The data below represent the number of days​ absent, x, and the final​ grade, y, for a sample of college students at a large university. Complete parts​ (a) through​ (e) below.

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The data below represent the number of days​ absent, x, and the final​ grade, y, for a sample of college students at a large university. Complete parts​ (a) through​ (e) below.

The data below represent the number of days​ absent, x, and the final​ grade, y, for a sample of college students at a large university. Complete parts​ (a) through​ (e) below.
/ Stu dent: P atric k J o hnson D ate : 0 1/3 0/2 1 In str u cto r: C heban A chary a C ours e : S TA TIS TIC AL M ETH O DS I (A chary a – S ectio n 8 11 – S prin g 2 021) A ssig nm ent: 4 .2 L east S quare s R egre ssio n The d ata b elo w re pre sent th e n um ber o f d ays a bsent, x , a nd th e fin al g ra de, y , fo r a s am ple o f c olle ge s tu dents a t a la rg e univ e rs it y . C om ple te p arts (a ) th ro ugh (e ) b elo w . N o. o f a bse nce s, x 0 1 2 3 4 5 6 7 8 9 F in al g ra de, y 92.5 89.0 85.4 82.3 78.7 73.6 63.5 67.3 63.8 60.3 (a ) Fin d th e le ast-s quare s re gre ssio n lin e tre atin g th e n um ber o f a bsences, x , a s th e e xpla nato ry v a ria ble a nd th e fin al g ra de, y , a s th e re sponse v a ria ble . The e quatio n o f th e le ast-s quare s re gre ssio n lin e is g iv e n b y w here is th e s lo pe o f th e le ast-s quare s re gre ssio n lin e a nd is th e y -in te rc ept o f th e le ast-s quare s re gre ssio n lin e. = b x+ b y 1 0 b = r• 1 sy sx b = − b 0 y 1x F ir s t fin d th e c orre la tio n c oeffic ie nt, r. T he fo rm ula b elo w c an b e u sed to fin d th e s am ple c orre la tio n c oeffic ie nt w here is th e s am ple m ean o f th e e xpla nato ry v a ria ble , is th e s ta ndard d evia tio n o f th e e xpla nato ry v a ria ble , is th e s am ple m ean o f th e re sponse v a ria ble , is th e sam ple s ta ndard d evia tio n o f th e re sponse v a ria ble , a nd n is th e s am ple s iz e. x sx y sy r= ∑ xi− x sx yi− y sy n− 1 C alc ula te th e m ean o f th e x -v a ria ble a nd y -v a ria ble . x = 4.5 y = 75.64 C alc ula te th e s am ple s ta ndard d evia tio n o f th e x -v a ria ble a nd y -v a ria ble , ro undin g to th re e d ecim al p la ces, sx = 3.028 sy = 11.587 N ow c alc ula te and fo r e ach o bserv a tio n. S ta rt w it h th e fir s t o bserv a tio n, and . C om pute fo r th is o bserv a tio n, ro undin g to s ix d ecim al p la ces. xi− x sx yi− y sy x = 0 1 y = 92.5 1 xi− x sx xi− x sx = 0− 4.5 3.028 = − 1.486129 C om pute fo r th is o bserv a tio n, ro undin g to s ix d ecim al p la ces. yi− y sy yi− y sy = 92.5− 75.64 11.587 = 1.455079 P erfo rm th e s am e c alc ula tio ns fo r th e re m ain in g o bserv a tio ns u sin g , , , and , ro undin g th e re sult s to s ix d ecim al p la ces. x= 4.5 sx= 3.028 y= 75.64 sy= 11.587 / xi yi xi− x sx yi− y sy 0 92.5 − 1.486129 1.455079 1 89.0 − 1.155878 1.153016 2 85.4 − 0.825627 0.842323 3 82.3 − 0.495376 0.574782 4 78.7 − 0.165125 0.264089 5 73.6 0.165125 − 0.176059 6 63.5 0.495376 − 1.047726 7 67.3 0.825627 − 0.719772 8 63.8 1.155878 − 1.021835 9 60.3 1.486129 − 1.323897 Next, m ult ip ly th e v a lu es in th e th ir d c olu m n b y th ose in th e fo urth c olu m n to c re ate a fift h c olu m n, ro undin g e ach re sult to s ix d ecim al p la ces. xi yi xi− x sx yi− y sy xi− x sx yi− y sy 0 92.5 − 1.486129 1.455079 − 2.162435 1 89.0 − 1.155878 1.153016 − 1.332746 2 85.4 − 0.825627 0.842323 − 0.695445 3 82.3 − 0.495376 0.574782 − 0.284733 4 78.7 − 0.165125 0.264089 − 0.043608 C ontin ue fillin g in th e ta ble , ro undin g e ach re sult to s ix d ecim al p la ces. xi yi xi− x sx yi− y sy xi− x sx yi− y sy 5 73.6 0.165125 − 0.176059 − 0.029072 6 63.5 0.495376 − 1.047726 − 0.519018 7 67.3 0.825627 − 0.719772 − 0.594263 8 63.8 1.155878 − 1.021835 − 1.181117 9 60.3 1.486129 − 1.323897 − 1.967482 N ow , ta ke th e s um o f th e v a lu es in th e fift h c olu m n. … − 2.162435− 1.332746+ − 1.181117− 1.967482= − 8.809919 F in d th e c orre la tio n c oeffic ie nt b y d iv id in g th is s um b y n 1, ro undin g to s ix d ecim al p la ces. − r= ∑ xi− x