AP Statistics Notes Unit Three: Exploring Relationships Between Variables

Transcription

1 AP Statistics Ntes Unit Three: Explring Relatinships Between Variables Syllabus Objectives: 1.12 The student will analyze patterns in scatterplts The student will assess the linearity f bivariate data. Scatterplts This is the mst effective way t display the relatinship between tw QUANTITATIVE variables. It shws the RELATIONSHIP between tw quantitative variables (bivariate data) measured n the same individuals. Determine the explanatry and respnse variables. A respnse variable measures an utcme f a study. The respnse variable is als called the dependent variable and is graphed n the vertical (y) axis. An explanatry variable attempts t explain the bserved utcmes. The explanatry variable is called the independent variable and is graphed n the hrizntal (x) axis. If there is n explanatry-respnse distinctin between the variables, either variable can g n either axis. Plt the individual rder pairs. Interpret the scatterplt. Ex: A sample f ne-way Greyhund bus fares frm Rchester NY t cities less than 750 miles was taken frm the Greyhund website. The fllwing table gives the destinatin city, the distance and the ne-way fare. Distance is the explanatry variable and Fare is the respnse variable. Fare shuld depend n distance. Destinatin City Distance Standard One-Way Fare Albany, NY Baltimre, MD Buffal, NY Chicag, IL Cleveland, OH Mntreal, QU New Yrk City, NY Ottawa, ON Philadelphia, PA Ptsdam, NY Syracuse, NY Trnt, ON Washingtn, DC ***Cmments: The axes need nt intersect at (0,0). Fr each f the axes, the scale shuld be chsen s that the minimum and maximum values n the scare are cnvenient and the values t be pltted are between the tw values. Fr this example, the x axis (distance) runs frm 50 t 650 miles where the data pints are between 69 and 607. The y axis (fare) runs frm $10 t $100 where the data pints are between $17 and $96. Each axis shuld als be apprpriate labeled. 1

2 Interpreting Scatterplts Lk fr the verall pattern and fr any striking deviatins frm that pattern. An imprtant kind f deviatin is an utlier, an individual rdered pair that falls utside the verall pattern n the relatinship. Describe the verall pattern by the frm, directin and strength f the relatinship. Frm is it linear r curved? Directin des the verall pattern mves frm upper left t lwer right, r vice versa? Tw variables are psitively assciated when as ne variable increases, the ther increases as well. The pattern mves frm lwer left t upper right. Tw variables are negatively assciated if ne variable increases and the ther decreases. The pattern mves frm upper left t lwer right. Psitive Assciatin Negative Assciatin N Assciatin Strength Hw clsely d the pints fllw a clear frm? Linear strength is nrmally described as weak, mderate r strng. The mre linear, the strnger it is. Categrical variables can be added t scatterplts. Use different clrs r symbls t plt pints when adding a categrical variable (t see a pattern in males r females separately plt males as a dt and females as a square.) Ex: D heavier peple burn mre energy? Metablic rate, the rat at which the bdy cnsumes energy, is imprtant in studies f weight gain, dieting and exercise. Data n the lean bdy mass and resting metablic rate fr 12 wmen and 7 men wh were subjects in a study f dieting were input in the TI-84 graphing calculatr. The lean bdy mass fr males was placed in L 1, the male metablic rate was placed in L 2. The mass and metablic rate fr females were placed in L3 and L 4. Press and turn tw f the plts n. The first plt will graph the male data as squares and the secnd plt will graph the female data as plus signs. Hit Zm 9 r set the windw t display the scatterplt. R a t e (cal) Bdy Mass (kg) 2

3 Syllabus Objective: 1.14 The student will calculate the cefficients f crrelatin and determinatin. Measuring Linear Assciatin: Crrelatin Crrelatin measures the strength and directin f the linear assciatin between tw quantitative variables. The variable that represents crrelatin is r. r measures hw tightly the pints n a scatterplt cluster abut a straight line. r is called the crrelatin cefficient. Frmula: zz x y x r = = n 1 n 1 ( x x)( y y) s s **Ntice that the frmula begins by standardizing the bservatins. r is the average f the prducts f the tw standardized frmulas. Therefre, r des nt have any units. y Ex: x = s = x y= s = y x y x-x s x y-y x-x y-y s y sx sy r = =

