Statistics Lecture 9 Hypothesis testing: Parametric tests •David Bartl •Statistics •INM/BASTA Outline of the lecture •z-tests for the means — type I and type II error — significance level α — power of the test 1 − β •t-tests for the means •χ2-test for the variance •z-test for the population proportion •p-value of the test Theorems & Confidence intervals Theorems & Confidence intervals Theorems & Confidence intervals Theorems & Confidence intervals Theorems & Confidence intervals z-tests for the means •One-sample z-test for the population mean •Paired-sample z-test for the difference of the population means •Two-sample z-test for the difference of the population means • One-sample z-test for the population mean One-sample z-test for the population mean One-sample z-test for the population mean One-sample z-test for the population mean One-sample z-test for the population mean One-sample z-test for the population mean One-sample z-test for the population mean One-sample z-test for the population mean One-sample z-test for the population mean One-sample z-test for the population mean One-sample z-test for the population mean One-sample z-test for the population mean Type I and Type II error Type I and Type II error Type I and Type II error Type I and Type II error Type I and Type II error One-sample z-test for the population mean One-sample z-test for the population mean One-sample z-test for the population mean One-sample z-test for the population mean Paired-sample z-test for the difference of the pop.means Paired-sample z-test for the difference of the pop.means Paired-sample z-test for the difference of the pop.means Paired-sample z-test for the difference of the pop.means Paired-sample z-test for the difference of the pop.means •We have thus reduced the paired-sample z-test for the difference of the population means to the one-sample z-test for the population mean, which we already know. Paired-sample z-test for the difference of the pop.means Paired-sample z-test for the difference of the pop.means Paired-sample z-test for the difference of the pop.means Paired-sample z-test for the difference of the pop.means Paired-sample z-test for the difference of the pop.means Paired-sample z-test for the difference of the pop.means Paired-sample z-test for the difference of the pop.means Two-sample z-test for the difference of the popul. means Two-sample z-test for the difference of the popul. means and Two-sample z-test for the difference of the popul. means and Two-sample z-test for the difference of the popul. means Two-sample z-test for the difference of the popul. means Two-sample z-test for the difference of the popul. means Two-sample z-test for the difference of the popul. means Two-sample z-test for the difference of the popul. means Two-sample z-test for the difference of the popul. means Two-sample z-test for the difference of the popul. means Two-sample z-test for the difference of the popul. means Two-sample z-test for the difference of the popul. means t-tests for the means •One-sample t-test for the population mean •Paired-sample t-test for the difference of the population means •Two-sample t-test for the difference of the population means • One-sample t-test for the population mean One-sample t-test for the population mean One-sample t-test for the population mean One-sample t-test for the population mean One-sample t-test for the population mean One-sample t-test for the population mean One-sample t-test for the population mean One-sample t-test for the population mean One-sample t-test for the population mean One-sample t-test for the population mean The gamma function The gamma function – another definition (due to Euler) One-sample t-test for the population mean One-sample t-test for the population mean One-sample t-test for the population mean Type I and Type II error One-sample t-test for the population mean One-sample t-test for the population mean One-sample t-test for the population mean One-sample t-test for the population mean Paired-sample t-test for the difference of the pop.means Paired-sample t-test for the difference of the pop.means Paired-sample t-test for the difference of the pop.means Paired-sample t-test for the difference of the pop.means Paired-sample t-test for the difference of the pop.means •We have thus reduced the paired-sample t-test for the difference of the population means to the one-sample t-test for the population mean, which we already know. Paired-sample t-test for the difference of the pop.means Paired-sample t-test for the difference of the pop.means Paired-sample t-test for the difference of the pop.means Paired-sample t-test for the difference of the pop.means Paired-sample t-test for the difference of the pop.means Paired-sample t-test for the difference of the pop.means Paired-sample t-test for the difference of the pop.means Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY and Two-sample t-test for the diff. of the pop. means // σX=σY and Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX=σY Two-sample t-test for the diff. of the pop. means // σX≠σY Theorem (Satterthwaite’s approximation): Two-sample t-test for the diff. of the pop. means // σX≠σY •Exercise: • •Use the last Theorem (Satterthwaite’s approximation) to formulate a statistical two-sample t-test for the difference of the population means with two-sided / one-sided alternative hypothesis (not assuming the same variance). χ2-test for the variance •χ2-test for the variance • χ2-test for the population variance χ2-test for the population variance χ2-test for the population variance χ2-test for the population variance χ2-test for the population variance χ2-test for the population variance χ2-test for the population variance χ2-test for the population variance χ2-test for the population variance χ2-test for the population variance χ2-test for the population variance z-test for the population proportion •One-sample binomial test for the population proportion •One-sample z-test for the population proportion • One-sample z-test for the population proportion One-sample z-test for the population proportion One-sample z-test for the population proportion One-sample binomial test for the population proportion One-sample binomial test for the population proportion One-sample binomial test for the population proportion One-sample binomial test for the population proportion One-sample z-test for the population proportion One-sample z-test for the population proportion One-sample z-test for the population proportion One-sample z-test for the population proportion One-sample z-test for the population proportion p-value of the test •The general outline of a statistical hypothesis test •The p-value of a test • The general outline of a statistical hypothesis test The general outline of a statistical hypothesis test The general outline of a statistical hypothesis test The general outline of a statistical hypothesis test The p-value of the test hypothesis.)