9.2 Objectives Describe the sampling distribution of a sample proportion. (Remember that “describe” means to write about the shape, center, and spread.) Compute the mean and standard deviation for the sampling distribution of p-hat. Identify the “rule of thumb” that justifies the use of the recipe for the standard deviation of p-hat. Identify the conditions necessary to use a Normal approximation to the sampling distribution of p-hat. Use a Normal approximation to the sampling distribution of p-hat to solve probability problems involving p-hat.

9.3 Objectives Give the mean and standard deviation of a population, calculate the mean and standard deviation for the sampling distribution of a sample mean. Identify the shape of the sampling distribution of a sample mean drawn from a population that has a Normal distribution. State the central limit theorem. Use the central limit theorem to solve probability problems for the sampling distribution of a sample mean.

In 9.2, we found that p-hat is approximately Normal under the right conditions. What were those? Wouldn’t it be nice if we could say something similar about the sampling distribution x-bar?

Categorical variables  sampling proportions Quantitative variables  sampling distribution stats such as median, mean and standard deviation.

Fig (p592) The Figures emphasize a principle that will be made precise in this section: Means of random samples are less variable than individual observations. Means of random samples are more Normal than individual observations. In this section, we are still considering distributions of sample statistics, but we are shifting our attention to x-bar. (a) The distribution of returns for a NYSE common stocks in (b) The distributions of returns for portfolios of 5 stocks in 1987.

The behavior of x-bar in repeated samples is similar to that of sample proportion p-hat. The sample mean x-bar is an unbiased estimator of the population mean mu. The values of x-bar are less spread out for larger sample. Their standard deviation decreases at the rate sqrt(n). You will need to take a sample four times as large in order to half the stdev. Use sigma/sqrt(n) for the stdev of x-bar only when the population is at least 10 times the sample. (This is almost always the case.)

FYI

“Describe” Describing the behavior of ANY distribution means to talk about –SHAPE –CENTER and –SPREAD

Fig 9.16 (p 595)

Practice: P & 33

Central Limit Theorem Watch the videos as homework CLT discusses the SHAPE (& only the shape) of the sampling distribution of x-bar when the sample is sufficiently large. If n is not large enough, the shape of the sampling distribution of x-bar more closely resembles the shape of the original population.

Thus there are 3 situations to consider when discussing the shape of the sampling distribution The population has a Normal distribution—shape of sampling distribution: Normal, regardless of sample size Any population shape, small n—shape of sampling distribution: similar to shape of parent population. Any population shape, large n—shape of sampling distribution: close to Normal (CLT)

Practice: 35, 37, 38 & 47