Chapter 11 - Multiple Regression - Statistics for Business | Trường Đại học Quốc tế, Đại học Quốc gia Thành phố HCM

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Môn: Statistics for Business (BAO8OIU)

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Profile Mai Nguyệt
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Stat ist ics for Business | Chapt er 11: M ult iple Regression
1
STATISTICS FOR BUSINESS [IUBA]
CHAPTER 11
M ULTIPLE REGRESSION
STRUCTURE OF PAPER
PART I - M ULTIPLE REGRESSION M ODEL
PART II - M EASURES OF PERFORM ANCE OF A REGRESSION M ODEL AND THE ANOVA TABLE
PART III - THE F – TEST OF A M ULTIPLE REGRESSION M ODEL
PART IV - TESTS OF THE SIGNIFICANCE OF INDIVIDUAL REGRESSION PARAM ETERS
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Stat ist ics for Business | Chapt er 11: M ult iple Regression
2
PART I
M ULTIPLE REGRESSION M ODEL
The population regression model of a dependent variable on a set of independent
variables
,
,… ,
is given by
=
+
+
+ ⋯+
+
where
is t he int ercept of t he regression surface and each
, = 1, … , is t he slope
of the regression surface – sometimes called the response surface – with respect t o variable
.
M odel Assum ptions:
1. For each observat ion, the error term is normally distributed wit h m ean zero and st andard
deviation and is independent of the error t erms associated wit h all ot her observat ions.
That is,
~
(
0,
)
for all = 1, 2,…,
independent of ot her errors.
2. In t he context of regression analysis, t he variables
are considered fix quant it ies, although
in t he cont ext of correlational analysis, t hey are random variab les. In any case,
  ℎ . When w e assume that
are fixed quant it ies, we are
assum ing t hat we have realizat ion of varibles
and t hat the only random ness in com es
from the error t erm .
The Estimat ed Regression Relationship
The estim ated regression relationship is
=
+
+
+ ⋯+
where
is t he predict ed value of , the value lying on t he estim ated regression surface.
The term s
, = 0, …, , are the least-squares est imat es of the p opulation regression
par ameters
.
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Stat ist ics for Business | Chapt er 11: M ult iple Regression
3
PART II
M EASURES OF PERFORM AM CE OF A REGRESSION M ODEL
AND THE ANOVA TABLE
M ean Square Error
(

)
Standard Error of Estimate
(
)
 and are measur es of how well the regression fit s the dat a

=

−
(
+
)
=
√

M ultiple Correlation Coefficient
(
)
M ultiple Coefficient of Det ermination
(
)
Adjusted M ultiple Coefficient of Det ermination
(
)
,
, and
are measures of how well t he regression m odel fits the dat a. In ot her
words, t hey measure t he percentage of variation in t he dependent variable explained
by the independent variables
M ult iple Correlat ion Coefficient
=
M ult iple Coefficient of Determination
=


= −


Adjusted M ult iple Coefficient of Determination
= −
/
[
−
(
+
) ]

/
(
−
)
=
−
(
−
)
−
−
(
+
)
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Stat ist ics for Business | Chapt er 11: M ult iple Regression
4
ANOVA Table for M ultiple Regression M odel
Source of Variation
Sum of Squares
(

)

M ean Square
(

)
F. ratio
(
)
Regression
(
)


=

=


Error
(
)

−
(
+
1
)

=

−
(
+
1
)
Total
(
)

1
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Stat ist ics for Business | Chapt er 11: M ult iple Regression
5
PART III
THE F – TEST OF A M ULTIPLE REGRESSION M ODEL
F – test of a mult iple regression model is a statist ical hypot hesis test for the existence
of a linear relationship between and any of the
F – test of a M ultiple Regression M odel
HYPOTHESIS TESTING PROCESS:
STEP 01: Determine t he null and alt ernatio n hypothesis:
=
=
=
= ⋯=
=
=   
(
= , …,
)
 
STEP 02: Const ruct the ANOVA Table for the multiple regression m odel
Source of Variation
Sum of Squares
(

)

M ean Square
(

)
F. ratio
(
)
Regression
(
)


=

=


Error
(
)

−
(
+
1
)

=

−
(
+
1
)
Total
(
)

1
STEP 03: Comput e the test statistic value
(
)
(based on the ANOVA table) and t he
critical value
(
)
(based on the level of significance)
The t est st at istic value:
= −=


