Section 9.3 Notes complete

Math 1442 (Statistics for Life) Ch 9 Sec 9.3 Section 9.3 Two Means: Dependent Samples This section presents methods for testing hypotheses and constructing confidence intervals involving the mean of the differences of the values from two populations that are dependent in the sense that the data consist of matched pairs. The pairs must be matched according to some relationship, such as before/after measurements from the same subjects or IQ scores of husbands and wives. Good Experimental Design When designing an experiment or planning an observational study, using dependent samples with matched pairs is generally better than using two independent samples.  Inferences About Differences from Matched Pairs  Objectives 1. Hypothesis Test: Use the differences from two dependent samples (matched pairs) to test a claim about the mean of the population of all such differences. 2. Confidence Interval: Use the differences from two dependent samples (matched pairs) to construct a confidence interval estimate of the mean of the population of all such differences.  Notation d = individual difference between the two values in a single matched pair µ d = mean value of the differences d for the population of all matched pairs of data d = ¿ mean value of the difference d for the paired sample data. s d = standard deviation of the differences d for the paired sample data n = number of pairs of sample data  Requirements • The sample data are dependent (matched pairs). • The matched pairs are a simple random sample. • Either or both of these conditions are satisfied: The number of pairs of sample data is large ( n > 30) or the pairs of values have differences that are from a population having a distribution that is approximately normal. These methods are robust against departures for normality, so the normality requirement is loose.

Math 1442 (Statistics for Life) Ch 9 Sec 9.3  HYPOTHESIS TEST About Differences from Matched Pairs:  Null and Alternate Hypotheses for Tests of Means Right-Tailed Test Left-Tailed Test Two-Tailed Test H 0 : μ d = 0 H 0 : μ d = 0 H 0 : μ d = 0 H 1 : μ d > 0 H 1 : μ d < 0 H 1 : μ d ≠ 0  Test Statistic for Matched Pairs: Dependent Samples (with H 0 : µ d = 0) P -Values: P -values are automatically provided by technology or the t distribution in Table A-3 can be used. Critical Values: Use Table A-3 ( t distribution). For degrees of freedom, use df = n − 1.  Confidence Interval Estimate of µ d : Dependent Samples Procedures for Inferences with Dependent Samples 1. Verify that the sample data consist of dependent samples (or matched pairs), and verify that the requirements in the preceding slides are satisfied. 2. Find the difference d for each pair of sample values. 3. Find the value of d and s d . 4. For hypothesis tests and confidence intervals, use the same t test procedures used for a single population mean. 5. For hypothesis tests and confidence intervals, use the same t test procedures used for a single population mean.

Math 1442 (Statistics for Life) Ch 9 Sec 9.3 Example 2: Are Best Actresses Generally Younger Than Best Actors? Data lists ages of actresses when they won Oscars in the category of Best Actress, along with the ages of actors when they won Oscars in the category of Best Actor. The ages are matched according to the year that the awards were presented. This is a small random selection of the available data so that we can better illustrate the procedures of this section. (a) Use the sample data with a 0.05 significance level to test the claim that for the population of ages of Best Actresses and Best Actors, the differences have a mean less than 0. (b) Using the same sample data, construct a 90% confidence interval estimate of µ d , which is the mean of the age differences. Actress (years) 28 28 31 29 35 Actor (years) 62 37 36 38 29 Difference d

Math 1442 (Statistics for Life) Ch 9 Sec 9.3

Math 1442 (Statistics for Life) Ch 9 Sec 9.3 Subject A B C D E F G H Before 6.6 6.5 9.0 10.3 11.3 8.1 6.3 11.6 After 6.8 2.4 7.4 8.5 8.1 6.1 3.4 2.0