AP Statistics Ch 3 Examining Relationships
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1 Introducton To tud relatonhp between varable, we mut meaure the varable on the ame group of ndvdual. If we thnk a varable ma eplan or even caue change n another varable, then the eplanator varable and the repone varable. Eample Breat Cancer o The mot common treatment for breat cancer wa once removal of the breat. It now uual to remove onl the tumor and nearb lmph node, followed b radaton. The change n polc wa due to a large medcal eperment that compared the two treatment. Some breat cancer patent, choen at random, were gven each treatment. The patent were cloel followed to ee how long the lved followng urger. o What are the eplanator and repone varable? Are the categorcal or quanttatve? Ch 3.1 Scatterplot A catterplot how the relatonhp between two quanttatve varable meaured on the ame ndvdual. In eamnng a catterplot, look for an overall pattern howng the form, drecton, and trength of the relatonhp, and then for outler. Poble form of a catterplot nclude lnear, curved, or clutered. The drecton of a catterplot can be potvel aocated when hgh value of each varable tend to occur together, or negatvel aocated when hgh value of one varable tend to occur wth low value of the other. The trength of a catterplot determned b how cloel the data pont follow the form. page 1
2 Eample ACT Score at CMA-B o The followng a catterplot of Math veru Englh ACT core of the 2007 graduatng cla of Chcago Mltar Academ-Bronzevlle. o I there a clear eplanator varable and repone varable? o Dcu the form, drecton, and trength of the catterplot. Can an Englh core be reaonabl predcted from a Math core? page 2
3 Ch 3.2 Correlaton The correlaton r meaure the drecton and trength of the lnear relatonhp between two quanttatve varable. The correlaton r between and r = 1 n n 1 =1 Correlaton r an average of the product of the tandardzed varable and. Correlaton make no dtncton between eplanator and repone varable. Correlaton requre that both varable be quanttatve. Becaue r ue the tandardzed value of the obervaton, r doe not change when we change the unt of meaurement of and. Potve r ndcate potve aocaton between varable, and negatve r ndcate negatve aocaton. The correlaton r alwa a number between -1 and 1. Value of r near 0 ndcate a ver weak relatonhp. Correlaton onl meaure the trength of a lnear relatonhp between two value. Correlaton not retant; r trongl affected b a few outler. page 3
4 Eample Heght v Weght o Ue the formula to calculate the correlaton between heght and weght of tudent n the AP Stattc cla. = = = = Heght (n) Weght (lb) Sum = = = n n r o Interpret r n the contet of the problem. page 4
5 Ch 3.3 Leat-Square Regreon The Leat-Square Regreon Lne A regreon lne a traght lne that decrbe how a repone varable change a an eplanator varable change. The leat-quare regreon lne of on the lne that make the um of the quare of the vertcal dtance of the data pont from the lne a mall a poble. Equaton: wth lope b = r ˆ = a + b and ntercept a = b. The equaton for the lope a that a change n one tandard devaton n correpond to a change of r tandard devaton n. Ever leat-quare regreon lne (LSRL) pae through the pont (, ). The lope the rate of change, the amount of change n ˆ when ncreae b 1. The ntercept of the regreon lne the value of ˆ when = 0. The ntercept tattcall meanngful onl when can actuall take on value cloe to zero. Eample IQ and School GPA o The fgure plot GPA agant IQ tet core for 78 eventh-grade tudent. page 1
6 o Decrbe the form, drecton, and trength of the relatonhp between IQ core and GPA. o Calculaton how that the mean and tandard devaton of the IQ core are = and = For the GPA, = and = The correlaton between IQ and GPA r = o Fnd the equaton of the leat-quare lne for predctng GPA from IQ. o Ue the equaton ou found to predct the GPA of a tudent wth a 70 IQ and a tudent wth a 140 IQ. Plot the lne on the catterplot b drawng a lne connectng thee two pont. o Interpret the lope and ntercept of the lne n the contet of th problem. page 2
7 Eample A Growng Chld o Sarah parent are concerned that he eem hort for her age. Ther doctor ha the followng record of Sarah heght: Age (month) Heght (cm) o Scatterplot o Ung our calculator, fnd the equaton of the LSRL of heght on age. o Predct Sarah heght at 40 month and at 60 month. Ue our reult to draw the regreon lne on our catterplot. o What Sarah rate of growth, n centmeter per month? Normall growng grl gan about 6 cm n heght between age 4 (48 month) and 5 (60 month). What rate of growth th n centmeter per month? I Sarah growng more lowl than normal? page 3
8 The Role of r 2 n Regreon The coeffcent of determnaton r 2 the fracton of the varaton n the value of that eplaned b the leat-quare regreon of on. Eample A Growng Chld (contnued) o Fnd the coeffcent of correlaton and the coeffcent of determnaton. Interpret the coeffcent of determnaton n the contet of the problem. Eample Heght v Weght (contnued) o What wa the correlaton coeffcent that ou calculated? Now calculate the coeffcent of determnaton and nterpret t n the contet of the problem. Redual A redual the dfference between an oberved value of the repone varable and the value predcted b the regreon lne. redual = oberved predcted redual = ˆ Becaue the redual how how far the data fall from the regreon lne, eamnng the redual help ae how well the lne decrbe the data. The mean of the leat-quare redual alwa zero. A redual plot a catterplot of the regreon redual agant the eplanator varable. If the regreon lne capture the overall relatonhp between and, the redual hould have no tematc pattern. page 4
9 A curved pattern how that the relatonhp not lnear. Increang pread about the lne a ncreae ndcate that predcton of wll be le accurate for larger. Indvdual pont wth large redual are outler n the vertcal () drecton becaue the le far from the lne that decrbe the overall pattern. Thee are called regreon outler. An obervaton nfluental for a tattcal calculaton f removng t would markedl change the reult of the calculaton. Pont that are outler n the drecton of a catterplot are often nfluental for the leat-quare regreon lne. page 5
10 Eample Drvng Speed and Fuel Conumpton o How doe fuel conumpton of a car change a t peed ncreae? Here are the data for a Brth Ford Ecort. Speed () (km/hr) L1 Fuel Ued () (lter/100 km) L2 Predcted Fuel Ued L3 Redual L4 = L2 L o Ue the calculator to fnd the LSRL equaton. Then ue the equaton to fll n the predcted column and then fll n the redual column. Confrm that the redual column um to zero. o Make a plot of the redual agant the value of. I a traght lne an approprate model for thee data? page 6
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.3 Leat-Square regreon Eample.10 How do chldren grow? The pattern of growth vare from chld to chld, o we can bet undertandng the general pattern b followng the average heght of a number of chldren. Here
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