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VCU STAT 210 - Lecture38(2) (1)

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Slide 1Practice ProblemsAdditional Reading and ExamplesTest 7Slide 5Statistical InferenceStatistical InferenceMotivating ExampleMotivating ExamplePoint EstimateSampling DistributionSampling DistributionSampling DistributionFinding ProbabilitiesExample 99Example 99Example 99Example 99Example 99Confidence Interval for m1 - m2Confidence Interval for m1 - m2Confidence Interval for m1 - m2Confidence Interval for m1 - m2Confidence Interval for m1 - m2TI-83/84 CalculatorExample 100Example 100Example 100Example 100Example 100Example 100Example 100Example 100Example 100Example 100Example 100Example 100Example 101Example 101Example 101Example 101Example 101Example 101Confidence Interval for m1 - m2Slide 45Confidence Interval for m1 - m2Confidence Interval for m1 - m2TI-83/84 CalculatorExample 102Example 102Example 102Example 102Example 102Example 102Example 102Example 102Example 102Example 102Example 102Example 103Example 103Example 103Example 103Example 103Example 103Example 103Example 103Example 103Example 103Motivating ExampleMotivating ExampleMotivating ExampleMotivating Example SolutionSee you tomorrow!STAT 210Lecture 38Confidence Intervals for m1 – m2November 30, 2016Practice ProblemsPages 286 through 291Relevant problems: X.4 (a) and (c), X.5, X.6, X.8 and X.10 (a)Recommended problems: X.5 and X.6Additional 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.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.Motivating ExampleThe Main Event of the 2016 World Series of Poker concluded on October 30 through November 1 (shown live on ESPN). 6,737 players paid $10,000 each to enter the event, and the winner, Qui Nguyen, received $8,005,310.Suppose we want to estimate the difference in the mean number of hands played by all professional players and the mean number of hands played by all amateur players.What are the populations of interest?The populations include all professional players in the 2016 World Series of Poker Main Event and all amateur players in the 2016 World Series of Poker Main Event.Motivating ExampleThe Main Event of the 2016 World Series of Poker concluded on October 30 through November 1 (shown live on ESPN). 6,737 players paid $10,000 each to enter the event, and the winner, Qui Nguyen, received $8,005,310.Suppose we want to estimate the difference in the mean number of hands played by all professional players and the mean number of hands played by all amateur players.What is the parameter of interest?The parameter of interest is m1 – m2 = the difference in the mean number of hands played by all professional players and the mean number of hands played by all amateur players.Point EstimateWe take a simple random sample of n1 subjects from the firstpopulation and an independent simple random sample of n2 subjects from the second population.A point estimate of m1 - m2 is the difference in the samplemeans: X1 - X2.Sampling DistributionAssumptions:(1) 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 appliesSampling DistributionCenter: the mean of the sampling distribution is mX1-X2 = m1 – m2Spread: the standard deviation of the sampling distribution is sX1-X2 = s12 + s22 n1 n2Shape: the shape of the sampling distribution will be normalSampling DistributionIf the assumptions are satisfied then the sampling distribution for the difference in the sample means, X1 - X2, isX1 - X2 ~ N m1 - m2 , s12 + s22 n1 n2Note the + signFinding ProbabilitiesThe sampling distribution and the Z-score transformation can be used to find probabilities for X1 – X2.Z = (X1 - X2) - (m1 - m2) s12 + s22 n1 n2Example 99From example 97, X1 – X2 ~ N(3, 1.226)Find P(X1 – X2 < 0)Example 99From example 97, X1 – X2 ~ N(3, 1.226)Find P(X1 – X2 < 0) = P ( Z < 0 – 3 ) 1.226Example 99From example 97, X1 – X2 ~ N(3, 1.226)Find P(X1 – X2 < 0) = P ( Z < 0 – 3 ) 1.226 = P(Z < -2.45)Example 99From example 97, X1 – X2 ~ N(3, 1.226)Find P(X1 – X2 < 0) = P ( Z < 0 – 3 ) 1.226 = P(Z < -2.45) = .0071Example 99From example 97, X1 – X2 ~ N(3, 1.226)Find P(X1 – X2 < 0) = P ( Z < 0 – 3 ) 1.226 = P(Z < -2.45) = .0071Calculator: normalcdf(-1E99,0,3,1.226)Confidence Interval for m1 - m2Since the populations means are unknown they must be estimated, and in this section we discuss confidence interval estimates of the difference between the population means: m1 - m2.We will first determine confidence intervals when the population standard deviations s1 and s2 are known, and then when the population standard deviations are unknown.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 [Clicker] to and from the point estimate to create the confidence interval.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:


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VCU STAT 210 - Lecture38(2) (1)

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