More on Inference

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

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

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

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

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

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

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.

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)

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.

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.