Stat 401 B – Lecture 611Simple Linear RegressionQuestionIs annual carbon dioxide concentration related to annual global temperature?2Simple Linear RegressionResponse variable, Y.Annual global temperature (oC).Explanatory (predictor) variable, x.Annual atmospheric CO2concentration.3Regression modelεμ+=xyY|xy|μ•Y represents a value of the response variable. • represents the population mean response for a given value of the explanatory variable, x.• represents the random error εStat 401 B – Lecture 624Linear Modelεββεμ++=+= xYxy 10|0β1βThe Y-intercept parameter.The slope parameter.5ConditionsThe relationship is linear.The random error term, , isIndependentIdentically distributedNormally distributed with standard deviation, .εσ613.514.014.515.0Temp300 350 400CO2Stat 401 B – Lecture 637Describe the plot.Direction – positive/negative.Form – linear/non-linear.Strength.Unusual points?8Method of Least SquaresFind estimates of such that the sum of squared vertical deviations from the estimated straight line is the smallest possible.10 and ββ9Least Squares Estimates()()()xyxyxxyyxx101021ˆˆˆ ˆˆˆβββββ+=−=−−−=∑∑Stat 401 B – Lecture 641013.514.014.515.0Temp300 350 400CO2Linear FitBivariate Fit of Temp By CO211Linear FitPredicted Temp = 9.8815 + 0.012584*CO2xy10ˆˆˆββ+=12InterpretationEstimated Y-intercept.This does not have an interpretation within the context of the problem. Having no CO2in the atmosphere is not reasonable given the data.Stat 401 B – Lecture 6513InterpretationEstimated slope.For each additional 1 ppmvof CO2, the annual global temperature goes up 0.012584 oC, on average.1413.514.014.515.0Temp300 350 400CO2Linear FitBivariate Fit of Temp By CO215How Strong?The strength of a linear relationship can be measured by R2, the coefficient of determination.RSquare in JMP output.Stat 401 B – Lecture 6616How Strong?806.099450.080145.022===RSSSSRTotalModel17Interpretation80.6% of the variation in the global temperature can be explained by the linear relationship with carbon dioxide concentration.19.4% is unexplained.18InterpretationThere is a fairly strong positive linear relationship between carbon dioxide concentration and global temperature.Cause and effect?Stat 401 B – Lecture 6719Cause and Effect?There is a strong positive linear relationship between the number of 2ndgraders in communities and the number of crimes committed in those communities.20Connection to CorrelationIf you square the correlation coefficient, r, relating carbon dioxide to global temperature you get R2, the coefficient of determination.898.0806.02+=+=±= Rr21Connection to Correlation x values theofdeviation standard theis y values theofdeviation standard theis ˆ
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