Stat 401 B – Lecture 511Two Independent SamplesQuestionIn 2000, did men and women differ in terms of their body mass index?21. Female 2. MalePopulationsSamplesrandom selectionrandom selectionInference3Body Mass IndexFemales Males860.7484.2750111===sYn215.7868.2650222===sYn544.7=psStat 401 B – Lecture 52495% Confidence Interval()2 df with table- tfrom 1121*21*21−+=+±−nntnnstYYp595% Confidence Interval()()()()3.61 to38.2 995.2616.0 509.19845.1616.0 501501544.79845.1868.26484.2711 21*21−±±+±−+±−nnstYYp6InterpretationWe are 95% confident that the difference in population mean BMI for women compared to men is between –2.38 and 3.61.Women could have a mean BMI as much as 2.38 lower than men or as much as 3.61 higher than men.Stat 401 B – Lecture 537Difference?Because zero is in the confidence interval, there could be no difference in population mean BMI for women compared to men.This agrees with the test of hypothesis.8Two-sample modelεμ+=iYiμ•Y represents a value of the variable of interest• represents the ithpopulation mean• represents the random error associated with an observationε9ConditionsThe random error term, , isIndependentIdentically distributedNormally distributed with standard deviation, εσStat 401 B – Lecture 5410ResidualsEstimate of error(Observation – Fit)ResidualiYY −=εˆ11Checking ConditionsIndependence.Hard to check this but the fact that we obtained the data through separate random samples of women and men assures us that the statistical methods should work.12Checking 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.Stat 401 B – Lecture 5513Checking 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.14.01.05.10.25.50.75.90.95.99-3-2-10123Normal Quantile Plot51015202530Count-20 -15 -10 -5 0 5 10 15 20BMI centered by GenderDistributions15BMI Residuals Histogram is skewed left and mounded to the right of zero. Box plot is fairly symmetric with two potential outliers on the high side. Normal quantile plot has points following the diagonal line for the first part but then wiggles around for larger values.Stat 401 B – Lecture 5616BMI ResidualsThe conditions for statistical inference may not be met for these data.17ConsequencesThe P-value for the test may not be correct.Even so, there is not much of a difference between women and men, and I would not change my conclusion from the test of hypthesis.18ConsequencesThe stated confidence level may not give the true coverage rate.I would still use the confidence interval but recognize that the true coverage rate is probably less than
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