MATHEMATICAL STUDIES PROJECT-IA.docx

8/18/2019 MATHEMATICAL STUDIES PROJECT-IA.docx

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Candidate Name: Pranav, Kethavarapu CandidateNumber: 001350-0036

International Baccalaureate Diploma programme

Good Shepherd International School

Internal Assessment

May 2016

Mathematical Studies-Standard level

opic title! Modeling the temperature "or a #ee$ "or various cities in the #orld%

Candidate name: pranav, kethavarapu

Candidate number: 001350-0036

School name: Good shepherd international school

Supervisor: Mrs !n"elin

#ord count:

Mathematics Studies SL-Ma !016


2/32


Ac$no#ledgements

$ %ould like to sincerel& ackno%led"e:

• Mrs !n"elin, m& math teacher %ho "uided me throu"h the pro'ect and

research process

• M& $( coordinator, Mrs )imala *ani +acob or providin" me %ith revie%s on

m& %ork



3/32


Introduction!


able o contents:

.o opic /a"e .o

1 Ac$no#ledgements 2

a&le o" contents '

3 Introduction (

)ollection o" data

5

6

2

4

10

11

1

13


4/32


$ have decided to do m& pro'ect on the topic: modelin" the temperature or seven dierent

cities $ have picked various cities across the "lobe or m& statistical pro'ect randoml& $ have

started recordin" the temperature o various cities at seven dierent points on the "lobe $

chose this topic to check %hether there is a model euation or the temperature chan"es

across the "lobe $t %as interestin" or me to ind that all m& "raphs sho%ed either a cubic or

uartic shape o the curve, %hich actuall& surprised me $ had done correlation and re"ression

to check the linearit& o the euation but mostl& it is not a positive one So $ had tried doin"

C($C re"ression or m& curve o best it hen to check i the population bet%een the

temperatures %ere eual or not $ have carried out the one-%a& anova i it is used to i"ure out

i there are an& note%orth& contrasts bet%een the means or three or more independent

"roups Conseuentl&, the h&pothesis is accepted, %hich e7presses that the population means

are all euivalent here %as not much contrast in the means or the temperatures $ there is

an euation ound like this then $ think the uture predictions o climate shit can be ound

%ell in advance

$ am "oin" to collect temperatures o seven dierent cities rom dierent time 8ones $

proceeded %ith m& pro'ect %ork throu"h the belo% sta"es:

SAG* 1!



5/32


$n the irst sta"e o m& pro'ect $ have recorded the temperature o various cities dail&,

calculated in Celsius #hile choosin" the dierent cities $ had taken consideration o the

dierent time 8ones and picked cities rom dierent time 8ones $ have accumulated the data

rom http:99%%%timeanddatecom9%eather9

SAG* 2!

$n the second sta"e o m& pro'ect rom the data $ have accumulated $ have done -

$M;.S$A1@ to the

ne7t Saturda&>A@ respectivel& hen $ calculated the mean or A1, A and the hi"hest

C


6/32


hen a scatter "raph or each da& comparin" %ith Saturda& o the data belo% has been plotted

to anal&8e %hether it ollo%s linear or cubic re"ression anal&sis

Saturd

a

Sunda

M"nd

a

#uesd

a

$ednesd

a

#hursd

a %rida

Saturd

aKua&aLumpur !' !6 !' !6 !' !( !( !)

!' !6 !' !6 !' !( !' !)

!6 !6 !6 !6 !' !' !' !(

!6 !6 !6 !6 !6 !' !6 !(

!6 !6 !6 !5 !6 !' !6 !(

!6 !6 !6 !5 !6 !' !5 !'

!6 !5 !6 !5 !6 !' !5 !'

!6 !5 !6 !5 !6 !' !5 !'

!6 !6 !' !5 !6 !' !6 !6

!' !' !( !' !( !' !( !'

!( 30 30 !) 30 !( 31 !)

30 31 30 31 31 30 3! 31

31 33 31 3! 33 !( 33 3!

33 3* 30 33 3* 30 3* 33

3* 35 3! 3* 3* 31 3* 33

3* 35 31 33 30 3! 3* 3!

3* 31 !' 3! !) 33 33 !(

!6 30 !' 30 !) 3! 33 !(

!* 30 !' !) !) 31 3! !)

!5 !) !' !' !) 31 31 !(

!5 !) !6 !' !( 30 30 !(

!6 !) !6 !' !( 30 30 !(

!6 !( !6 !' !( !) !) !'