4 Finding the crrelatin cefficient n the TI-84 graphing calculatr: Enter the explanatry variable in L1 and enter the respnse variable in L. 2 Make sure yur Diagnstics are n. This is NOT the calculatr s default. Press and scrll dwn until yu find DiagnsticOn. Hit enter twice. Nw the Diagnstics are n. G t the Stat menu. Entry 1 and Entry 8 will perfrm linear regressin. We will be using 8. Put in the tw lists and hit enter. r will be displayed n the next screen. Nte that r = , the same value fund n the previus page. Prperties f r: The value f r des nt depend n which f the tw variables is labeled x (the explanatry variable). The value f r des nt depend n the unit f measurement fr each value. The value f r is between -1 and +1 The crrelatin cefficient is nly when all the pints lie n a dwnward-slping line (perfect negative relatinship) nly when all the pints lie n an upward-slping line (perfect psitive relatinship). The value f r is a measure f the extent t which x and y are linearly related. Values f r near 0 indicate a very weak linear relatinship. The strength f the relatinship increases as r mves away frm 0 and tward -1 and +1. Crrelatin measures the strength f nly a LINEAR relatinship between the tw variables. Crrelatin is nt resistant it is greatly affected by utliers. Rugh guidelines value f r between 0.5 and + 0.5are cnsidered weak, mderate values are frm 0.5 t 0.8 (psitive and negative) and values f r 0.8 and abve are cnsidered strng. 4

5 Remember even if r = 0, that des nt mean there is n relatinship between the variables, it just means that there is n LINEAR relatinship. There culd be anther relatinship relating the variables, like an expnential r parablic relatinship. Ex: Fr ur value f r n the last example, , we wuld say that there is a strng, psitive linear assciatin between the tw variables. Nte that in ur sentence, we describe the strength, frm and directin!! Ex: Hurricanes develp lw pressure at their centers. This pulls in mist air, pumps up their rtatin, and generates high winds. After lking at the data f the Maximum Wind Speed versus Central Pressure fr 163 hurricanes that have hit the U.S. since 1851, the crrelatin cefficient was fund t be r = This means that there is a strng, negative linear assciatin between the wind speeds f hurricanes and their central pressures. Mre Examples: 5

6 Syllabus Objectives: 1.15 The student will determine the equatin f the least squares line The student will make predictins using the least squares regressin line. The Least Squares Regressin Line We can mdel the relatinship with a line and give its equatin. The linear mdel is just an equatin f a straight line thrugh the data. A regressin line is a line that describes hw a respnse variable y changes as an explanatry variable x changes. Regressin Lines as Mathematical Mdels Frm: y = a + bx b is the slpe, the amunt by which y changes as x increases by ne unit. a is the y-intercept, the value f y when x = 0. y 1 y = y increases by b = 3 x increases by 1 5 a = x We can use a regressin line t predict the respnse y fr a specific value f the explanatry variable x. The estimate made frm the mdel is called the predicted value and is written as ŷ (read as y-hat). ŷ is the predictin f y resulting frm the substitutin f a particular value int the equatin. The difference between the bserved value and its assciated predicted value is called the residual. The residual value tells us hw far ff the mdel s predictin is at that pint. The residual is seen belw as the vertical distance between the tw values. residual = bserved value predicted value 6

7 The least-squares regressin line f y n x is the line that makes the sum f the squared residuals the smallest. The line f best fit takes all f the sum f the squared residuals and makes them as small as pssible. Here is a gemetric representatin f the squared (vertical distances) residuals. Equatin f the Least-Squares Regressin Line (LSRL): ŷ = a + bx sy The slpe frmula: b = r, where r is the crrelatin cefficient and sy and s x s x are the standard deviatin f the tw variables. The y-intercept frmula: a = y bx, where x and y are the tw sample means. This frmula wrks because the line always passes thrugh the pint ( x, y ). Facts abut the LSRL: The distinctin between explanatry and respnse variables is essential! If yu mix them up, yur LSRL will be wrng, but yur crrelatin cefficient will be unchanged. There is a clse cnnectin between crrelatin (r) and slpe (b). Ntice the frmula - they will always have the same sign (psitive r negative). The LSRL can be fund with the TI-84. Ntice n the example n page 4, Stat Calc 8 will prduce the LSRL. We can use the regressin line t predict the respnse y fr a specific value f the explanatry variable x. Extraplatin is the use f a regressin line fr predictin utside the range f values f the explanatry variable x used t btain the line. Such predictins are ften nt accurate. Ex: Cnsider the fllwing data frm the article, The Carbnatin f Cncrete Structures in the Trpical Envirnment f Singapre. The explanatry variable is carbnatin depth in cncrete (mm) and the respnse variable is the strength f the cncrete (Mpa) x y First, input the data int the calculatr and draw the scatterplt. 7

