More on Inference

1. More on Inference

2. Confidence Interval • A level C confidence interval for a parameter is an interval computed from sample data by a method that has probability C of producing an interval containing the true value of the parameter. • Twenty-five samples from the same population provides 25 95% confidence intervals. • In the long run, 95% of all samples give an interval that covers

4. CI for population mean • Choose an SRS of size n from a population having unknown mean and known standard deviation . • A level C confidence interval for is

5. Margin of Error • A small margin of error says that we have pinned down the parameter quite precisely. • What if the margin of error is too large? • Use a lower level of confidence • Increase the sample size • Reduce

6. Choosing the Sample Size • The confidence interval for a population mean will have a specified margin of error m when the sample size is

7. Cautions • Any formula for inference is correct only in specific circumstances • The margin of error in a confidence interval covers only random sampling errors. • Review other cautions on page 426

8. Test Statistic for Hypothesis Testing • A test statistic measures compatibility between the null hypothesis and the data. • It is a random variable with a distribution that we know. • When testing the mean with a known variance (or standard deviation), we use the following test statistic

9. P-value • The probability, computed assuming that Ho is true, that the test statistic would take a value as extreme or more extreme than that actually observed is called the p-value of the test. • The smaller the p-value, the stronger the evidence against Ho provided by the data. • If the p-value is as small or smaller than alpha, we say that the data are statistically significant at level alpha.

10. CIs and 2-sided Tests • A level alpha 2-sided significance tests rejects a hypothesis exactly when the value falls outside a level 1 – alpha confidence interval for • Fixed alpha tests use the table of standard normal critical values (Table D)

11. Use and Abuse • P-values are more informative than the results of a fixed level alpha test. • Beware of placing too much weight on traditional values of alpha. • Very small effects can be highly significant, especially when a test is based on a large sample. • Lack of significance does not imply that Ho is true, especially when the test has low power. • Significance tests are not always valid.

12. Power • The probability that a fixed level alpha significance test will reject Ho when a particular alternative value of the parameter is true is called the power of the test to detect that alternative. • One way to increase power is to increase sample size. • Other suggestions are on page 472.