!6 !( !6 !' !( !( !) !'



7/32


!! !* !6 !( 30 3! 3* 36

0

10

!0

30

*0

f(x) = - 0.06x^3 + 4.95x^2 - 144.83x + 1426.01

R² = 0.79

Kua&a Lumpur

Sunda P"&n"mia& +Sunda,

M"nda #uesda

$ednesda #hursda

%rida Saturda

he above "raph even thou"h sho%s a pattern o cubic re"ression its not so clear So,

Separate scatter "raphs o each da& compared %ith Saturda& has been plotted belo%:-

!! !* !6 !( 30 3! 3* 360

5

10

15

!0

!5

30

35

*0

+., / - 005.3 2 *'3.! - 13(31. 2 136!)

4 / 0'!

Kua&a Lumpur-Saturda-Sunda

Saturda

Sunda



8/32


!! !* !6 !( 30 3! 3* 36

0

10

!0

30

*0

f(x) = - 0.03x^3 + 2.67x^2 - 74.99x + 720.8

R² = 0.74

Kua&a Lumpur-Saturda-M"nda

M"nda P"&n"mia& +M"nda,

Saturda

M"nda

!! !* !6 !( 30 3! 3* 36

0

10

!0

30

*0

f(x) = - 0.05x^3 + 4.26x^2 - 123.83x + 1213.9

R² = 0.86

Kua&a Lumpur-Saturda-#uesda

#uesda P"&n"mia& +#uesda,

Saturda

#uesda



9/32


!! !* !6 !( 30 3! 3* 36

0

10

!0

30

*0

f(x) = - 0.06x^3 + 5.15x^2 - 147.84x + 1429.24

R² = 0.76

Kua&a Lumpur-Saturda-$ednesda

$ednesda P"&n"mia& +$ednesda,

Saturda

$ednesda

!! !* !6 !( 30 3! 3* 360

10

!0

30

*0

f(x) = - 0.02x^3 + 1.5x̂ 2 - 47.4x + 520.57

R² = 0.55

Kua&a Lumpur- Saturda-#hursda

#hursda P"&n"mia& +#hursda,

Saturda

#hursda



10/32


!! !* !6 !( 30 3! 3* 36

0

10

!0

30

*0

f(x) = - 0.06x^3 + 5.68x^2 - 166.28x + 1635.34

R² = 0.65

Kua&a Lumpur-Saturda-%rida

%rida P"&n"mia& +%rida,

Saturda

%rida

;ach "raph above sho%s a clear cubic re"ression euation as a best it or the data



11/32


#hen an online calculator is used to check %hether the "raph values plotted above are correct

or not $t "ave the same perect value

!(=;:

he table contains the temperatures recorded or a %eek in !msterdam he R2

values

o each o the "raph is also sho%in" positive correlation .o%, to check %hether the same

happens to other countries, some random "raphs had been plotted belo%

Saturda Sunda

M"nda

#uesda

$ednesda

#hursda %rida

Saturda

msterdam 11 10 ( 3 5 5 6 '

11 11 ( 3 * 5 ' 6

11 10 ( 3 5 5 ' 611 11 ( ! 5 * ' 511 11 ' ! 5 * 6 5

11 11 ' ! 5 5 ' 511 11 6 1 5 5 ' 511 11 6 1 * 5 ' *

11 11 6 1 * ' 6 *

11 11 * 1 * ' 6 *11 11 * 1 * ' ' 311 11 * ! * ' 5 *11 11 3 3 * 6 5 *

11 11 * * 5 6 6 *11 11 5 * 5 ' ' 5

11 11 ' * 5 ( ( 511 11 ( 5 6 ) ( '11 11 ( 5 ' ) ) )

11 11 ) 5 ( 10 ) (11 10 10 ' ) 10 ) )1! 10 10 ( 10 11 10 10

13 10 ) ( 11 ) 10 1113 10 11 ) 10 1! 10 11

13 10 11 10 11 1! 11 1!13 10 1! 11 10 1! 10 1!



12/32


13 10 10 11 11 1! 11 1113 10 1! 11 11 13 1! 13

1* 11 1! 11 11 13 11 131* 10 13 ) 11 1! 1! 1*13 10 11 ) ) 13 11 13

13 10 1! 10 ) 13 1! 1*

13 10 11 10 10 1* 1! 1*13 10 11 11 11 13 1! 1*15 10 11 11 11 13 11 1*15 10 10 10 10 1! 11 13

1* 10 11 11 11 1! 1! 1315 10 10 10 10 1! 11 1!1* 10 ) 10 10 1! 10 13

1* 10 ) 10 10 11 11 1313 ) ( ) ) 10 10 1!

