Stat 401 B – Lecture 411Population – all items of interest.Example: All vehicles madeIn 2004.Parameter – numerical summary of the entire population.Example: population meanfuel economy (MPG).Sample –afew items from the population.Example: 36vehicles.Statistic – numericalsummary of the sample.Example: sample meanfuel economy (MPG).2One-sample modelεμ+=Yμ•Y represents a value of the variable of interest• represents the population mean• represents the random error associated with an observationε3ConditionsThe random error term, , isIndependentIdentically distributedNormally distributed with standard deviation, εσStat 401 B – Lecture 424ErrorsModelErrorμε−= Yεμ+=Y5ResidualsEstimate of error(Observation – Fit)ResidualYY −=εˆ6ResidualsExamine the residuals to see if the conditions for statistical inference are met.Stat 401 B – Lecture 437Checking ConditionsIndependence.Hard to check this but the fact that we obtained the data through random sampling assures us that the statistical methods should work.8Checking ConditionsIdentically distributed.Check using an outlier box plot. Unusual points may come from a different distributionCheck using a histogram. Bi-modal shape could indicate two different distributions.9Checking ConditionsNormally distributed.Check with a histogram. Symmetric and mounded in the middle.Check with a normal quantile plot. Points falling close to a diagonal line.Stat 401 B – Lecture 4410.01.05.10.25.50.75.90.95.99-3-2-10123Normal Quantile Plot246810Count-7.5 -5 -2.5 0 2.5 5 7.5ResidualDistributions11MPG ResidualsHistogram is symmetric and mounded in the middle.Box plot is symmetric with no outliers.Normal quantile plot has points following the diagonal line.12MPG ResidualsThe conditions for statistical inference appear to be satisfied.Stat 401 B – Lecture 4513Two Independent SamplesQuestionIn 2000, did men and women differ in terms of their body mass index?141. Female 2. MalePopulationsSamplesrandom selectionrandom selectionInference15Two-sample modelεμ+=iYiμ•Y represents a value of the variable of interest• represents the ithpopulation mean• represents the random error associated with an observationεStat 401 B – Lecture 4616ConditionsThe random error term, , isIndependentIdentically distributedNormally distributed with standard deviation, εσ17Testing HypothesesQuestionIn 2000, did men and women differ in terms of their body mass index, on average?18Step 1 - Hypotheses0or :0or :212121210≠−≠=−=μμμμμμμμAHHStat 401 B – Lecture 4719Step 2 – Test Statistic()()684.0 value-P408.0509.1616.0501501544.7868.26484.27112121===+−=+−=tnnsYYtp20Step 3 – DecisionFail to reject the null hypothesis because the P-value is larger than 0.05.21Step 4 – ConclusionOn average, men and women in 2000 could have had the same BMI.The difference between males’ and females’ average BMI’s is not statistically
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