1. Sampling Distributions
Chapter 9:
Sampling Distributions

2. Chapter Objectives Define a sampling distribution
Contrast bias and variability
Describe the sampling distribution of a sample proportion (shape, center, spread)
Use a Normal approximation to solve probability problems involving the sampling distribution of a sample proportion.
Describe the sampling distribution of a sample mean.
State the central limit theorem
Solve probability problems involving the sampling distribution of a sample mean.

3. 9.1 Sampling Distributions
Major topics:
Sampling variability
Describing sampling distributions
Bias of a statistic
Variability of a statistic

4. 9.1 Objectives Compare and contrast parameter and statistic
Explain what is meant by sampling variability
Define the sampling distribution of a statistic
Explain how to describe a sampling distribution
Define an unbiased statistic and an unbiased estimator
Describe what is meant by the variability of a statistic
Explain how bias and variability are related to estimating with a sample.

6. Parameter v statistic Population Sample Parameter Usually Greek letter
Fixed
Sample
Statistic
Usually Latin letter
Vary

7. Example 9.2: (respondents)/(sample size) = p-hat
p-hat is the sample proportion
p is the population proportion

8. Sampling variability
Take a large number of sample from the same population
Calculate the sample mean (x-bar) or sample proportion (p-hat) for each sample.
Make a histogram of the values of x-bar or p-hat.
Examine the distribution displayed in the histogram for shape, center, and spread, as well as outliers or other deviations.

9. …the distribution
of all possible samples
of the same size
from the same population.

10. Simulation v sampling distribution
Do not mistake a simulation of a sampling distribution for the actual sampling distribution. They are different. (One is a bunch of simulations, the other is a bunch of actual samples.)

12. Text exercises

13. Describing Sampling Distributions
Shape
Center
Spread

14. “Describe”…
Since chapter 1, when you have read the word “describe” you have been expected to discuss (and numerically support) shape, center, and spread. It is the same here.
Don’t jump to conclusions! Watch the labels on the axes. Often you are presented the same or very similar data with very different scales. (Perhaps, to trick you. It is more likely done to see how closely you are paying attention to the scales or persuade you. This is true in school, but more often in the media.)

15. A little off topic (but absolutely connected)… Elements of a persuasive argument/presentation/discussion
Since you brought up current events, look for a “weighted” presentation or argument.
Why is it so dependent on ONE or TWO elements?
Is the third so weak that the presenter doesn’t want to talk about it?
Ethos: ethics, right v wrong
Logos: logic
Pathos: passionate, emotional appeal
For more on this topic, look up Aristotle’s elements of persuasion.

16. The Bias of a Statistic
Sampling distributions allow us to describe bias more precisely by speaking of the bias of a statistic rather than the sampling method.
Bias concerns the center of the sampling distribution.
See p 573

19. Variability of a Statistic
Depends only on sample size, not the size of the population.
For example, it implies that a survey of (say) 1,200 people will have the same variability (that is, margin of error) whether the population being sampled is the City of Milwaukee or the entire United States.
In other words, larger samples does not mean better estimates for larger populations. We are going to look at “how much is enough?”

21. Bias and Variability

22. ____________ bias and _______________ variability

23. ____________ bias and _______________ variability

24. ____________ bias and _______________ variability

25. ____________ bias and _______________ variability

26. Text exercises

27. 9.1 Cooperative Assessment