Chapter 18 Sampling Distribution Models *For Means

Central Limit Theorem The mean of a random sample has a sampling distribution whose shape can be approximated by a Normal model. The larger the sample, the better the approximation will be. Means have smaller SD than individuals As the sample size increases the SD decreases

Sampling Distribution Model for Means The distribution of the means of random samples of size n follows the Normal model

Which Model to Use?? Check the type of data you have – Categorical: Proportions – Quantitative: Means

CLT Conditions All we need to use the Central Limit Theorem is: – the observations to be independent – collected with randomization Conditions: – Randomization Condition: data values must be sampled randomly – Independence Assumption: sampled values must be independent. When not replacing, check the 10% condition

– Large Enough Sample Condition: if population is roughly symmetric and unimodal then a small sample is ok to use if population is skewed in either direction, a larger sample is needed – Think about the distribution of the population and if your sample size is large enough to use the normal model. Tell whether you think the condition has been met.

Summary The statistic itself is a random quantity Sample to sample variability is what generates the sampling distribution Sampling distribution shows us the distribution of possible values the statistic could have had CLT – tells us that we can model their sampling distribution directly with Normal model

Beware! Don’t confuse the sampling distribution with the distribution of the sample Always check for independence. CLT depends crucially on the assumption of independence Watch out for small samples from skewed populations