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6
At the level of significance, the crit ical value:
=
,, 
( )
STEP 04: CONCLUSION
+ Situation 01: We cannot reject the null hypo thesis
since
<
For the instance that the null hypothesis is tr ue, no linear relationship
exists between t he dependent variable and any of the independent
variables
in the proposed regression m odel.
+ Situation 02: We can reject t he null hypothesis
since
>
For t he inst ance that we can reject the null hypot hesis, there is
st atist ical evidence to conclude that a regression relationship exists
between t he dependent var iable and at least one of the independent
variables
proposed in the regression model.
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7
PART IV
TESTS OF THE SIGNIFICANCE OF INDIVIDUAL REGRESSION PARAM ETERS
A test for t he significance of an individual parameters is important because it tells us
not only w hether there is evidence t hat variable
(
= 1,…,
)
has a linear
relationship with Y but also whet her there is stat istical evidence that variable
has
explanatory power with respect to the dependent variable .
Tests of the Significance of Individual Regression Parameters
HYPOTHESIS TESTING PROCESS:
STEP 01: Determine t he null and alt ernative hypo theses:
(Note: They are two tailed-testing)
(1)
:
=
0
:
≠
0
(2)
:
=
0
:
≠
0
… …
(k)
:
=
0
:
≠
0
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STEP 02: Comput e the test statistic value
(
/
)
and t he crit ical values
(
/
)
based on the level of significance
+ Sit uation 01: If , we use −
(
+ 1
)
< 30 −
For test
(
= 1,…,
)
, t he test st at istic value:
=
−
(
)
At the level of significance
(
)
, the critical values:
±
= ±
,
( )
+ Sit uation 02: If , we use −
(
+ 1
)
≥30 −
For test
(
= 1,…,
)
, t he test st at istic value:
=
−
(
)
At the level of significance
(
)
, the critical values:
±
= ±
󰇡
󰇢
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9
STEP 03: CONCLUSION
At the level of significance, for each test,
+ Sit uat ion 01: We cannot reject
since
∈
[
−
,
]
or
∈
[
−
,
]
For the instance that t he null hypot hesis is true t hat t he slope
par ameter of
is non – significant and no linear relationship exists
between the dependent variable and the independent variable
+ Sit uat ion 02: We can reject
since
∉
[
−
,
]
or
∉
[
−
,
]
For the instance that we can reject the null hypothesis, t he variable
is significant. It m eans that t here is stat ist ical evidence t hat variable
has a linear relationship w ith and explanat ory power wit h respect to
the dependent variable.
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10
Example: (Case of M ultiple Regression Analysis)
PROBLEM 01: (The Form of M ult iple-Choice Questions)
The sample data size 12 taken from four populat ions. The SPSS output for regression
analysis is as follow s which are missing some values:
n = 12 1 = 3, k = 4 −
ANOVA
M odel Sum of Squares Df M ean Square F Sig.
1 Regression 80.117 3 26.706 56.700 0.000
Residual 0.471 3.768 8
Total 83.885 11
a Predict ors: (Co nst ant ), X1, X2, X3
b Dependent Variable: Y
Coefficients
M odel Coefficient s Std. Error t Sig.
1 (Constant ) 45.56 5.674
X1 2.754 0.775 0.0075 3.5535
X2 3.56 1.107 0.0123 3.2159
X3 1.85 1.065 0.1206 1.7371
Dependent Variable: Y = 0.05
1. Fill in the ANOVA table and coefficients table by the relevant values at the suit cell.
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11
2. Com ment on t he result of Regression:
SOLUSION
:
=
=
= 0
:  ℎ
(
= 1,2,3
)
 