8 Strength (M pa) Depth (mm) Interpret: There is a strng, negative linear relatinship between depth f crrsin and cncrete strength. As the depth increases, the strength decreases at a cnstant rate. Next, find the equatin f the LSRL that mdels the relatinship between crrsin and strength. yˆ = x, where x is the depth and ŷ is the predicted strength. r = 0.968, telling us that there is a strng, negative linear relatinship between depth f crrsin and strength f cncrete. What des the slpe tell us? Fr every increase f 1 mm in depth f crrsin, we predict n average, a Mpa decrease in strength f the cncrete. Here the LSRL is drawn n the scatterplt. Use the LSRL t find the predicted strength f cncrete with a crrsin depth f 25 mm. yˆ = (25) = yˆ = Mpa What is the predicted strength f cncrete with a crrsin depth f 40 mm? yˆ = (40) = yˆ = Mpa Hw des this predictin cmpare with the bserved strength at a crrsin depth f 40 mm? The bserved strength was 12.4 The predictin did nt match the bservatin. That is, there was an errr r residual between ur predictin and actual bservatin. Our mdel predicted t high the actual pint is belw the LSRL. residual = bserved predicted residual = =

9 Cefficient f Determinatin (r 2 ) It is the prprtin f variatin in y that can be explained by the least-squares regressin f y n x. It is the crrelatin cefficient squared. 2 SST SSE 2 2 Frmula: r = where SST = ( y y) and SSE = ( y yˆ ) SST SST (SST in cmputer utput) is the ttal sum f squares and SSE (SSResid in utput) is the residual sum f squares. These can be fund n cmputer utputs and yu are nt required t find them. The clser the r 2 is t 1, the better the fit. If r 2 = 1, then 100% f the variatin in y can be explained by the linear regressin between the tw variables. That means there wuld be a perfect fit, with n scatter arund the line. All f the variance is accunted fr by the mdel and nne is left in the residuals at all. Of curse, this is t gd t be true. The cefficient determinatin will range frm 0 t 1 and it measures the success f the regressin mdel. We can find the cefficient f determinatin by squaring r r it can be fund n cmputer printuts and yur TI-84. Ex: Frm ur previus example, ntice the calculatr screen shws: 2 r = This states that 93.75% f the variability in predicted strength can be explained by the LSRL n depth. Linear Regressin Example: Animal-waste lagns and spray fields near aquatic envirnments may significantly degrade water quality and endanger health. The Natinal Atmspheric Depsitin Prgram has mnitred the atmspheric ammnia at swine farms since The data n the swine ppulatin size (in thusands) and atmspheric ammnia (in parts per millin) fr ne decade are given belw. Year Swine Ppulatin Atmspheric Ammnia (a) Cnstruct a scatterplt fr these data. Cllectin Scatter Plt Sw ine_pp_size (b) The value fr the crrelatin cefficient fr these data is Interpret this value. There is a strng, psitive, linear relatinship between swine ppulatin size and atmspheric ammnia. 9

10 (c) Based n the scatterplt in part (a) and the value f the crrelatin cefficient in part (b), des it appear that the amunt f atmspheric ammnia is linearly related t the swine ppulatin size? Explain. Bth the value f the crrelatin cefficient and the pattern in the scatterplt indicate that there is a psitive linear relatinship between the size f the swine ppulatin and atmspheric ammnia. Cllectin Scatter Plt Sw ine_pp_size Atm_Ammnia = 0.133Sw ine_pp_size ; r 2 = 0.72 (d) What percent f the variability in atmspheric ammnia can be explained by swine ppulatin size? 2 r = Cmputer Printuts Minitab utput fr regressin. Often, the regressin equatin is given at the tp. The slpe b can als be fund under the Cef clumn acrss frm the explanatry variable (Distance). The y-intercept a is als under the Cef clumn acrss frm Cnstant. The cefficient f determinatin (r 2 ) is als given. a b Regressin Analysis: Standard Fare versus Distance The regressin equatin is Standard Fare = Distance Least squares regressin line Predictr Cef SE Cef T P Cnstant Distance S = R-Sq = 93.5% R-Sq(adj) = 92.9% Analysis f Variance r 2 Surce DF SS MS F P Regressin Residual Errr Ttal SSResid SST

