Sampling Distribution Models Chapter 18 Sampling Distribution Models Copyright © 2009 Pearson Education, Inc.

Symbols Review Statistic (Sample) vs. Parameter (Population) MEAN SD PROPORTION

Assumptions and Conditions Randomization Condition: The sample should be a simple random sample of the population. 10% Condition: If sampling has not been made with replacement, then the sample size, n, must be no larger than 10% of the population. *10n is less than the population size Normality Check: (Success/Failure Condition) The sample size has to be big enough so that both np and n(1 – p) are at least 10.

The Sampling Distribution Model for a Proportion Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of is modeled by a Normal model with Mean: Standard deviation:

The Central Limit Theorem for Sample Proportions When working with proportions, knowing the mean of the sampling distribution automatically gives us the standard deviation as well so, the distribution of the sample proportions is modeled with a probability model that is:

The Central Limit Theorem for Sample Proportions A picture of what we just discussed is as follows:

What About Quantitative Data? Proportions summarize categorical variables. The Normal sampling distribution model looks like it will be very useful. Can we do something similar with quantitative data? We can indeed. Even more remarkable, not only can we use all of the same concepts, but almost the same model.

Assumptions and Conditions We can’t check these directly, but we can think about whether the Independence Assumption is plausible. We can also check some related conditions: Randomization Condition: The data values must be sampled randomly. 10% Condition: When the sample is drawn without replacement, the sample size, n, should be no more than 10% of the population. *10n is less than the population size Normality Check: (Large Enough Sample Condition) The CLT doesn’t tell us how large a sample we need. General Rule: the sample size (n) must be at least 30.

But Which Normal?  

Notation for Normal Distributions  

Standard Deviation of Sampling Dist. Both of the sampling distributions we’ve looked at are Normal. For proportions For means

Standard Error When we don’t know p or σ, we’re stuck, right? Nope. We will use sample statistics to estimate these population parameters. Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error.

Standard Error For a sample proportion, the standard error is For the sample mean, the standard error is