simple linear regression answers

7/23/2019 Simple Linear Regression Answers

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1a.

X Y XY X 2

45 55 2 475 2 025

46 54 2 484 2 116

48 57 2 736 2 304

50 65 3 250 2 500

45 57 2 565 2 025

49 58 2 842 2 401

48 60 2 880 2 304

55 67 3 685 3 025

57 57 3 249 3 249

52 62 3 224 2 704

58 70 4 060 3 364

53 64 3 392 2 809

47 59 2 773 2 209

46 46 2 116 2 116

50 65 3 250 2 500

∑ X = 749 ∑ Y = 896 ∑ XY = 44 981 ∑ X 2 = 37 651

Y’ = b X + a

Y’ = 0.96 X + 11.80

b) When X = 43, Y’ = 0.96(43) + 11.80

= 53.08

c)

1

.).2(96.0

749)37651(15

)896)(749()44981(15

)(

2

22

pd

X X n

Y X XY nb

=

−

−=

∑−∑

∑∑−∑=

.).2(80.1115

)749(96.0896 pd

n

X bY a

=−

=

∑−∑=



X Y Y’ = 0.96 X + 11.80 Y – Y’ (Y – Y’)2

45 55 55.00 0.00 0.00

46 54 55.96 -1.96 3.84

48 57 57.88 -0.88 0.77

50 65 59.80 5.20 27.04

45 57 55.00 2.00 4.00

49 58 58.84 -0.84 0.71

48 60 57.88 2.12 4.49

55 67 64.60 2.40 5.76

57 57 66.52 -9.52 90.63

52 62 61.72 0.28 0.08

58 70 67.48 2.52 6.35

53 64 62.68 1.32 1.74

47 59 56.92 2.08 4.33

46 46 55.96 -9.96 99.20

50 65 59.80 5.20 27.04

∑ (Y – Y’)2= 275.98

d(i)

2

.).2(61.4

215

98.275

2

2

pd

n

)Y' -(Y se

=

−=

−

∑=



ddoaaeacsscoe, s c s s c o e ,

d(ii)

4446485052545658AdditionalMathematicsscore,X455055606570 P h y s i c s s c o r e , Y RSqLinear =0.456

d(iii)

Coefficients(a)

3



Model

UnstandardizedCoefficients

StandardizedCoefficients

t Sig. Std. !rror eta

" #Constant$ "".8%0 "4.57% .8"2 .4%2

&dditionalMat'e(aticsscore) *

.+5+ .2+" .675 %.2+8 .006

a ,e-endent aria/le 1'sics score) 3

Y’ = b X + a

Y’ = 0.959 X + 11.830

d(iv)

Model Summary

Model R R Sqare &dsted R

SqareStd. !rror of t'e !sti(ate

" .675#a$ .456 .4"4 4.608

a 1redictors #Constant$) &dditional Mat'e(atics score) *

se = 4.608

e)

Step 1 State the n!"" and a"te#native h$p%the&e&

'% β = 0 (he #e#e&&i%n c%e**icient in the p%p!"ati%n e!a"& e#%)

'1 β ≠ 0 (he #e#e&&i%n c%e**icient in the p%p!"ati%n i& n%t e!a" t% e#%)

#

'% dditi%na" /atheatic& &c%#e i& n%t a &tati&tica""$ &ini*icant p#edict%# %* h$&ic& &c%#e.

'1 dditi%na" /atheatic& &c%#e i& a &tati&tica""$ &ini*icant p#edict%# %* h$&ic& &c%#e.

Step 2 Set the c#ite#i%n *%# #eectin the n!"" h$p%the&i&

eect '% i* p 0.05

Step 3 a##$ %!t the ana"$&i& !&in SSS

Model Summary

4





" .675#a$ .456 .4"4 4.608

a 1redictors #Constant$) &dditional Mat'e(atics score) *

Coefficients(a)

Model




" #Constant$ "".8%0 "4.57% .8"2 .4%2

&dditionalMat'e(aticsscore) *

.+5+ .2+" .675 %.2+8 .006

a ,e-endent aria/le 1'sics score) 3

Step 4 /ae a deci&i%n b$ app"$in the c#ite#i%n *%# #eectin the n!"" h$p%the&i&

#% the SSS %!tp!t, p = 0.006

(he p#%babi"it$ %* c%ittin a $pe e##%# that i&, the "ie"ih%%d %* #eectin the n!""

h$p%the&i& hen it i& t#!e i& 0.006)

he#e*%#e, #eect '% beca!&e p 0.05

Step 5 /ae a c%nc"!&i%n in the c%nte:t %* the p#%b"e

dditi%na" /atheatic& &c%#e i& a &tati&tica""$ &ini*icant p#edict%# %* h$&ic& &c%#e,

t (14) = 3.298, p .05

(hat i&, n%"ede %* dditi%na" /atheatic& &c%#e& enhance& the p#edicti%n %* h$&ic&

&c%#e&)

he #e#e&&i%n e!ati%n i& a& *%""%&

Y’ = 0.959 X + 11.830

Predicted Physics score = 0.959 ( Additional Mathematics score) + 11.830

r 2 = 0.456

45.6; %* the va#iance in h$&ic& &c%#e& can be a&&%ciated ith (e:p"ained b$) the va#iance in

dditi%na" /atheatic& &c%#e&.

2a.