11 Ex: A scatterplt shws that there is a strng linear relatinship between the average utside temperature (measured by heating degree-days) in a mnth and the average amunt f natural gas that a husehld uses per day during the mnth. As the average number f heating degree-days per day increases, the average amunt f gas cnsumed per day in hundreds f cubic feet als increases. The Minitab printut shws the fllwing statistics. The regressin equatin is: yˆ = x. The slpe in this example says, that n average, each additinal degree-day predicts cnsumptin f mre hundreds f cubic feet f natural gas per day. 2 r = Over 99% f the variatin in gas cnsumptin is accunted fr by the linear relatinship with degree-days. r = = This shws us that there is a strng, psitive linear relatinship between gas used and D-days. Using the equatin t predict gas cnsumptin at 20 degree-days, y ˆ = (0.1890)(20) = Fr 20 degree-days, gas cnsumptin wuld be abut hundreds f cubic feet. Linear Regressin Example: Cmmercial airlines need t knw the perating cst per hur f flight fr each plane in their fleet. In a study f the relatinship between perating cst per hur and number f passenger seats, investigatrs cmputed the regressin f perating cst per hur n the number f passenger seats. The 12 sample aircraft used in the study included planes with as few as 216 passenger seats and planes with as many as 410 passenger seats. Operating cst per hur ranged between $3,600 and $7,800. Sme cmputer utput frm a regressin analysis f these data is shwn belw. 11

12 (a) What is the equatin f the least squares regressin line that describes the relatinship between perating cst per hur and number f passenger seats in the plane? Define any variables used in this equatin. yˆ = x, where x = number f passenger seats and ŷ = the predicted perating cst per hur. (b) What is the value f the crrelatin cefficient fr perating cst per hur and number f passenger seats in the plane? Interpret this crrelatin. The value f the crrelatin cefficient is r = = r is psitive because the scatterplt shws a psitive assciatin and the slpe is psitive. There is a mderate, psitive linear relatinship between perating csts per hur and number f passenger seats. (c) Suppse that yu want t describe the relatinship between perating cst per hur and number f passenger seats in the plane fr planes nly in the range f 250 t 350 seats. Des this line shwn in the scatterplt still prvide the best descriptin f the relatinship fr data in this range? Why r why nt? N. The equatin f the least-squares regressin line is influenced by the three pints in the upper right-hand crner and the tw pints in the lwer left-hand crner f the scatterplt. The seven remaining pints (with number f seats in the 250 t 350 range) wuld have a negative crrelatin. Hence, the slpe f the recalculated leastsquares regressin line is negative. 12

13 Syllabus Objectives: 1.17 The student will analyze residual plts fr patterns The student will analyze the effect f utliers and influential pints n the least squares regressin line. A residual plt is a scatterplt f the regressin residuals against the explanatry variable, x. Residual plts help us t assess the fit f a regressin line. We are lking fr NO PATTERN r CURVATURE. Unifrm r randm scatter in the residual plt tells us that a linear mdel is apprpriate. If there is curvature, increasing r decreasing spread, r lts f pints with large residuals, this is an indicatr that the linear regressin is nt a gd fit fr the data. The residuals are fund fr each data pint and pltted n the vertical axis. The explanatry variable is pltted n the hrizntal axis. N need t d by hand. Each time Linear Regressin (Stat, Calc, 8) is perfrmed n the graphing calculatr, the list named Resid is created. This list cntains all f the residuals. Find this list under the named lists and plt the scatterplt. Ex: Returning t ur running example f crrsin depth and strength f cncrete. Find the named list Resid and draw the scatterplt. r e s i d u a l s depth(mm) There appears t be n pattern in the residual plt. The LSRL may be ur best predictin mdel. Examples f residual plts where a straight line may nt be the best mdel. The first shws curvature, the secnd shws a fanned pattern. Outliers and Influential Observatins An utlier is an bservatin that lies utside the verall pattern f the ther bservatins. Pints that are utliers in the y directin f a scatterplt have large regressin residuals, but ther utliers need nt have large residuals. An bservatin is influential fr a statistical calculatin if remving it wuld markedly change the result f the calculatin. Pints that are utliers in the x directin f a scatterplt are ften influential fr the LSRL. We say that a pint is influential if mitting it frm the analysis gives a very different mdel. 13

