Chapter 9 Sampling Distributions Chapter Objectives Define a
Chapter 9: Sampling Distributions
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.
9. 1 Sampling Distributions Major topics: • Sampling variability • Describing sampling distributions • Bias of a statistic • Variability of a statistic
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.
Parameter v statistic • Population – Parameter – Usually Greek letter – Fixed • Sample – Statistic – Usually Latin letter – Vary
• Example 9. 2: (respondents)/(sample size) = p-hat • p-hat is the sample proportion • p is the population proportion
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.
…the distribution of all possible samples of the same size from the same population.
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. )
Text exercises
Describing Sampling Distributions • Shape • Center • Spread
“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. )
A little off topic (but absolutely connected)… Elements of a persuasive argument/presentation/discussion Ethos: ethics, right v wrong Logos: logic Since you brought up current events, look for a “weighted” presentation or argument. Why is it so dependent on ONE or TWO elements? Pathos: passionate, emotional appeal Is the third so weak that the presenter doesn’t want to talk about it? For more on this topic, look up Aristotle’s elements of persuasion.
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
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? ”
Bias and Variability
______ bias and ________ variability
______ bias and ________ variability
______ bias and ________ variability
______ bias and ________ variability
Text exercises
9. 1 Cooperative Assessment
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