13 ) ' ) ) ) ) 1!1! ) ' ( ( ( ( 1111 ) 6 ' ' ( ( 11

10 ) 6 ' ' ' ( 1010 ) 5 ' ' ' ( 11

) 10 11 1! 13 1* 15 160

5

10

15

f(x) = 0.12x^3 - 4.64x^2 + 58.16x - 230.25

R² = 0.33

msterdam-Saturda-Sunda


Saturda

Sunda



13/32


) 10 11 1! 13 1* 15 160

5

10

15

f(x) = - 0.5x^3 + 18.66x^2 - 229.6x + 933.07

R² = 0.8

msterdam-Saturda-#uesda


Saturda

#uesda

) 10 11 1! 13 1* 15 160

5

10

15

f(x) = - 0.31x^3 + 11.67x^2 - 142.15x + 574.01

R² = 0.8

msterdam-Saturda-$ednesda

$ednesda P"&n"mia& +$ednesda,

Saturda

Sunda

$t is ver& clear that it ollo%s a cubic re"ression euation %ith moderate positive correlation

!(=;3:

he table 3 contains the temperatures recorded or a %eek in =ondon

Saturda

Sunda

M"nda

#uesda

$ednesda

#hursda

%rida

Saturda

L"nd"

n 11 11 6 6 ' 5 ' '11 10 5 5 ' 5 6 5



14/32


11 ) * * ' 5 6 5

11 ) * * ' 5 6 6

11 ( 3 3 ' * 6 6

11 ( 3 3 ' * 6 6

11 ' * * ( 6 6 6

11 ' 5 5 ) ' ' '

1! ( ' ' ) ) ( (

13 ) ) ) 11 11 ) )

1* ) ) ) ) 11 10 10

16 10 ) 11 10 1! 11 1!

16 10 10 1! ) 1* 1! 13

1' 10 1! 1! 11 1* 1! 1*

1' 10 1! 1! 1! 13 1! 1*

1( 11 1! 13 13 13 1! 13

1' 11 1! 1! 1* 13 1! 13

16 11 1! 1! 13 13 11 1!

15 11 11 11 1! 1! 11 11

13 ) 11 10 11 1! 10 11

1! ' ) ) ) 10 ) 10

1! ' ( ) ( 10 ) )

1! 6 ' ( ' ( ( )

11 6 6 ( 6 ' ' (

10 11 1! 13 1* 15 16 1' 1( 1)0

5

10

15

+., / - 005.3 2 !1).! - 30'1. 2 1*(13

4 / 051

L"nd"n-Saturda-Sunda


Saturda

Sundau



15/32


10 11 1! 13 1* 15 16 1' 1( 1)0

5

10

15

+. / 00*.3 - 166.! 2 !*)). - 11631

4 / 0'6

L"nd"n-Saturda-$ednesda

$ednesda P"&n"mia& +$ednesda

Saturda

$ednesda

10 11 1! 13 1* 15 16 1' 1( 1)0

5

10

15

+. / 006.3 - !(6.! 2 *55. - !!((3

4 / 0)3

L"nd"n-Saturda-#hursda

#hursda P"&n"mia& +#hursda

Saturda

#hursda

;ven in this case the euation is a cubic one but there is a sli"ht decrease in the value o co

eicient o determination %hich sa&s that it has a ver& moderate positive correlation

!(=;:

he table contains the temperatures recorded or a %eek in Madrid

Saturda

Sunda

M"nda

#uesda

$ednesda

#hursda %rida

Saturda

Madrid 16 15 1* 16 1) 1' 1( 1'

16 13 1* 16 1' 1' 16 1'

15 13 13 16 1' 16 1* 15

1' 16 1( 1( 1( 1( 1( 1(

!1 1) !1 !3 !1 !0 !! !1!3 !! !3 !3 !* !* !3 !!



16/32


1( !1 !0 !3 !! !! 1( !1

16 16 1' !0 1) 1( 1' !0

1* 15 16 1' 1( 1) !0 !1 !! !3 !*0

!0

*0

+., / 00*.3 - !5(.! 2 5!!(. - 3360(

4 / 0(!

