Slide 1Practice ProblemsAdditional Reading and ExamplesTest 7Course EvaluationsSlide 6Statistical InferenceStatistical InferenceConfidence Interval for m1 - m2Confidence Interval for m1 - m2Confidence Interval for m1 - m2Confidence Interval for m1 - m2Confidence Interval for m1 - m2Example 103Example 103Example 103Example 103Example 103Example 103Example 103Example 103Example 103Example 103Tests of Significance for m1 - m2Tests of Signfiicance for m1-m2Slide 26Tests of Significance for m1 – m2Tests of SignificanceTests of SignificanceTests of SignificanceTests of SignificanceTests of Significance for m1-m2Tests of Significance for m1-m2Tests of Significance for m1-m2TI-83/84 CalculatorExample 104Example 104Example 104Example 104Example 104Example 104Example 104Example 104Example 104Example 104Example 104Example 104Example 104Tests of Significance on m1 - m2Tests of SignificanceTests of Significance on m1 - m2Tests of Significance on m1 - m2TI-83/84 CalculatorExample 105Example 105Example 105Example 105Example 105Example 105Example 105Example 105Example 105Example 105Example 105Example 105Slide 66Example 105Example 105Example 105Example 105Example 106Example 106Example 106Example 106Example 106Example 106Example 106Example 106Example 106Slide 80Example 106Example 106Example 106Example 106Example 106Motivating ExampleMotivating ExampleMotivating ExampleMotivating ExampleMotivating ExampleMotivating ExampleMotivating ExampleMotivating ExampleMotivating Example SolutionMotivating Example SolutionHave a nice weekend!STAT 210Lecture 39Tests of Significance for m1 – m2December 1, 2016Practice ProblemsPages 286 through 291Relevant problems: X.3, X.4 (a) and (b), X.7, X.9, X.10 (b), X.11 and X.12Recommended problems: X.7 and X.11Additional 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.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!!!ClickerStatistical 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.Confidence Interval for m1 - m2Suppose we want to estimate the difference between the population means: m1 - m2The point estimate is X1 - X2, which should be close to m1 - m2 but is likely not equal to m1 - m2 exactly.Hence we want a C% confidence interval estimate, and we subtract and add a margin of error to and from the point estimate to create the confidence interval.Confidence Interval for m1 - m2 Assumptions:(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) The populations are possibly nonnormal but both sample sizes are large enough such that the Central Limit Theorem applies(3) The population standard deviations s1 and s2 are known.Confidence Interval for m1 - m2 If the assumptions are satisfied, then a C% confidence interval for m1 - m2 can be determined using: (X1 - X2) + Z* s12 + s22 n1 n2z* is found from the bottom two lines of the t-table on page 340, as before.Confidence Interval for m1 - m2When s1 and s2 are unknown, the use of the Z distribution is not appropriate.Instead we must use the t distribution:Confidence Interval for m1 - m2Then a C% confidence interval for m1 - m2 is (X1 - X2) + t* S12 + S22 n1 n2which follows a t distribution with degrees of freedom equal to thesmaller of n1 - 1 and n2 - 1.All the definitions, interpretations, and properties of our confidence interval remain the same as before.Example 103Populations of interest: ???Parameter of interest: ???Example 103Populations of interest: all vehicles produced before the new standards went into place and all vehicles produced after the new standards went into place.Parameter of interest: m1 – m2where m1 = mean miles per gallon in all vehicles produced before the standards went into placeand m2 = mean miles per gallon in all vehicles produced after the standards went into placeExample 103Assumptions: ???Example 103Assumptions: 1. Independent simple random samplesExample 103Assumptions: 1. Independent simple random samples 2. Sample sizes are both large enough for the Central Limit Theorem to apply.Example 103Assumptions: 1. Independent simple random samples 2. Sample sizes are both large enough for the Central Limit Theorem to apply. 3. Population standard deviations unknown,(only sample standard deviations given)so we’ll use the t distributionExample 103Summary statistics:Before: n1 = 57 x1 = 23.1 s1 = 6.3 After: n2 = 51 x2 = 25.2 s2 = 6.5Example 103Summary statistics:Before: n1 = 57 x1 = 23.1 s1 = 6.3 After: n2 = 51 x2 = 25.2 s2 = 6.5n1 – 1 = 57 - 1 = 56 n2 – 1 = 51 - 1 = 50 df = minimum of 56 and 50 = 50For a 90% CI with df = 50, t* = 1.676Example 103Summary statistics:Before: n1 = 57 x1 = 23.1 s1 = 6.3 After: n2 = 51 x2 = 25.2 s2 = 6.5So t* = 1.676 (x1 - x2) + t* s12 + s22 = (23.1 – 25.2) + 1.676 6.32 + 6.52 n1 n2 57 51= -2.1 + 1.676 (1.235) = -2.1 + 2.07= (-4.17, -0.03)margin of errorExample 103Interpretation: We have 90% confidence that the difference in the mean miles per gallon in vehicles produced before the new standards went into place and in vehicles produced after the new standards went into place is between -4.17 mpg and -0.03 mpg.Tests of Significance for m1 - m2We hypothesize that the difference between the population means equals some specified value m0 and
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