Since t he test stat ist ic value is to o large
(
F
= 56.7
)
, w e can strongly reject at all H
level of significance. It means that based on the ANOVA t able for regression model
and t he hypot hesis test ing, we have enough evidence to pr ove that t here is a
regression relat ionship bet ween the dependent variable Y and the independent
variables Xi.
3. Using = 0.05, X3 is a signif icant predictor : (1 point )
a. True
b. False
SOLUTION: 3b. False
H
: β
= 0
H
: β
≠0
Based on t he table of coefficient s, the p-value is lar ge, that is, the t est stat istic value
falls in t he non-rejection region. So, we cannot reject at 0.05 level of significance.
H
It means t hat based on t he table and t he hypothesis testing, we can believe t hat the
variable X3 is not significant predictor .
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4. What is this model predict with X1=10, X2=15, X3=50?
a. 160
b. 219
c. 238
d. Other ……………………………………………
SOLUTION: 4b. 219
Based on the table o f coefficient s, we can est im ate the multiple regression model f or
pr edictor as followings:
Y
= 45.560 + 2.754X
+ 3.56X
+ 1.85X
Y
= 45.560 + 2.754 × 10 + 3.56 × 15 + 1.85 × 15 = 219
5. Com pute t he value of R: 97.700%
SOLUTION
R =
SSR
SST
=
80.117
83.885
= 0.977 = 97.700%
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PROBLEM 02: (The Form of Writing Questions)
A grocery st ore forecasts the monthly demand (Y) for their products using mult iple-
regression. Three independent var iables used are X
1
, X
2
, and X
3
. The data for last 12 months
of the year 2010 are collected. The regression results are shown below:
ANOVA table:
Source SS df M S F
Regression
Residual Error
Total 625.667
Coefficients
Predictors Coefficients S.E. of coefficients t
Const ant -29.743 12.903
X1 1.104 0.283
X2 1.106 0.205
X3 -0.169 0.198
R
2
= 94.76%. Level of significance is α = 0.05.
1. Fill up the ANOVA table; give t he com ments on relat ionships among variables.
SOLUTION
Source SS df M S F
Regression 593.132 3 197.711 48.615
Residual Error 32.535 8 4.067
Total 625.667 11
R
= 1 −


or SSE = SST
(
1 −R
)
= 625.667
(
1 −0.948
)
= 32.535
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:
=
=
= 0
:  ℎ
(
= 1,2,3
)
 
