1. Statistics 270– Lecture 25

2. Cautions about Z-Tests Data must be a random sample Outliers can distort results Shape of the population distribution matters (large and small samples?) For significance tests, there is a difference between practical and statistical significance

3. Types of Errors in a Significance Test Suppose that the null hypothesis is true, we could collect a sample that suggests that we reject H 0 Suppose that H 0 is not true, we could fail to reject the null hypothesis

4. Inference About the Population Mean To make inference about the population mean, , we have used the z-test Key feature:  is known Most often,  is unknown!  must be estimated from the data Use to estimate 

5. Inference About the Population Mean To make probability statements about the the sample mean, we have used the Z-statistic when  is known When  is unknown, we use the one-sample t-statistic with (n-1) degrees of freedom

6. Inference About the Population Mean Standard error: Degrees of freedom:

7. One Sample t-Test for a Population Mean Data: random sample x 1, x 2, …, x n Mean, , is unknown Standard deviation is unknown For testing the hypothesis H 0 :  =  0 Test Statistic: Degrees of freedom: t has a t-distribution with n-1 degrees of freedom

8. One Sample t-Test for a Population Mean Computing p-value depends on the alternate hypothesis: P-values are exact if the population distribution is normal and approximately correct for large samples in other cases

9. One Sample t-Test for a Population Mean Rule of Thumb For n<15, use t-test if data appear approximately normal For n 15, can use t-test when no outliers For large n (say n 40), can safely use t-test Note: always require random sample!

10. Example Composition of earth’s atmosphere has changed over time Gas bubbles in ancient amber are examined to study the nature of the atmosphere long ago Measurements on specimens of amber from the Cretaceous period (75-95 million years ago) give the following percentages of nitrogen 63.465.064.4 63.354.864.5 60.849.151.0

11. Example Assume the data are a random sample from the Cretaceous period To see if there is a difference with today’s 78.1% nitrogen, conduct a hypothesis test using these data

12. Example Hypotheses: Test Statistic P-value Conclusion