9.2 Objectives

1. 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. • Use a Normal approximation to the sampling distribution of p-hat to solve probability problems involving p-hat.

2. 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.

3. 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?

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

5. Fig. 9.15 (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 1987. (b) The distributions of returns for portfolios of 5 stocks in 1987.

6. 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.)

7. FYI

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

10. Fig 9.16 (p 595)

11. Practice: P 595 31 & 33

12. 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.

13. 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)

21. Practice: 35, 37, 38 & 47