14 Ex: Des the age at which a child begins t talk predict later scre n a test f mental ability? A study f the develpment f yung children recrded the age in mnths at which each f 21 children spke their first wrd and their Gesell Adaptive scre. Nte: Child 18 and Child 19. We can see frm the residual plt that Child 19 is an utlier with a very large residual. Child 18 is an influential bservatin that des nt have a large residual. If Child 18 is remved frm the data, it greatly affects the LSRL. This figure shws tw least-squares regressin lines. The slid line is calculated frm all the data. The dashed line is calculated leaving ut Child 18. Child 18 is influential because it mves the regressin line quite a bit. Influential Observatin Example: A simple randm sample f 9 students was selected frm a large university. Each f these students reprted the number f hurs he r she had allcated t studying and the number f hurs allcated t wrk each week. A least squares regressin was perfrmed and part f the resulting cmputer utput is shwn belw. Predictr Cef StDev T P Cnstant Wrk S = R-Sq = 47.6% R-Sq (adj) = 40.1% The scatterplt t the right displays the data that were cllected frm the 9 students Wrk (a) After pint P, labeled n the previus graph, was remved frm the data, a secnd linear regressin was perfrmed and the cmputer utput is shwn belw. Study Predictr Cef StDev T P Cnstant Wrk S = R-Sq = 2.5% R-Sq (adj) = 0.0% P 14

15 Des pint P exercise a large influence n the regressin line? Explain. The pint P des have a large influence n the regressin line. When P is remved frm the data set, the slpe f the line changes frm t , the intercept changes frm t , and the value f R 2 drps frm 47.6% t 2.5%. Als, the shape is significantly different frm 0 when the pint P is included in the data set and is nt significantly different frm 0 when the pint P is excluded frm the data set. Ex: Lydia and Bb were searching the Internet t find infrmatin n air travel in the United States. They fund data n the number f cmmercial aircraft flying in the United States during the years The dates were recrded as years since Thus, the year 1990 was recrded as year 0. They fit a least-squares regressin line t the data. The graph f the residuals and part f the cmputer utput fr their regressin are given belw. (a) Is a line an apprpriate mdel t use fr these data? What infrmatin tells yu this? Yes. The residual plt shws n pattern, indicating a linear mdel is apprpriate. (b) What is the value f the slpe f the least squares regressin line? Interpret the slpe in the cntext f this situatin. Slpe = aircraft/year On average, the number f cmmercial aircraft flying in the U.S. increased by apprximately each year. Predictr Cef Stdev t-rati p Cnstant Years s = (c) What is the value f the intercept f the least squares regressin line? Interpret the intercept in the cntext f this situatin. Intercept = aircraft. Predicted number f cmmercial aircraft that were flying in 1990 (since x=0 crrespnds t year 1990) was (d) What is the predicted number f cmmercial aircraft flying in 1992? Fr 1992, x = 2, s predicted number f cmmercial aircraft flying is (2) = aircraft. (e) What is the actual number f cmmercial aircraft flying in 1992? Frm the residual plt, the residual fr 1992 is +40, 40 = actual predicted. actual = = aircraft. Since actual number flying must be an integer, actual must have been

16 Syllabus Objective: 1.19 The student will transfrm bivariate data t achieve linearity, including lgarithmic and pwer transfrmatins. Transfrming t achieve linearity Applying a functin such as a lgarithm r pwer t a quantitative variable is called transfrming r reexpressing the data. This helps us t straighten nnlinear patterns. Once the curved data is straightened, we can use the tls f linear regressin t summarize and analyze ur data. T decide which transfrmatin t use, the prcess is ften ne f trial and errr. There are tw majr types after the mdel is recgnized, the transfrmatin prcess becmes much simpler. Steps: Make a scatterplt f the data and als find the residual plt. If the scatterplt is curved r if the residual plt is curved, a linear mdel is nt apprpriate and the data must be transfrmed. Make ne f the transfrmatins described belw. After the data has been reexpressed, graph the scatterplt t check fr linearity and als check the residual plt fr randm scatter. If bth illustrate the new mdel is linear, run the linear regressin n the new data. When predicting, apply the inverse lg peratin t islate the variables. Expnential grwth In linear grwth, a fixed increment is added t the variable in each equal time perid. Expnential grwth r decay ccurs when a variable is multiplied by a fixed number in each equal time perid. T straighten an expnential mdel, find the lgarithm f the y-values. Ln r Lg may be applied. Ex. f Expnential grwth: A researcher bserves the grwth f a particular bacteria and recrds the fllwing results: Time (hr) Ppulatin (in thusands) Cllectin Scatter Plt Time Cllectin 2 Scatter Plt Time Time Ppulatin = 2.71Time - 3.4; r 2 = 0.88 The scatterplt des nt fllw a linear pattern. Althugh r 2 is high, the curvature in the scatterplt and the residual plt tell us that a linear mdel is nt apprpriate. The data appear t fllw an expnential mdel s a transfrmatin f y lg yis apprpriate. Time (hr) Ppulatin Lg Pp

