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# 370 Note for STAT 501 at PSU

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Date Created: 02/06/15

Introductory Examples Statistics 501 Dr Heckard Example 1 Teen birth rate and poverty level The data are for the 50 states and the District of Columbia in the United States the variables are y year 2002 birth rate per 1000 females 15 to 17 years old andx poverty rate the year 2000 percent of the state s population living in households with incomes below the federally de ned poverty level Data source Mind On Statistics 3ml ed Utts and Heckard A plot of the data birth rate on the vertical shows a generally linear relationship on average with a positive slope As the poverty level increases the birth rate for 15 to 17 yearold females tends to increase as well Scatterplot of Brth15t017 vs PovPct 50 o 40 I Q a o a I I O a 30 I n o g Q O t O o o a 20 39 39 o s 39 o 39 o 0 O t o I o 3 on 10 o 39 o 5 10 15 20 25 Rde The following plot created in Minitab using StatgtRegressiongtFitted Line Plot shows a regression line superimposed on the data The equation is given near the top of the plot Minitab should have written that the equation is for the average birth rate or predicted birth rate would be okay too because a regression equation describes the average value of y as a function of one or more Xvariables In statistical notation the equation could be written y 4267 1373X Fitted Line Plot Brth15t017 4267 1373 PovPct 50 S 5 55057 Rqu 53 3 O R Sqad 52 4 40 N H a 30 H t E 20 10 5 10 15 20 25 o The interpretation of the slope value 1373 is that the 15 to 17 yearold birth rate increases 1373 units on average for each one unit one percent increase in the poverty rate 0 The interpretation of the intercept value 4267 is that if there were states with poverty rate 0 the predicted average15 to 17 yearold birth rate for would be 4267 for those states In the graph with a regression line present we also see the information that s 555057 and R2 533 o The value of s tells us roughly the average difference between the yvalues of individual observations and predictions of y based on the regression line 0 The value of R2 can be interpreted to mean that poverty rates explain 533 of the observed variation in the 15 to 17 year old birth rates of the states The R2 adj value 524 is an adjustment to R2 based on the number of xvariables in the model only one here and the sample size With only one xvariable the adjusted R2 is not important Example 2 Lung function in 6 to 10 year old children The data are from ri 345 children between 6 and 10 years old The variables are y forced exhalation volume FEV a measure of how much air somebody can forcibly exhale from their lungs and x age in years Data source The data here are a part of dataset given in Kahn Michael 2005 An Exhalent Problem for Teaching Statistics T he Journal of Statistical Education 132 and also given at httpwwwsciusqeduaustaffdunnDatasets Following is a plot of the data with a straightline regression line superimposed The plot was done in Minitab and as in Example 1 we point out that the word average should come before the yvariable name 0 The estimated regression equation is that average FEV 00498 02621 Age For instance for a 8 year olds we can use the equation to estimate that the average FEV 00498 026218 21466 0 The interpretation of the slope is that average FEV increases 02621 for each oneyear increase in age in the observed age range Fitted Line Plot FEV 00498 02621 Age 6 7 8 9 10 Age An interesting and possibly important feature of these data is that the variance of individual yvalues from the regression line increases as age increases This feature of data is called non constant variance For example the FEV values of 10year olds are more variable than FEV value of 6year olds This is seen by looking at the vertical ranges of the data in the plot

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