Slide 1Practice ProblemsAdditional Reading and ExamplesTest 7OPTIONAL Final ExamCourse EvaluationsSlide 7Statistical InferenceStatistical InferenceTests of Significance for m1 - m2Tests of Signfiicance for m1-m2Tests of SignificanceTests of Significance for m1-m2Tests of Significance on m1 - m2Example 106Example 106Example 106Example 106Example 106Example 106Example 106Example 106Example 106Slide 24Example 106Example 106Example 106Example 106Example 106Statistical InferencePaired t-TestPaired t-TestPaired t-TestPaired t-TestPaired t-TestTI-83/84 CalculatorExample 107Example 107Example 107Example 107Example 107Example 107Example 107Example 107Example 107Example 107Example 107Example 107Example 107Example 107Slide 51Example 107Example 107Example 107Example 107Example 107Example 108Example 108Example 108Example 108Example 108Example 108Example 108Example 108Example 108Example 108Example 108Example 108Slide 69Example 108Example 108Example 108Example 108Example 108See you Wednesday for the test!STAT 210Lecture 40Tests of Significance for m1 – m2 Paired DataDecember 5, 2016Practice ProblemsPages 286 through 291Relevant problems: X.13 and X.14Recommended problems: X.13 and X.14Additional Reading and ExamplesRead pages 281 through 283Test 7Wednesday, December 7Questions for the first 10 minutes, then test – papers due promptly at the end of class!Covers chapter 10 (pages 261 – 291)Combination of multiple choice questions and written/short answer problems.Formulas and tables provided; Bring a calculator!Practice Tests and Formula Sheet on Blackboard.OPTIONAL Final ExamMonday, December 12 4:00 – 6:50 pmHelp Sessions: Times will be posted on BlackboardCUMULATIVE! (Covers Chapters 1 – 10)Combination of multiple choice questions and written/short answer problemsFormulas and tables provided; Bring a calculator!Practice Tests and Formula Sheet on Blackboard.Course EvaluationsCourse evaluations have begun online, and will be available until December 10th. Please take a few minutes to log on to http://go.vcu.edu/eval and complete the evaluation form. I really appreciate any thoughts or feedback that you have about the course or my teaching! THANKS!!!Clicker2Statistical InferenceStatistical inference involves using statistics computed from sample data to make statements about unknown population parameters.This includes estimation (confidence intervals) and significance tests.The past two chapters we made inferences about the population mean m and about the population proportion p.Statistical InferenceNow suppose we have two populations with population means m1 and m2, respectively, and of interest is to make statistical inferences about the difference between the two population means: m1 - m2.Tests of Significance for m1 - m2We hypothesize that the difference between the population means equals some specified value m0 and we use data from our samples to test whether this value is reasonable or whether the mean difference is actually greater than m0, less than m0, or not equal to m0.Null Hypothesis: H0: m1 - m2 = m0 Ha: m1 - m2 > m0Alternative Hypothesis: Ha: m1 - m2 < m0Ha: m1 - m2 = m0Tests of Signfiicance for m1-m2In many problems m0 = 0, and hence the hypotheses are often written as follows:H0: m1 - m2 = 0 H0: m1 = m2Ha: m1 - m2 > 0 Ha: m1 > m2 Ha: m1 - m2 < 0 Ha: m1 < m2Ha: m1 - m2 = 0 Ha: m1 = m2Tests of SignificanceAssumptions:(1) Suppose we have two independent simple random samples.(2) (i) Either both populations are normally distributed: X1 ~ N(m1, s1) and X2 ~ N(m2, s2)or (ii) Both sample sizes are large enough such that the Central Limit Theorem appliesTests of Significance for m1-m2 If the population standard deviations and are known, the test statistic for testing whether the mean difference isas specified is:Z = (X1 - X2) - m0 s12 + s22 n1 n2Tests of Significance on m1 - m2If the population standard deviations and are unknown (meaning we only know the sample standard deviations s1 and s2), the test statistic for testing whether the mean difference isas specified is: t = (X1 - X2) - m0 S12 + S22n1 n2This test statistic follows a t distribution with degrees of freedomequal to the smaller of n1 - 1 and n2 - 1.Example 106Populations of interest: ??? Parameter of interest: ???Example 106The populations of interest are all people living in urban settings and all people living in rural settings.Parameter of interest = m1 - m2m1 = mean loss in investments among all people who lived in urban settingsm2 = mean loss in investments among all people who lived in rural settings.Example 106m1 = mean loss in investments among all people living in urban settingsm2 = mean loss in investments among all people living in rural settings(1) H0: m1 - m2 = 0 H0: m1 = m2 Ha: m1 - m2 = 0 Ha: m1 = m2Example 106m1 = mean loss in investments among all people living in urban settingsm2 = mean loss in investments among all people living in rural settings(1) H0: m1 - m2 = 0 H0: m1 = m2 Ha: m1 - m2 = 0 Ha: m1 = m2 a = .10Example 106m1 = mean loss in investments among all people living in urban settingsm2 = mean loss in investments among all people living in rural settings(1) H0: m1 - m2 = 0 H0: m1 = m2 Ha: m1 - m2 = 0 Ha: m1 = m2 Assumptions: (1) We have independent, simple random samples (2) The sample sizes are large enough for the Central Limit Theorem to apply.Example 106m1 = mean loss for all people living in urban settingsm2 = mean loss for all people living in rural settings(1) H0: m1 - m2 = 0 H0: m1 = m2 Ha: m1 - m2 = 0 Ha: m1 = m2 (2) n1 = 100 x1 = 26048 s1 = 21219 n2 = 100 x2 = 31295 s2 = 18365 t = (x1 - x2) - m0 = (26048 - 31295) - 0 = -5247 = -1.870 2806.3 s12 + s22 (21219)2 + (18365)2 n1 n2 100 100Example 106m1 = mean loss for all people living in urban settingsm2 = mean loss for all people living in rural settings(1) H0: m1 - m2 = 0 H0: m1 = m2 Ha: m1 - m2 = 0 Ha: m1 = m2 (2) n1 = 100 x1 = 26048 s1 = 21219 n2 = 100 x2 = 31295 s2 = 18365 t = (x1 - x2) - m0 = (26048 - 31295) - 0 = -5247 = -1.870 2806.3 s12 + s22 (21219)2 + (18365)2 n1 n2 100 100(3) df = smaller of (100 - 1 = 99 and 100 - 1 = 99) = 99Example 106m1 = mean loss for all
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