Since the test statistic value is too large
(
F
= 48.615
)
, w e can strongly reject at H
all level of significance. It means that based on the ANOVA table for regression
model and the hypothesis t esting, w e have enough evidence to prove t hat t here is a
regression relat ionship between t he dependent variable Y and the independent
variables
X
(
i = 1,2,3
)
.
2. Write the regression equation. What predict or sho uld be removed from the equation?
SOLUTION
Predictors Coefficients S.E. of coefficients t
Const ant -29.743 12.903
X1 1.104 0.283 3.901
X2 1.106 0.205 5.395
X3 -0.169 0.198
0.854
From the t able, w e can set up the regression equat ion as followings:
Y = −29.743 + 1.104X
+ 1.106X
−0.169X
+ ε
To test w het her t he variables of t he regression modal are signif icant , w e have to
conduct the t-test of individual regression paramet ers.
Our null and alternat ive hypothesizes of each variable:
:
β
=
0
:
β
≠
0
:
β
=
0
:
β
≠
0
H
:
β
=
0
H
:
β
≠
0
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Based on the table o f coefficient s, we can compute the t est stat ist ic value of each
variable as follow ings:
t
=
b
−
0
s
(
b
)
=
1.104
0.283
= 3.901
t
=
b
−
0
s
(
b
)
=
1.106
0.205
= 5.395
t
=
b
−
0
s
(
b
)
=
−0.169
0.198
= −0.854
df = n −
(
k + 1
)
= 8
α= 0.05, / 2 = 0.05/ 2 = 0.025
The crit ical value:
± t
= ± t
,/
= ± t
,.
= ± 2.306
Thus, at 0.05 level of signif icance, for the instance of and
X
X
, w e can reject H
.
On t he other hand, we cannot r eject t he null hypothesis of since
X
t
belon g t o
the non-r ejection region. It means t hat based on t he hypot hesis testing, we have
enough evidence t o prove t hat the var iables
X
and X
are significan t. How ever, X
is not signif icant and should be rem oved from t he regression equation. And w e
should conduct the multiple regression model again.
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International University IU
STATISTICS FOR BUSINESS [IUBA] CHAPTER 11 M ULTIPLE REGRESSION STRUCTURE OF PAPER
PART I - M ULTIPLE REGRESSION M ODEL
PART II - M EASURES OF PERFORM ANCE OF A REGRESSION M ODEL AND THE ANOVA TABLE
PART III - THE F – TEST OF A M ULTIPLE REGRESSION M ODEL
PART IV - TESTS OF THE SIGNIFICANCE OF INDIVIDUAL REGRESSION PARAM ETERS n io s s re g e R le ltip u : M 1 r 1 te ap h C s | es sin u r B s fo istic tat S
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International University IU PART I
M ULTIPLE REGRESSION M ODEL
The populat ion r egression m odel of a dependent variable  on a set of  independent
var iables ,,… , is given by
 =  +  +  + ⋯ +  + 
w here  is t he  intercept of the regression surface and each ,  = 1,… , is the slope
of t he regression sur face – som et im es called t he response sur face – w it h respect t o var iable . M odel Assum ptions:
1. For each observat ion , t he er ror t erm  is normally distribut ed wit h mean zero and standard
deviat ion  and is independent of the error terms associated wit h all ot her observations. That is,
~ (0,) for all  = 1,2,…, independent of ot her errors. n
2. In t he cont ext of regression analysis, t he var iables  io
 are considered fix quantit ies, although s s
in t he cont ext of correlat ional analysis, t hey are random variab les. In any case, re g 
   ℎ  . When we assume that  are fixed quantities, we are e R
assum ing t hat w e have realizat ion of  varibles  and that the only randomness in  comes le from t he error t erm . ltip u : M 1 r 1
The Est imat ed Regression Relat ionship te ap
The est im at ed regression relat ionship is h C s |
 =  +  +  + ⋯ +  es sin u
w here  is t he predicted value of , the value lying on t he estimated regression surface. r B
The t erm s ,  = 0,…,, are the least-squares estimates of the population regression s fo par am et er s . istic tat S
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International University IU PART II
M EASURES OF PERFORM AM CE OF A REGRESSION M ODEL AND THE ANOVA TABLE
M ean Square Error ( )
St andard Error of Estimate ( )
 and  are measures of how well the regression fits the data   =  −  = √ ( + )
M ultiple Correlation Coefficient ( )
M ult iple Coefficient of Det erm inat ion ( )
Adjusted M ult iple Coefficient of Det erm inat ion (  )
, , and  are measures of how well the regression model fits the data. In other n
w ords, t hey m easure t he per cent age of var iat ion in t he dependent var iable explained io s
by t he indepen dent var iables  sre g e
 M ult iple Correlat ion Coefficient R le ltip  =  u : M
 M ult iple Coefficient of Determination 1 r 1   te  = ap h
 =  −  C  s |
Adjusted M ult iple Coefficient of Determination es sin
/ [ − ( + )]  −  u  =  − =  − ( − ) r B / ( − )  − ( + ) s fo istic tat S
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ANOVA Table for M ultiple Regression M odel Sum of Squares M ean Square F. ratio Source of Variat ion  ( ) ( ) () Regression      = ()   =  Error    − ( + 1)  = ( )  − ( + 1) Tot al   − 1 ( ) n io s s re g e R le ltip u : M 1 r 1 te ap h C s | es sin u r B s fo istic tat S