Madrid-Saturda-Sunda


Saturda

Sunda

1* 15 16 1' 1( 1) !0 !1 !! !3 !*0

!0

*0

+., / - 0.3 2 00).! 2 !). - 31*1

4 / 0''

Madrid-Saturda-#uesda


Saturda

#uesda

1* 15 16 1' 1( 1) !0 !1 !! !3 !*0

!0

*0

+., / 005.3 - 30).! 2 5)'. - 3656*

4 / 0(5

Madrid-Saturda-#hursda

#hursda P"&n"mia& +#hursda,

Saturda

#hursda



17/32


Dere a"ain, Madrid is havin" a stron" positive correlation but ollo%in" the same cubic

re"ression

able 5:

he table 5 contains the temperatures recorded or a %eek in Melbourne

Saturda

Sunda

M"nda

#uesda

$ednesda

#hursda %rida

Saturda

Me&b"urne 13 11 10 10 ( ( ( 13

13 11 10 10 ' ( ( 13

13 11 10 10 ' ' ( 1313 11 10 10 ( ' ( 13

13 11 10 10 ) ( ( 161! 10 10 ) ) 6 ( 1*



18/32


1! 10 10 ) ) 6 ' 151! 10 10 ) ( 6 ' 15

11 10 10 ( ( 6 ' 1511 10 10 ( ( 6 6 15

10 10 10 ( ( 6 5 1511 ) 10 ) ' 6 5 15

10 ) 10 ) 6 6 ' 1510 10 10 ) 6 6 ) 1610 10 10 ) 6 ( 1* 15

10 10 10 ) ' ) 1' 1510 10 11 ) 11 11 1) 15

11 10 11 10 11 13 !0 1511 11 11 10 13 13 !1 16

11 11 13 11 13 1* !1 1611 1! 13 1! 1* 15 !1 16

1! 1! 1* 1! 1* 16 !! 1'1! 13 1* 1* 15 1' !! 16

13 1! 1* 1* 16 1( !! 1'

13 13 1* 1* 16 1( !! 1)13 15 1* 1* 16 1( !! 1)1* 1* 1* 1* 16 1) !! !0

1* 15 1* 1* 1' 1) !1 !01* 1* 1* 1* 1' 1) !0 !0

13 1* 1* 15 1' 1( 1) !113 1* 1* 1* 16 1( 1( !1

13 13 1* 13 15 1' 1' !!13 13 1* 1* 15 15 16 !!13 13 13 13 15 1* 16 !1

13 1! 13 1! 13 13 13 !013 1! 13 11 1! 13 13 1)

13 1! 13 1! 11 1! 13 1(13 11 1! 1! 11 1! 1! 1(

13 11 1! 1! 11 11 13 1'13 11 11 11 11 10 13 1'

13 11 10 1! ) 10 13 151! 11 11 11 ( 10 15 1'

1! 10 11 ) ( ) 1* 151! 10 11 ) ( 10 1* 1*

11 10 11 ( ( ) 1* 1311 10 11 ' ' ( 13 13



19/32


)5 10 105 11 115 1! 1!5 13 135 1* 1*50

5

10

15

!0

+., / 00(.3 - !*5.! 2 !6*1. - (6!34 / 05(

Me&b"urne-Saturda-Sunda


Saturda

Sunda

)5 10 105 11 115 1! 1!5 13 135 1* 1*50

5

10

15

+. / 013.3 - *3(.! 2 510*. - 1((6)

4 / 031

Me&b"urne-Saturda-M"nda

M"nda P"&n"mia& +M"nda

Saturda

M"nda

)5 10 105 11 115 1! 1!5 13 135 1* 1*50

5

10

15

!0

+. / 03(.3 - 1336.! 2 155'1. - 5)'*'

4 / 03(

Me&b"urne-Saturda-$ednesda


Saturda

$ednesda



20/32


able 6:

he table 6 contains the temperatures recorded or a %eek in (an"alore

Saturda

Sunda

M"nda

#uesda

$ednesda

#hursda %rida

Saturda

an7a&"re !1 !0 !! !3 !* !1 !1 !*

!! !0 !1 !3 !3 !1 !0 !!

!) !* !3 !* !5 !* !* !*

30 !' !( !( !) !) 30 !)

!0 30 30 31 3! 3! 33 3!

!0 !! !) 30 !5 !! 31 !!1) !3 !6 !' !* !1 !' !3

1( !! !* !5 !! !1 !5 !!

