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### Chapter 9

Section 9.1 – Sampling Distributions

Introduction

- The process of statistical inference involves using information from a sample to draw conclusions about a wider population.
- Different random samples yield different statistics. We need to be able to describe the sampling distribution of possible statistic values in order to perform statistical inference.
- We can think of a statistic as a random variable because it takes numerical values that describe the outcomes of the random sampling process. Therefore, we can examine its probability distribution using what we learned in Chapter 7

Parameters and Statistics

- As we begin to use sample data to draw conclusions about a wider population, we must be clear about whether a number describes a sample or a population.
Remember s and p:

statistics come from samples and

parameters come from populations

- See examples 9.1 & 9.2 on p.488-489

Sampling Variability

- How can be an accurate estimate of ? After all, different random samples would produce different values of .
- This basic fact is called sampling variability: the value of a statistic varies in repeated random sampling.
- To make sense of sampling variability, we ask, “What would happen if we took many samples?”
- See example 9.3 on p.490

Sampling Distribution

- The following diagram represents the distribution of the sample proportion from 1000 SRSs of size 100 drawn from a population with p = 0.7.
- Strictly speaking, the sampling distribution is the ideal pattern that would emerge if we looked at all possible samples of the same size from the population. A fixed number, like the 1000 trials in the figure above is only an approximation to the sampling distribution.

Describing Sampling Distributions

- Describe the shape, center, spread, and any outliers of a sampling distribution to help answer the question, “How trustworthy is a statistic as an estimator of the parameter?”
- See example 9.5 on p.494-495
- The overall shape of the distribution is
symmetric and approximately normal

- The center of the distribution is very
close to the true value of p = .37 for the

population from which samples were drawn.

- The values of have a large spread ranging from .22 to .54. Because the distribution is close to normal we can use standard deviation to describe the spread, which is about .05
- There are no outliers or other important deviations from the overall pattern.

- The overall shape of the distribution is

This sampling distribution represents

1000 SRSs of size 100

The mean of all of the ’s is .372 and

the median is exactly .37

This sampling distribution represents 1000 SRSs of size 1000

The range is a lot less and almost all ’s are close to the population parameter p = .37

This is the same sampling distribution as the previous, but with a different scale so you can see the shape better

- Homework: p.489 #’s 1-4

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