17 After taking the lg f the y-values, nte that the scatterplt des fllw a linear pattern and there is n pattern apparent in the residual plt. The cmputer utput shws the leastsquares regressin line and a very high cefficient f determinatin. Nw the line can be used fr predictin. Suppse we want t predict the ppulatin at 15 hurs. lg yˆ = x lg yˆ = (15) = lg yˆ = = yˆ Cllectin Time lg_ppulatin = 0.111Time ; r 2 = 0.99 Our predictin fr the ppulatin f this bacteria at 15 hurs is thusands f bacteria, r apprximately 98,855 bacteria. Pwer mdel Examples f the pwer (square, cube) mdel weight, vlume, area T prduce a linear relatinship frm a pwer law mdel, apply the lgarithm transfrmatin t bth variables. Again, natural lgarithms (ln) r Base 10 lgs (lg) may be applied Scatter Plt Time Ex. f Pwer Mdel: Scientists lked at the heart weight (in grams) f 7 mammals and the length f the cavity f the left ventricle f their hearts (in centimeters) t discver if there was sme kind f relatinship. Mammal Muse Rat Rabbit Dg Sheep Ox Hrse Cavity Length (cm) Heart Wt. (gms) The scatterplt (abve left) shws curvature and the residual plt (abve right) als shws curvature. A linear mdel is clearly nt apprpriate. Because weight implies this might be a pwer mdel, apply the lgarithm t bth variables. The lg f cavity length was stred in L3 and the lg f heart weight was stred in L 4. A scatterplt f the new transfrmed variables shws a very linear pattern. lg (heart weight) lg(cavity length) 17

18 Residual plt The new residual plt shws randm scatter implying the pwer mdel is the apprpriate mdel. The least-squares regressin line fr the new mdel is lg yˆ = lg x. We can nw use the mdel t make predictins. T predict the heart weight fr a mammal with a cavity length f 10 centimeters: lg yˆ = lg(10) = (1) = lg yˆ = = yˆ Our predictin fr the heart weight f a mammal with a left ventricle cavity length f 10 cm is apprximately 1006 grams. Example: The Earth's Mn has many impact craters that were created when the inner slar system was subjected t heavy bmbardment f small celestial bdies. Scientists studied 11 impact craters n the Mn t determine whether there was any relatinship between the age f the craters (based n radiactive dating f lunar rcks) and the impact rate (as deduced frm the density f the craters). The data are displayed in the scatterplt. (a) Describe the nature f the relatinship between impact rate and age. There is a strng nnlinear relatinship between impact rate and age. Impact rate declines rapidly with age ver the age range frm 0.4 t abut 0.7 billin years, and then seems t level ut. Prir t fitting a linear regressin mdel, the researchers transfrmed bth impact rate and age by using lgarithms. The fllwing cmputer utput and residual plt were prduced. Regressin Equatin: ln(rate) = ln(age) Predictr Cef SE Cef T P Cnstant ln(age) S = R-Sq = 89.4% R-Sq (adj) = 88.2% 18

19 (b) Interpret the value f r % f the variability in ln(impact rate) can be explained by a linear r straight line, relatinship between ln(impact rate) and ln(age). (c) Cmment n the apprpriateness f this linear regressin fr mdeling the relatinship between the transfrmed variables. There is a nticeable curved pattern in the residual plt, which indicates that the linear mdel is nt the best chice fr describing the relatinship between ln(impact rate) and ln(age). 19