The Basics A population is the entire group

on which we would like to have information.

A sample is a smaller group, selected somehow from the population, on which we do have information.

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The Basics A parameter is a number (such as the

mean or standard deviation) which describes a population

A statistic is a number (such as the mean or standard deviation) which describes a sample

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sx

Sampling Distributions Suppose you select a gazillion different

random samples from the same population From each sample, you compute a certain

statistic The distribution of these gazillion different

statistics is called a sampling distribution

Sampling variability Take a large number of samples from the same

population.

Calculate the sample statistic for each sample.

Make a histogram of the values of the statistic.

Examine the histogram’s shape, center, and spread.

Sampling distributions The sampling distribution of a statistic is the

distribution of values that statistic would take if you sampled all possible samples of size N.

For example, suppose 60% of all Americans don’t like to shop for clothes.

If we draw a random sample of 100 Americans, how many of them don’t like to shop for clothes?

Sampling distribution for p = 0.60 and N = 100

•63%•66%•63%•59%•54%•61%•60%•57%•61%•63%•59%•62%

Sampling distribution for p = 0.60 and N = 100•56% 66% 62% 59% 61% 56% 54% 58% 68% 62% 62% 67% 56% 55% 51% 63% 58% 61% 61% 70% 65% 61% 56% 60% 57% 59% 64% 56% 63% 56% 50% 63% 60% 65% 62% 56% 51% 62% 59% 65% 53% 57% 55% 64% 66% 61% 56% 63% 57% 59% 61% 54% 52% 60% 55% 63% 60% 67% 50% 56% 60% 55% 58% 57% 60% 63% 59% 61% 64% 58% 60% 61% 69% 69% 61% 56% 64% 70% 61% 62% 65% 61% 56% 60% 60% 57% 62% 58% 59% 58% 58% 56% 53% 49% 56% 58% 53% 72% 63% 61% 60% 51% 69% 57% 65% 62% 65% 51% 57% 57% 56% 58% 58% 58% 53% 58% 56% 56% 57% 61% 65% 65% 58% 57% 66% 62% 58% 54% 65% 62% 59% 56% 59% 61% 53% 60% 61% 59% 62% 58% 59% 58% 58% 61% 61% 67% 59% 61% 58% 65% 63% 59% 54% 61% 60% 57% 61% 63% 59% 64% 61% 60% 60% 61% 56% 62% 60% 64% 56% 61% 56% 60% 61% 56% 58% 68% 67% 56% 55% 62% 64% 52% 58% 53%

AFTER A THOUSAND SAMPLES:Sampling distribution for p = 0.60 and N = 100

40% 44% 48% 52% 56% 60% 64% 68% 72% 76% 80%

•Mean = 59.97% •St Dev = 4.74%

AFTER A GAZILLION SAMPLES:Sampling distribution for p = 0.60 and N = 100

•Mean = 60% Standard Deviation = 4.899%

40% 45% 50% 55% 60% 65% 70% 75% 80%

The bias of a statistic

A statistic is unbiased if the mean of its sampling distribution is expected to be equal to the population parameter it is estimating.

But if the statistic over-estimates or under-estimates the parameter, then that statistic is biased.

The variability of a statistic

• The spread of the sampling distribution shows how variable the statistic is, from one sample to another.

• Usually, the spread of the sampling distibution is smaller for large samples, and larger for small samples.