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International University IU PART III
THE F – TEST OF A M ULTIPLE REGRESSION M ODEL
F – test of a mult iple regression model is a st at ist ical hypot hesis test for the existence
of a linear relat ionship bet ween and any of the 
F – test of a M ultiple Regression M odel
HYPOTHESIS TESTING PROCESS: STEP 01:
Det er mine t he nu ll and alt ernat io n hypot hesis:
 =  =  =  = ⋯ =  = 
 =    ( = ,…,)  STEP 02:
Const r uct t he ANOVA Table f or t he m ult iple regression m odel n Sum of Squares M ean Square F. ratio io s Source of Variat ion  s ( ) ( ) () re g Regression   e    = R ()   =  le Error  ltip   − ( + 1)  = ( u )  − ( + 1) Tot al : M   − 1 1 ( ) r 1 te ap h STEP 03:
Com put e t he t est st at ist ic value (  C
) (based on the ANOVA table) and t he s |
cr it ical value ( ) (based on the level of significance) es sin The t est st at ist ic value: u r B  s fo
 =  −  =  istic tat S
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At t he level of significance, t he cr it ical value:
 = ,,() STEP 04: CONCLUSION
+ Sit uat ion 01: We cannot reject t he null hypo t hesis  since  < 
For t he inst ance t hat t he null hypo t hesis is t r ue, no linear r elat ionship
exist s bet w een t he dependent var iable  and any of t he independent
var iables  in the proposed regression model.
+ Sit uat ion 02: We can reject t he null hypot hesis  since  > 
For t he inst ance t hat w e can reject t he null hypot hesis, t her e is
st at ist ical evidence t o conclude t hat a regr ession r elat ion ship exist s
bet w een t he dependent var iable  and at least one of the independent
var iables  proposed in the regression model. n io s s re g e R le ltip u : M 1 r 1 te ap h C s | es sin u r B s fo istic tat S
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TESTS OF THE SIGNIFICANCE OF INDIVIDUAL REGRESSION PARAM ETERS
A test for t he significance of an individual parameters is important because it tells us
not only whether there is evidence that variable  ( = 1, …, ) has a linear
relationship w it h Y but also whet her there is st at istical evidence t hat variable  has
explanat ory power wit h respect to the dependent variable .
Tests of the Significance of Individual Regression Paramet ers
HYPOTHESIS TESTING PROCESS: STEP 01:
Det er mine t he nu ll and alt ernat ive hypo t heses:
(Not e: They a re t w o t ailed-test ing) (1) :  = 0 :  ≠ 0 n (2)  io :  = 0 s s :  ≠ 0 re g e R … … le ltip (k)  u :  = 0 :  ≠ 0 : M 1 r 1 te ap h C s | es sin u r B s fo istic tat S
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Com put e t he t est st at ist ic value ( / ) and the critical values ( / )
based on t he level of signif icance
+ Sit uation 01: If  − ( + 1) < 30, w e use  − 
For t est  ( = 1, …, ), t he test st atistic value:    −   =  ( )
At t he level of significance ( ) , the critical values: ±  = ±  ,()
+ Sit uation 02: If  − ( + 1) ≥ 30, w e use  − 
For t est  ( = 1, …, ), t he test st atistic value:  −  n  = io ( s ) s re g
At t he level of significance ( ) , the critical values: e R le ±  = ± 󰇡 ltip 󰇢 u : M 1 r 1 te ap h C s | es sin u r B s fo istic tat S
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At t he level of significance, for each t est ,
+ Sit uat ion 01: We cannot reject  since  ∈ [−,] or  ∈ [−,]
For t he inst ance t hat t he null hypot hesis is t r ue t hat t he slope
par am et er of  is non – significant and no linear relationship exists
bet w een t he dependent var iable  and the independent variable 
+ Sit uat ion 02: We can reject  since  ∉ [−, ] or  ∉ [−,]
For t he inst ance t hat w e can r eject t he nu ll h ypot hesis, t he var iable 
is significant . It m eans t hat t her e is st at ist ical evidence t hat variable 
has a linear r elat ionship w it h  and explanat ory pow er wit h respect t o t he dependent var iable. n io s s re g e R le ltip u : M 1 r 1 te ap h C s | es sin u r B s fo istic tat S
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Example: (Case of M ultiple Regression Analysis)
PROBLEM 01: (The Form of M ultiple-Choice Questions)
The sam ple dat a size 12 t aken fr om f our populat ions. The SPSS out p ut for regression
analysis is as f ollow s w h ich ar e m issin g som e values: n = 12, k = 4 − 1 = 3 ANOVA M odel Sum of Squares Df M ean Square F Sig. 1 Regression 80.117 3 26.706 56.700 0.000 Residual 3.768 8 0.471 Tot al 83.885 11
a Predict or s: (Co nst ant ), X1, X2, X3 b Dependent Variable: Y Coefficient s n M odel Coefficient s Std. Error t Sig. io s s 1 (Const ant ) 45.56 5.674 re g X1 2.754 0.775 3.5535 0.0075 e R X2 3.56 1.107 3.2159 0.0123 le X3 1.85 1.065 1.7371 0.1206 ltipu Depen dent Var iable: Y = 0.05 : M 1
