Review Topic 14: Relationships between two numerical variables
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1 Review Topi 14: Reltionships etween two numeril vriles Multiple hoie 1. Whih of the following stterplots est demonstrtes line of est fit? A B C D E 2. The regression line eqution for the following grph is losest to: A y= x B y = x C y = x D y= x E y= x 3. The type of orreltion shown in the grph for question 2 would est e desried s: A wek, positive orreltion B moderte, positive orreltion John Wiley & Sons Austrli, Ltd 1
2 C D E strong, positive orreltion no orreltion moderte, negtive orreltion 4. Wht type of orreltion does n r-vlue of 0.64 indite? A Strong, positive orreltion B Strong, negtive orreltion C Moderte, positive orreltion D Moderte, negtive orreltion E Wek, positive orreltion 5. A grdener trks orreltion oeffiient of 0.79 etween the growth rte of his trees nd the mount of fertiliser used. Wht n the grdener onlude from this result? A An inrese in tree growth inreses the use of fertiliser. B An inrese in the use of fertiliser inreses the helth of the tress. C The growth rte of the trees is influened y the mount of fertiliser used. D The growth rte of the trees influenes the qulity of the fertiliser used. E There is no orreltion etween the growth rte of the tress nd the mount of fertiliser used. 6. When y= x, the vlue of y when x = 2.5 is: A B C D E The grph for the regression line eqution y= 85-4x is most likely to e: A B C D E John Wiley & Sons Austrli, Ltd 2
3 8. A series of dt points reorded oeffiient of determintion vlue of Clulte the Person s produt-moment oeffiient. A 82% B 0.18 C 0.67 D 0.91 E 18% 9. For the following smple dt set, whih of the following is n exmple of interpolting dt? x y A Finding the vlue of x when y = 7 B Finding the vlue of y when x = 17 C Finding the vlue of x when y = 27 D Finding the vlue of y when x = 37 E Finding the vlue of x when y = For the dt set from question 9, the regression line eqution is: A y= 10 + x B y= x C y= x D y= x E y= x Short nswer 1. For eh of the following grphs, desrie the strength of orreltion etween the explntory nd response vriles. 2. For eh of the grphs in question 1, drw line of est fit where possile. John Wiley & Sons Austrli, Ltd 3
4 3. Identify the explntory nd response vrile for eh of the following senrios: In junior Siene lss, students plot the time tken to oil vrious quntities of wter. Extr uses re ordered to trnsport numer of students to the shool thletis rnivl. 4. Use the following dt to omplete this question. x y Plot the dt on stterplot. Comment on the diretion nd strength of the dt. Find Person s produt-moment orreltion oeffiient nd the oeffiient of determintion for the dt. d Use you nswer from prt to further disuss the reltionship etween the vriles. 5. Person s produt moment orreltion oeffiient for stterplot ws found to e Find the vlue of the oeffiient of determintion. Wht would these vlues indite to you out the strength of reltionship etween the two vriles? 6. During n interview investigting the link etween the sles of helthy snk foods (funtionl foods) nd the inresing onsumer demnd for these produts, n dvertising expert mde the following omment: There is orreltion ut it s not ustion our inresing need for helthy food nd our lziness hs resulted in mss innovtion of funtionl foods. Explin why he might hve stted there is no ustive link etween the sles of helthy foods nd lziness. Extended response 1. Dt on 15 people s shoe size nd the length of their hir ws olleted. d Shoe size Length of hir (m) Shoe size Length of hir (m) Shoe size Length of hir (m) Drw stterplot of this dt. Find the eqution of the line of est fit. Find Person s produt-moment orreltion oeffiient nd the oeffiient of determintion for the dt. Wht onlusions ould you drw from this dt? 2. An independent geny test-drove rndom smple of urrent model vehiles nd mesured their fuel tnk pity ginst the verge fuel onsumption. Along with the following stterplot, regression eqution of y= x ws estlished. Identify the response vrile in this sitution. Rewrite the eqution in terms of the explntory nd response vriles. It is often sid tht smller vehiles re more eonomil. Determine orret to 2 deiml ples the John Wiley & Sons Austrli, Ltd 4