16 1( !0 !! !* !6 !( 30 3!0

!0

*0

+., / 005.3 - 3!).! 2 '!'5. - 503*(

4 / 033

an7&"re-Saturda-Sunda


Saturda

Sunda

16 1( !0 !! !* !6 !( 30 3!0

!0

*0

+. / 00'.3 - *(*.! 2 110*1. - (0!3(

4 / 0*3

an7&"re-Saturda-$ednesda


Saturda

$ednesda



21/32


16 1( !0 !! !* !6 !( 30 3!0

!0

*0

+., / 01.3 - 6(!.! 2 15103. - 1066)3

4 / 05

an7&"re-Saturda-%rida


Saturda

%rida



22/32


able 2:

Saturda

Sunda

M"nda

#uesda

$ednesda

#hursda

%rida

Saturda

ei8in7 1* !1 !0 1' 16 !0 1) 15

1! 16 1) 1' 1* 1' 1( 1*

1) 1) !1 1) 1( !0 1) 1(

!6 !6 !* !! !! !5 !0 !!

31 30 !' !3 !6 !( !6 !3

30 30 !' !5 !6 !) !0 !3

!6 !6 !! !! !3 !6 15 !1

!* !3 1) 1( !1 !* 1* 1)

10 15 !0 !5 30 350

10

!0

30

*0

+., / 0.3 - 003.! 2 0(. 2 11

4 / 0)!

ei8in7-Saturda-Sunda


Saturda

Sunda

10 15 !0 !5 30 350

10

!0

30

+., / 0.3 - 01).! 2 35!. - 1'6

4 / 0('

ei8in7-Saturda-M"nda

M"nda P"&n"mia& +M"nda,

Saturda

M"nda



23/32


10 15 !0 !5 30 350

10

!0

30

+., / 001.3 - 06.! 2 113'. - *)!)

4 / 0'3

ei8in7-Saturda-%rida


Saturda

%rida

Since most o the scatter plots are ollo%in" cubic euation as a best curve o it he ne7t

step is to check %hether the coeicients o the cubic re"ression euation ound b& the ;7cel

sheet is correct or not So, Buala =umpurs data Saturda& and Sunda& has been chosen or

this calculation

Groupin" all the values o * or Buala =umpur in a table

Sat- sun , . 0%/1

Saturda&>7@

Sunda&>&@

2 6

2 66 6

6 6



24/32


6 6

6 6

6 5

6 5

6 6

2 2

30

30 31

31 33

33 3

3 35

3 35

3 31

6 30

30

5 4

5 4

6 4

6 6

=et the "eneral euation o a cubic euation is "iven b&:

y=

ax3

+bx

2

+cx+

d - ;uation !

Substitutin" or x=24∧ y=30∈the above ;uation !

24¿¿24¿¿

a¿

-;uation-1




25/32


27¿¿27¿¿

a¿

-;uation-

Substitutin" or x=31∧ y=33∈theabove ;uation !

31¿¿31¿

¿a¿

-;uation-3


33¿¿33

¿¿

a¿

-;uation-

by subtracting Equation1− Equation2 #e "et

−5859a−153b−3c=3


−10108a−232b−4c=−6


−6146 a−128b−2c=−1



26/32



−22113a−513b−9 c=−4

a=−11189

b=277

54c=

−268118

d=10182

7

d=1454.57

Dence the cubic euation

y=−0.058201 x3+5.12963 x2−148.944 x+1454.57

#hich matches the value in the "raph belo%

!! !* !6 !( 30 3! 3* 360

10

!0

30

*0

+., / - 005.3 2 *'3.! - 13(31. 2 136!)

4 / 0'!

Kua&a Lumpur

Saturda

Sunda

o check more on the relationship bet%een the temperature or the da&s o Buala =umpur

Mann-Whitney U test: is conducted to check %hether there is an& si"niicant dierence in

their temperatures Same data rom Buala =umpur is taken to test this behavior

=et the variables denote:

U = Mann−Whitney U variable .



27/32


n1=Sample¿ saturday .

n2=Sample¿ sunday.

Ri= Rank of the sample¿¿

H o=here is significant difference∈their temperatures.

H 1=here isno signifcant difference∈their temperatures .

x=¿ Di"hest rank total

! x=¿

The formula for Mann-Whitney U test is

U =n1

n2+

n2(n

2+1)

2− ∑

i=n1+1

n2

Ri

%"r the test 9rst ran;ed . and int" di va&ues "r the Mann-

$hitne > test "r t="-tai&ed test at 5 &eve& " si7ni9cance &eve&



28/32


&ar7erran;

#. 6!65

N. !*

U calculated=24"24+24"25

2−626.5

U calculated=249.5

5 levelof significance

U table value(24,24)=201¿

U #alc


29/32


H 1: &llthemeans are not equal .

