribution of values of y . The random variable represented by our measurements of y has an expected (mean) value of E( y ) = µ , equal to the population mean, and a standard deviation of € σ € SD( y ) = . n € € That is, the sampling distribution model for y is € € σ N µ , . n € Moreover, this normal model is a better description of the sampling distribution the larger the sample € size n is. Chapter 18 5 • standard error In practice, we won’t generally know the value of the standard deviation of the sampling distribution of our statistic – after all, it depends on population parameters: for instance, SD( pˆ ) = € SD( y ) = pq n σ n depends on p and q; depends on σ. € But we can estimate them. When we do, our € estimates are called standard errors: € ˆˆ pq ; n s SE( y ) = . n SE( pˆ ) = € € € €