5



X Y XY X 2

2.2 2.6 5.72 4.84

1.0 1.5 1.50 1.00

2.3 2.7 6.21 5.29

3.6 4.0 14.40 12.96

0.9 1.0 0.90 0.81

1.6 2.0 3.20 2.56

3.3 3.5 11.55 10.89

3.4 3.0 10.20 11.56

4.0 2.5 10.00 16.00

2.6 2.8 7.28 6.76

∑ X = 24.9 ∑ Y = 25.6 ∑ XY = 70.96 ∑ X 2 = 72.67

Y’ = b X + a

Y’ = 0.68 X + 0.87

b) When X = 3.5, Y’ = 0.68 (3.5) + 0.87

= 3.25

c.

X Y Y’ = 0.68 X + 0.87 Y – Y’ (Y – Y’)2

2.2 2.6 2.37 0.23 0.05

1.0 1.5 1.55 -0.05 0.00

6

.).2(68.09.24)67.72(10

)6.25)(9.24()96.70(10

)(

2

22

pd

X X n

Y X XY nb

=−

−=

∑−∑

∑∑−∑=

.).2(87.010

)9.24(68.06.25

pd

n

X bY a

=

−

=

∑−∑=



2.3 2.7 2.43 0.27 0.07

3.6 4.0 3.32 0.68 0.46

0.9 1.0 1.48 -0.48 0.23

1.6 2.0 1.96 0.04 0.00

3.3 3.5 3.11 0.39 0.15

3.4 3.0 3.18 -0.18 0.03

4.0 2.5 3.59 -1.09 1.19

2.6 2.8 2.64 0.16 0.03

∑ (Y – Y’)2= 2.21

d(i)

7

.).2(53.0

210

21.2

2

pd

n

)Y' -(Y s

2

e

=

−=

−

∑=



".02.0%.04.0Number of hour sspentstudyingdaily,X".0".52.02.5%.0%.54.0 C P A , Y

d(ii)

".. . .,"."......

, Linr.

d(iii)

8



Coefficients(a)

Model




" #Constant$ .876 .4%5 2.0"% .07+

(/er of'ors s-entstding dail)*

.676 ."6" .82+ 4."+" .00%

a ,e-endent aria/le C1&) 3

Y’ = b X + a

Y’ = 0.676 X + 0.876

d(iv)

Model Summary



" .82+#a$ .687 .648 .5272

a 1redictors #Constant$) (/er of 'ors s-ent stding dail) *

se = 0.5272

e)




#

'% <!be# %* h%!#& &pent &t!d$in dai"$ i& n%t a &tati&tica""$ &ini*icant p#edict%# %* .

'1 <!be# %* h%!#& &pent &t!d$in dai"$ i& a &tati&tica""$ &ini*icant p#edict%# %* .


eect '% i* p 0.01


Model Summary

9





" .82+#a$ .687 .648 .5272

a 1redictors #Constant$) (/er of 'ors s-ent stding dail) *

Coefficients(a)

Model




" #Constant$ .876 .4%5 2.0"% .07+

(/er of'ors s-entstding dail)*

.676 ."6" .82+ 4."+" .00%

a ,e-endent aria/le C1&) 3


#% the SSS %!tp!t, p = 0.003





<!be# %* h%!#& &pent &t!d$in dai"$ i& a &tati&tica""$ &ini*icant p#edict%# %* ,

t (9) = 4.191, p .01

(hat i&, n%"ede %* n!be# %* h%!#& &pent &t!d$in dai"$ enhance& the p#edicti%n %*

.)

he #e#e&&i%n e!ati%n i& a& *%""%&

Y’ = 0.676 X + 0.876

Predicted CGPA = 0.676 ( Number of hours spent studying daily) + 0.876

r 2 = 0.687

68.7; %* the va#iance in can be a&&%ciated ith (e:p"ained b$) the va#iance in n!be#

%* h%!#& &pent &t!d$in dai"$.

3a)

10



Coefficients(a)

Model




" #Constant$ 27.8%" .724 %8.42+ .000

&ge of a car#ear$) *

2.2+2 ."%8 .+86 "6.65+ .000

a ,e-endent aria/le 1rice of a car) 3

Y’ = b X + a

Y’ = -2.292 X + 27.831

When X = 3, Y’ = -2.292 (3) + 27.831

= 20.955 th%!&and init /a"a$&ia

b)




#

'% e %* a ca# i& n%t a &tati&tica""$ &ini*icant p#edict%# %* the p#ice %* a ca#.

'1 e %* a ca# i& a &tati&tica""$ &ini*icant p#edict%# %* the p#ice %* a ca#.


eect '% i* p 0.05


Model Summary

Model R R Sqare

&dsted R

Sqare

Std. !rror of

t'e !sti(ate" .+86#a$ .+72 .+68 .5657

11



a 1redictors #Constant$) &ge of a car #ear$) *

Coefficients(a)

Model

Unstandardized

Coefficients

Standardized

Coefficients


" #Constant$ 27.8%" .724 %8.42+ .000

&ge of a car#ear$) *

2.2+2 ."%8 .+86 "6.65+ .000

a ,e-endent aria/le 1rice of a car) 3


#% the SSS %!tp!t, p = 0.000





e %* a ca# i& a &tati&tica""$ &ini*icant p#edict%# %* the p#ice %* a ca#,

t (9) = -16.659, p .05

(hat i&, n%"ede %* the ae %* a ca# enhance& the p#edicti%n %* the p#ice %* a ca#.)

he #e#e&&i%n e!ati%n i& a& *%""%&

Y’ = -2.292 X + 27.831

Predicted price of a car = -2.292 ( Age of a car ) + 27.831

r2

= 0.972

97.2; %* the va#iance in the p#ice %* a ca# can be a&&%ciated ith (e:p"ained b$) the va#iance

in the ae %* a ca#.

12

simple linear regression answers

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