?Sum o each entr& in the table

' xi(2=∑ of eachentry squares∈theentry.

M = !umber of columns.

) = !umber of ro*s .

n=mk =columns"ro*s.

SS =∑ of the squares of the total.

SS+=∑of the squaresbet*eenthe days .

SSW =∑ of the squares*ithin the days .

MSS= Mean∑ of the squares.

% −ratio

df = ,egreesof freedom.

- =0.05=5 levelof significance.

K >

L

Saturda !' !' !6 !6 !6 !6 !6 !6 !6 !' !( 30 31 33 3* 3* 3* !6 !*

Sunda !6 !6 !6 !6 !6 !6 !5 !5 !6 !' 30 31 33 3* 35 35 31 30 30

M"nda !' !' !6 !6 !6 !6 !6 !6 !' !( 30 30 31 30 3! 31 !' !' !' #uesda !6 !6 !6 !6 !5 !5 !5 !5 !5 !' !) 31 3! 33 3* 33 3! 30 !)



30/32


L>

MP

$ednesda !' !' !' !6 !6 !6 !6 !6 !6 !( 30 31 33 3* 3* 30 !) !) !)

#hursda !( !( !' !' !' !' !' !' !' !' !( 30 !( 30 31 3! 33 3! 31

%rida !( !' !' !6 !6 !5 !5 !5 !6 !( 31 3! 33 3* 3* 3* 33 33 3!

Saturda !) !) !( !( !( !' !' !' !6 !' !) 31 3! 33 33 3! !( !( !)

#L?@% #A

?S>M@%SB>?

S

K >L

L>

MP>

Saturda'!)

'!)

6'6

6'6

6'6

6'6

6'6

6'6

6'6

'!)

'(* )00 )61

10()

1156

1156

Sunda6'6

6'6

6'6

6'6

6'6

6'6

6!5

6!5

6'6

'!)

)00 )61

10()

1156

1!!5

1!!5

M"nda'!)

'!)

6'6

6'6

6'6

6'6

6'6

6'6

'!)

'(*

)00 )00 )61 )00

10!* )61

#uesda6'6

6'6

6'6

6'6

6!5

6!5

6!5

6!5

6!5

'!)

(*1 )61

10!*

10()

1156

10()

$ednesda

'!)

'!)

'!)

6'6

6'6

6'6

6'6

6'6

6'6

'(*

)00 )61

10()

1156

1156 )00

#hursda

'(

*

'(

*

'!

)

'!

)

'!

)

'!

)

'!

)

'!

)

'!

)

'!

)

'(

* )00 '(* )00 )61

10!

*

%rida'(*

'!)

'!)

6'6

6'6

6!5

6!5

6!5

6'6

'(*

)61

10!*

10()

1156

1156

1156

Saturda(*1

(*1

'(*

'(*

'(*

'!)

'!)

'!)

6'6

'!)

(*1 )61

10!*

10()

10()

10!*



31/32


he table belo% denotes the number o ro%s and columns o the data above

! = M " )

SS =157082−5468"5468192=1358

SS+=(665 )2

24+

(691)2

24+

(661 )2

24+

(674)2

24+

(685)2

24+

(695 )2

24+

(708 )2

24+

(689 )2

24+

(5468 )2

192=73.3

SSW =1358−73.3=1258

Source o variation SS

MS

S *atio

(et%een Groups 2

233333333

3 105

15005241

3

#ithin Groups

1

15333

3 64

otal

14

1

135241666

2

degreesof freedom for *ithin groups= ! − ) =192−8=184

MSS for bet*een groups=73.33

7=10.5


M !*

K (

N1)!

Source o

variation

e"rees o

reedom >d@

Sum o

suares >SS@

Mean o

Suares

>MSS@

E-ratio

(et%een

temperaturesB −¿ 1 SS+ SS+

( ) −1)

SS+/( ) −1)

SSW /( ! − ) )

#ithin

temperatures . −¿ B SSW SSW

( ! − ) )

otal . −¿ 1 SS


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MSS for *ithin groups=1284.583

184=6.98

% − Ratio=10.5

6.98=1.500579213

- =0.05/criticalregionis% 7,184

>2.02

% calc=1.500579213

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