Gotta Keep It Correlati Correlatio.2 Learig Goals I this lesso, ou will: Determie the correlatio coefficiet usig a formula. Iterpret the correlatio coefficiet for a set of data. ew Stud Liks Dark Chocolate to Heart Health. Video Games Show to NBoost I.Q. College Graduates Live Loger, New Stud Fids. You have probabl see or heard headlies similar to these i magazies, o TV, ad olie. Each oe of these headlies is the result of a correlatioal stud. I a correlatioal stud, researchers compare two variables to see how the are associated. The do this through the use of surves or eve b researchig documets such as medical records. 2012 Caregie Learig What methods do ou thik researchers could have used to produce the results metioed i the headlies above? 533
Problem 1 Associate, Formulate, Correlate! Recall that data comparig two variables ca show a positive associatio, a egative associatio, or o associatio. 1. Describe the tpe of associatio betwee the idepedet ad depedet variables show o each scatterplot. The, draw a lie of best fit for each, if possible. a. Miles per Gallo Weight of Vehicle b. c. Height Grades o Algebra Test IQ Score Time Spet Studig 2012 Caregie Learig 534 Chapter Correlatio ad Residuals
A measure of how well a liear regressio lie fits a set of data is called correlatio. The correlatio coefficiet is a value betwee 21 ad 1 which idicates how close the data are to formig a straight lie. The closer the correlatio coefficiet is to 1 or 21, the stroger the liear relatioship is betwee the two variables. The variable r is used to represet the correlatio coefficiet. I remember that the correlatio coefficiet either falls betwee 1 ad 0 if the data show a egative associatio, or betwee 0 ad 1 if the data show a positive associatio. 2. Determie whether the poits i each scatter plot have a positive correlatio, a egative correlatio, or o correlatio. Four possible r-values are give. Circle the r-value ou thik is most appropriate. Eplai our reasoig for each. a. 8 7 r 5 0. r 5 20. r 5 0.0 r 5 20.0 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 The closer the r-value gets to 0, the less of a liear relatioship there is i the data! 2012 Caregie Learig b. 8 7 6 5 4 3 2 r 5 0.7 r 5 20.7 r 5 0.07 r 5 20.07 1 0 1 2 3 4 5 6 7 8.2 Correlatio 535
c. 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 r 5 1 r 5 0.5 r 5 0.01 You ca calculate the correlatio coefficiet of a data set usig this formula: ( i 2 )( i ) r 5 ( i 2 ) 2 ( i ) 2 i51 i51 Most of the pieces of this formula look familiar. I thik we used them i the formula for stadard deviatio! Let s determie the correlatio coefficiet of this data set usig the formula. (23, 23), (1, 2) ad (3, 4) Look at the umerator of the formula first. ( i 2 )( i 2 ) Determie the mea of the -values ad the mea of the -values. 5 1 3 5 1 Keep i mid that the otatio just tells ou that ou will be determiig the sum of all the data values. 2012 Caregie Learig 536 Chapter Correlatio ad Residuals
Notice these differeces are used throughout the formula. Determie the differece betwee each data value ad the mea for both the -coordiates ad the -coordiates. Determie the product of the differeces i each pair. The, determie the sum of those products. This is our umerator. ( i 2 ) ( 23 2 1 3 ) ( 1 2 1 3 ) 5 2 3 ( 3 2 1 3 ) 5 8 3 10 5 2 3 ( i 2 )( i ) ( i 2 ) (23 2 1) 5 24 (2 2 1) 5 1 (4 2 1) 5 3 ( 2 10 3? 24 )5 40 3 ( 2 3? 1 ) 5 2 3 40 3 1 2 1 8 5 22 3 ( 8 3? 3 ) 5 8 Now let s aalze the deomiator of the formula. ( i 2 ) 2 ( i ) 2 Determie the sum of the squares of the differeces betwee each value ad its mea. ( i 2 ) 2 ( 2 10 3 ) 2 5 100 ( 2 3 ) 2 5 4 ( 8 3 ) 2 5 64 5 56 3 ( i ) 2 (24) 2 5 16 (1) 2 5 1 5 26 (3) 2 5 2012 Caregie Learig Determie the square root of each sum. Determie the product of these two values. This is our deomiator. ( i 2 ) 2 ( i 2 ) 2 56 3 4.32 26 5.0 ( i 2 ) 2 ( i ) 2 (4.32)(5.0) 5 22.02768.2 Correlatio 537
3. Put the pieces together. Determie the correlatio coefficiet of the data set. 4. Iterpret the correlatio coefficiet of the data set. Problem 2 The Doctor Will See You Now The Ceter for Disease Cotrol collected data o the percet of childre, aged 12 to 1, that were cosidered obese betwee the ears 171 ad 2007. The data are give i the table. Year Percet of Obese Childre 171 6.4 What do ou otice as ou read through the data? 176 5.0 188 10.5 1 14.8 2001 16.7 2003 17.4 2005 17.8 2012 Caregie Learig 2007 18.1 538 Chapter Correlatio ad Residuals
1. Idetif the idepedet ad depedet quatities i this problem situatio. 2. Costruct a scatter plot of the data usig our graphig calculator. a. Sketch the scatter plot. Label the aes. b. Do ou thik a liear regressio equatio would best describe this situatio? Eplai our reasoig. 3. Use a graphig calculator to determie whether a lie of best fit is appropriate for these data. a. Determie ad iterpret the liear regressio equatio. Wait! There s a r ad a r 2 value o m calculator. Which oe do I use? 2012 Caregie Learig b. Determie the correlatio coefficiet. c. Would a lie of best fit be appropriate for this data set? Eplai our reasoig..2 Correlatio 53
4. The amout of atibiotic that remais i our bod over a period of time varies from oe drug to the et. The table give shows the amout of Atibiotic X that remais i our bod over a period of two das. Time (hours) 0 6 12 18 24 30 36 42 48 Amout of Atibiotic X i Bod (mg) 60 36 22 13 7.8 4.7 2.8 1.7 1 a. Determie ad iterpret a liear regressio equatio for this data set. b. Determie ad iterpret the correlatio coefficiet of this data set. c. Does it seem appropriate to use a lie of best fit? If o, eplai our reasoig. If es, determie ad iterpret the least squares regressio equatio. d. Sketch a scatter plot of the data. Amout of Atibiotic X i the Bod (mg) 0 80 70 60 50 40 30 20 10 0 5 10 15 20 25 30 35 40 45 Time (hours) e. Look at the graph of the data. Do ou still agree with our aswer to part (c)? Eplai our reasoig. 2012 Caregie Learig Be prepared to share our solutios ad methods. 540 Chapter Correlatio ad Residuals