sx yi− y sy n− 1 r= − 8.809919 10− 1 r= − 0.978880 U se th e c alc ula te d v a lu es to d ete rm in e . U se r , , and , ro undin g to s ix d ecim al p la ces. b1 = − 0.978880 sx= 3.028 sy= 11.587 b1 = r• sy sx b1 = − 0.978880• 11.587 3.028 b1 = − 3.745800 / Use th is v a lu e, , and to c alc ula te , ro undin g to s ix d ecim al p la ces. x= 4.5 y= 75.64 b0 b0 = − b y 1x b0 = 75.64− (− 3.745800)(4.5) b0 = 92.496100 W rit e th e e quatio n fo r th e le ast-s quare s re gre ssio n lin e, ro undin g to th re e d ecim al p la ces. x y= − 3.746 + 92.496 (b ) In te rp re t th e s lo pe a nd y -in te rc ept, if a ppro pria te . F ir s t in te rp re t th e s lo pe. T he s lo pe o f a lin e is . change in y change in x The s lo pe, , is th e c hange in fin al g ra de fo r a u nit c hange in d ays a bsent, o n a ve ra ge. − 3.746 There fo re , fo r e ve ry s tu dent’s a ddit io nal d ay a bsent, th e s tu dent’s fin al g ra de by , on a ve ra ge. falls 3.746 N ext, in te rp re t th e y -in te rc ept. T he y -in te rc ept o f a lin e is fo und b y le ttin g x e qual 0 a nd s olv in g fo r .y R ecall th at th e y -in te rc ept is . T he y -in te rc ept o f is th e p re dic te d v a lu e o f fin al g ra de w hen th e d ays a bsent is 0 . y= 92.496 92.496 N ote th at w hen a s tu dent is a bsent fo r 0 d ays, th e s tu dent is n ot a bsent a t a ll. T here fo re , th e s core o f a s tu dent th at is n ot absent is a bout . 92.496 (c ) P re dic t th e fin al g ra de fo r a s tu dent w ho m is ses cla ss p erio ds a nd c om pute th e re sid ual. Is th e o bserv e d fin al g ra de above o r b elo w a ve ra ge fo r th is n um ber o f a bsences? two To p re dic t th e fin al g ra de, s ubstit u te fo r x a nd s olv e fo r in th e le ast-s quare s re gre ssio n lin e. 2 y S ubstit u te fo r x a nd s olv e fo r , ro undin g to o ne d ecim al p la ce. 2 y y = − 3.746x+ 92.496 y = − 3.746(2)+ 92.496 y = 85.0 S o th e p re dic te d fin al g ra de fo r a s tu dent w ho m is ses cla ss p erio ds is . 2 85.0 The re sid ual is e qual to th e o bserv e d v a lu e o f th e fin al g ra de m in us th e p re dic te d v a lu e o f th e fin al g ra de o f a s tu dent m is sin g cla sses. two R ecall th at th e p re dic te d v a lu e o f th e fin al g ra de is . T he o bserv e d v a lu e is fo r th is s am ple . C om pute th e re sid ual. 85.0 85.4 r e sid ual = observ e d pre dic te d − re sid ual = 85.4− 85.0 re sid ual = 0.4 R ecall th at th e re sid ual c om pare s th e p re dic te d, o r a ve ra ge, v a lu e o f th e fin al g ra de to th e o bserv e d fin al g ra de. U se th e fo rm ula fo r th e re sid ual a nd it s s ig n to d ete rm in e w heth er th e o bserv e d v a lu e is a bove o r b elo w th e p re dic te d, o r a ve ra ge, v a lu e. (d ) D ra w th e le ast-s quare s re gre ssio n lin e o n th e s catte r d ia gra m o f th e d ata . A s catte r d ia gra m is a g ra ph th at s how s th e re la tio nship b etw een tw o q uantit a tiv e v a ria ble s m easure d o n th e s am e in div id ual. E ach in div id ual in th e d ata s et is re pre sente d b y a p oin t in th e s catte r d ia gra m . T he e xpla nato ry v a ria ble is p lo tte d o n th e horiz onta l a xis , a nd th e re sponse v a ria ble is p lo tte d o n th e v e rtic al a xis . / Dra w a s catte r d ia gra m tre atin g x a s th e e xpla nato ry v a ria ble a nd y a s th e re sponse va ria ble . T his is s how n to th e rig ht. 0 5 1 0 50 75 100 x y N ow d ra w th e le ast-s quare s re gre ssio n lin e o n th is s catte r d ia gra m . T his is s how n to th e rig ht. 0 5 1 0 50 75 100 x y (e ) W ould it b e re asonable to u se th e le ast-s quare s re gre ssio n lin e to p re dic t th e fin al g ra de fo r a s tu dent w ho h as m is sed 1 8 cla ss p erio ds? W hy o r w hy n ot? In o rd er to u se a le ast-s quare s re gre ssio n lin e to m ake a p re dic tio n, th e v a lu e in q uestio n m ust b e w it h in th e s cope o f th e m odel. U se th is in fo rm atio n to d ete rm in e if it is re asonable to u se th e le ast-s quare s re gre ssio n lin e to p re dic t th e fin al g ra de fo r a s tu dent w ho m is sed 1 8 c la ss p erio ds.

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