Inference About Means

Copyright © 2010 Pearson Education, Inc.

Getting Started

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Now that we know how to create confidence intervals and test hypotheses about proportions, it’d be nice to be able to do the same for means. Just as we did before, we will base both our confidence interval and our hypothesis test on the sampling distribution model. The Central Limit Theorem told us that the sampling distribution model for means is Normal s with mean μ and standard deviation SD ( y ) = n

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Slide 23 - 3

Getting Started (cont.)

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All we need is a random sample of quantitative data. And the true population standard deviation, σ. n Well, that’s a problem…

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Slide 23 - 4

Getting Started (cont.)

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Proportions have a link between the proportion value and the standard deviation of the sample proportion. This is not the case with means—knowing the sample mean tells us nothing about SD( y) We’ll do the best we can: estimate the population parameter σ with the sample statistic s. s Our resulting standard error is SE ( y ) = n

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Getting Started (cont.)

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We now have extra variation in our standard error from s, the sample standard deviation. n We need to allow for the extra variation so that it does not mess up the margin of error and P-value, especially for a small sample. And, the shape of the sampling model changes—the model is no longer Normal. So, what is the sampling model?

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Gosset’s t

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William S. Gosset, an employee of the Guinness Brewery in Dublin, Ireland, worked long and hard to find out what the sampling model was. The sampling model that Gosset found has been known as Student’s t. The Student’s t-models form a whole family of related distributions that depend on a parameter known as degrees of freedom. n We often denote degrees of freedom as df, and the model as tdf. Copyright © 2010 Pearson Education, Inc.

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A Confidence Interval for Means?

A practical sampling distribution model for means

When the conditions are met, the standardized sample mean

y -m t= SE ( y )

follows a Student’s t-model with n – 1 degrees of freedom. s We estimate the standard error with SE ( y ) = n

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Slide 23 - 8

A Confidence Interval for Means? (cont.)

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When Gosset corrected the model for the extra uncertainty, the margin of error got bigger. n Your confidence intervals will be just a bit wider and your P-values just a bit larger than they were with the Normal model. By using the t-model, you’ve compensated for the extra variability in precisely the right way.

Copyright © 2010 Pearson Education, Inc.

Slide 23 - 9

A Confidence Interval for Means? (cont.)

One-sample t-interval for the mean

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When the conditions are met, we are ready to find the confidence interval for the population mean, μ. The confidence interval is * n -1 where the standard error of the mean is

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y ±t

´ SE ( y )

s SE ( y ) = n

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* The critical value tn-1 depends on the particular confidence level, C, that you specify and on the number of degrees of freedom, n – 1, which we get from the sample size.

Copyright © 2010 Pearson Education, Inc.

Slide 23 - 10

A Confidence Interval for Means? (cont.)

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Student’s t-models are unimodal, symmetric, and bell shaped, just like the Normal. But t-models with only a few degrees of freedom have much fatter tails than the Normal. (That’s what makes the margin of error bigger.)

Copyright © 2010 Pearson Education, Inc.

Slide 23 - 11

A Confidence Interval for Means? (cont.)

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As the degrees of freedom increase, the t-models look more and more like the Normal. In fact, the t-model with infinite...