5 d e f g fuel onsumption of vehile tht hd 40-litre fuel tnk. Is your nswer to prt n exmple of interpoltion or extrpoltion? Explin your response. Clulte, orret to the nerest whole numer, the tnk size of vehile tht hd fuel onsumption rte of 10.2 L per 100 km. Person s produt-moment orreltion oeffiient for this dt is How n you use this vlue to evlute the reliility of your dt? List t lest two other ftors tht ould influene the dt. 3. The weight of top rnd runners ws trked ginst the reommended retil prie, nd the results were reorded in the following stterplot. d Identify the explntory vrile for this sitution. How would you desrie the reltionship etween these two vriles? The oeffiient of determintion for this dt is r 2 = Wht onlusions n e estlished from this result? Identify two externl ftors tht ould explin the distriution of the dt points. 4. The Bureu of Meteorology reords dt suh s mximum tempertures nd solr exposure on dily nd monthly sis. The following dt tle, for the Botnil Grdens in Melourne, shows the monthly verge mount of solr energy tht fll on horizontl surfe nd the monthly verge mximum temperture. (Note: The dt vlues hve een rounded to the nerest whole numer.) Month Jn Fe Mr Apr My Jun Jul Aug Sep Ot Nov De Averge solr exposure (MJ) Averge mx dily temp. ( C) d e f g h Identify the explntory nd response vriles for this sitution. Using CAS, plot the dt on stterplot. Desrie the trend of the dt. Clulte Person s produt-moment oeffiient nd oeffiient of determintion for this dt. Wht do these vlues tell you out the reliility of the dt? Plot the regression line for this dt nd write the eqution in terms of the vriles. Using your eqution, lulte the mount of solr exposure for monthly mximum temperture of 37 C. Extrpolte the dt to find the verge mximum temperture expeted for month tht reorded n verge solr exposure of 3 MJ. Explin why prt g is n exmple of extrpoltion. John Wiley & Sons Austrli, Ltd 5
6 Review nswers Multiple hoie 1 B 2 A 3 B 4 C 5 C 6 B 7 C 8 D 9 B 10 D Short nswer 1 Moderte positive orreltion No orreltion Strong negtive orreltion 2 No line of est fit possile Explntory vrile = quntity of wter Response vrile = time Explntory vrile = numer of students Response vrile = numer of uses required 4 Strong negtive orreltion r = ; r 2 = d Person s produt-moment orreltion oeffiient nd the oeffiient of determintion onfirm strong reltionship etween the two vriles There is moderte negtive reltionship etween the vriles. The oeffiient of determintion suggests tht 57% of the vrition in the y-vrile is due to hnges in the x-vrile, nd 43% is due to other ftors. 6 Although there ppers to e link etween the lziness of people nd the inrese in sles of helthy foods, there re lso mny other possile ftors esides lziness. Bsed on this oservtion lone, the use of n inrese in sles of helthy foods nnot e onluded to e due to lziness. John Wiley & Sons Austrli, Ltd 6
7 Extended response 1 y= x r = ; r 2 = d There is wek link etween the two vriles. Therefore, no solid onlusions n e mde to suggest hnge in person s shoe size will ffet the length of their hir. 2 Averge fuel onsumption Averge fuel onsumption = fuel tnk pity 5.17 d Extrpoltion, s the x-vlue is outside of the originl dt rnge. e 85 L f This vlue indites moderte reltionship etween the vriles. Therefore, the dt n e used, ut other ftors should lso e onsidered. g Vrious nswers re possile, e.g. the mnner in whih person drives the vehile, wether onditions nd rod onditions. 3 Weight (grms) No orreltion This supports the view tht there is no orreltion etween the vriles. Bsed on this vlue, no onlusions n e mde from the dt. d Vrious nswers re possile, e.g. populrity of the shoe or desired profits. 4 Explntory vrile = verge solr exposure Response vrile = mximum dily temperture e Strong positive orreltion d r = ; r 2 = These vlues indite strong reltionship etween the two vriles. The oeffiient of determintion suggests tht nerly 70% of the mximum dily temperture is due to the mount of solr exposure. John Wiley & Sons Austrli, Ltd 7
8 Mximum dily temperture = verge solr exposure f 22 MJ g 19 C h An x-vlue of 3 MJ is outside the originl dt set. John Wiley & Sons Austrli, Ltd 8
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