Ap Stats Notes
Chapter 23
Inference About Means
Copyright © 2010 Pearson Education, Inc.
Getting Started n n
n
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 Copyright © 2010 Pearson Education, Inc.
Slide 23 - 3
Getting Started (cont.) n n
All we need is a random sample of quantitative data. And the true population standard deviation, σ. n Well, that’s a problem…
Copyright © 2010 Pearson Education, Inc.
Slide 23 - 4
Getting Started (cont.) n n
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 Copyright © 2010 Pearson Education, Inc.
n
n
Slide 23 - 5
Getting Started (cont.) n n
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?
Copyright © 2010 Pearson Education, Inc.
Slide 23 - 6
Gosset’s t n n
n
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
Inference About Means
Copyright © 2010 Pearson Education, Inc.
Getting Started n n
n
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 Copyright © 2010 Pearson Education, Inc.
Slide 23 - 3
Getting Started (cont.) n n
All we need is a random sample of quantitative data. And the true population standard deviation, σ. n Well, that’s a problem…
Copyright © 2010 Pearson Education, Inc.
Slide 23 - 4
Getting Started (cont.) n n
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 Copyright © 2010 Pearson Education, Inc.
n
n
Slide 23 - 5
Getting Started (cont.) n n
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?
Copyright © 2010 Pearson Education, Inc.
Slide 23 - 6
Gosset’s t n n
n
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