1. Fill in the ANOVA table and coefficients table by the relevant values at the suit cell. r 1 te ap h C s | es sin u r B s fo istic tat S
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2. Comment on t he result of Regression: SOLUSION
:  =  =  = 0
:   ℎ  ( = 1,2,3)  
Since t he t est st at ist ic value is t o o large ( F = 56.7), we can strongly reject H at all 
level of signif icance. It m eans t hat based on t he ANOVA t able for regression m odel
and t he hypot hesis t est ing, w e have enough evidence t o pr ove t hat t her e is a
regression r elat ionship bet w een t he depen dent var iable Y and t he indepen dent var iables Xi.
3. Using  = 0.05, X3 is a significant predictor : (1 point ) a. True b. False SOLUTION : 3b. False H n : β = 0 io s s H: β ≠ 0 re g e
Based on t he t able of coef ficient s, t he p-value is lar ge, t hat is, t he t est st at ist ic value R le
falls in t he no n-r eject ion region. So, w e canno t reject H at 0.05 level of significance.  ltip u
It m eans t hat based on t he t able and t he hypot hesis t est ing, w e can believe t hat t he : M
var iable X3 is not signif icant predict or . 1 r 1 te ap h C s | es sin u r B s fo istic tat S
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4. What is this model predict with X1=10, X2=15, X3=50? a. 160 b. 219 c. 238
d. Other …………………………………………… SOLUTION : 4b. 219
Based on t he t able o f coeff icient s, w e can est im at e t he m ult iple regression m odel f or pr edict or as fo llow ings: Y
 = 45.560 + 2.754X + 3.56X + 1.85X Y
 = 45.560 + 2.754 × 10 + 3.56 × 15 + 1.85 × 15 = 219
5. Compute the value of R: 97.700% SOLUTION
R = SSR = 80.117 = 0.977 = 97.700% n SST 83.885 io s s re g e R le ltip u : M 1 r 1 te ap h C s | es sin u r B s fo istic tat S
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PROBLEM 02: (The Form of W riting Questions)
A gr ocery st ore forecast s t he m ont hly demand (Y) for t heir pr oduct s using m ult iple-
regr ession. Thr ee independent var iables used ar e X1, X2, and X3. The dat a for last 12 mont hs
of t he year 2010 are collect ed. The regression result s are show n below : ANOVA table: Source SS df M S F Regression Residual Err or Tot al 625.667 Coefficient s Predict ors Coefficient s S.E. of coefficient s t Const ant -29.743 12.903 X1 1.104 0.283 X2 1.106 0.205 X3 -0.169 0.198 n io s
R2 = 94.76%. Level of significance is α = 0.05. s re g e R le
1. Fill up the ANOVA table; give the comment s on relationships among variables. ltip u SOLUTION : M 1 r 1 Source SS df M S F te Regression 593.132 3 197.711 48.615 ap h Residual Err or 32.535 8 4.067 C Tot al 625.667 11 s | es sin u R = 1 or SSE = SST( 1 r B − 
− R) = 625.667(1 − 0.948) = 32.535  s fo istic tat S
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:  =  =  = 0
:   ℎ  ( = 1,2,3)  
Since t he t est st at ist ic value is t o o lar ge ( F = 48.615), we can strongly reject H at 
all level of signif icance. It m eans t hat based on t he ANOVA t ab le for regression
m odel an d t he hypot hesis t est ing, w e have enough evidence t o pr ove t hat t here is a
regression relat ionship bet w een t he dependent variable Y and the independent var iables X (i = 1,2,3) .
2. Write the regression equation. What predict or should be removed from the equation? SOLUTION Predict ors Coefficient s S.E. of coefficient s t Const ant -29.743 12.903 X1 1.104 0.283 3.901 X2 1.106 0.205 5.395 X3 -0.169 0.198  0.854 n io
From t he t ab le, w e can set up t he regression equat ion as f ollow in gs: s s re g Y = e
−29.743 + 1.104X + 1.106X − 0.169X + ε R le ltip u
To t est w het her t he var iables of t he regressio n m odal are signif icant , w e have t o : M 1
conduct t he t -t est of individual regressio n par am et ers. r 1 te
Our nu ll and alt ernat ive hypot hesizes of each variable: ap h C H : β = 0 H : β = 0 H: β = 0 s | H : β ≠ 0 H : β ≠ 0 H: β ≠ 0 es sin u r B s fo istic tat S
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Based on t he t able o f coeff icient s, w e can com put e t he t est st at ist ic value of each var iable as fo llow ings: b − 0 1.104 t  = = 3.901  =  s( b) 0.283 b − 0 1.106 t  =  = = 5.395  s( b) 0.205 b − 0 −0.169 t  =  = = −0.854  s( b) 0.198 df = n − (k + 1) = 8
α = 0.05,/ 2 = 0.05/ 2 = 0.025
The crit ical value: ± t  = ± t,/  = ± t,. = ± 2.306
Thus, at 0.05 level of signif icance, f or t he inst ance of X and  X, we can reject H. n
On t he ot her hand, w e cannot r eject t he null hypot hesis of X since  t  belong t o  io s s
t he non-r eject ion r egion. It m eans t hat based on t he hypot hesis t est ing, w e have re g
enough evidence t o prove t hat t he var iables X X e
 and X are significant. However,  R
is not signif icant and should be rem oved fr om t he r egression equat ion. An d w e le
should conduct t he mu lt iple regression m odel again. ltip u : M 1 r 1 te ap h C s | es sin u r B s fo istic tat S
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