Advnce Research Method

8/8/2019 Advnce Research Method

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Advance Research Project

Final Report


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+Group Members

n Mirza

n Fahad

n Rafia Alvi

n Nida Ghanchi

n Zeeshan Sangani


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+Q#1

n Part 1:

n Confidence interval for the 50 samples:

n Formulas used:n Z-test: x z /2 . S/ nn T-test:x t /2 (n 1) . S/ n


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+Confidence Interval

Lower 78.75 77.98 67.77 77.74

Upper 83.69 84.10 82.47 84.90

Lower 82.29 80.86 67.37 81.27

Upper 85.89 85.50 80.27 85.85


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+Q#2

n Confidence Interval for the difference of two PopulationMeans ( For 25 and 50 Sample Size)

n Formulas used:n Z-test: x 1 - x 2 z /2 ( 12/n 1 + 22/n 2)n T-test: x 1 - x 2 t /2 (n 1+n 2-2) Sp (1/n 1 + 1/n 2)


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+Confidence Interval 1-2

SP 6.744 13.291 7.457 13.649 8.078 14.015

Lower -3.66 -1.46 -4.34 -1.84 -4.87 -14.17

Upper 4.02 13.66 4.14 13.68 4.31 1.77

Lower -2.03 3.57 -2.38 2.50 -3.64 -16.59

Upper 3.85 16.97 3.44 16.22 2.88 -2.89


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+Q#3

n Test the hypothesis that population means of the marksin economics, BMS, Statistical Inference and Finance. Bytaking extreme values from Q#1 separately at 0.05 levelof significance (using full sample of 50 and taking first 25

as sample size)?


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+Formula

n For Z test:n (X-)/( / n)

n For T Testn (X-)/( / n)


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+Summary of 25 samples

summary ECO BMS SI FINLower: Ho Reject Reject Reject RejectUpper: Ho Reject Reject Reject Reject

Verica=onC.I is

verifiedC.I is

verifiedC.I is

verifiedC.I is

verified

Remarks lies in that

C.I lies in that

C.I lies in that

C.I lies in that

C.I


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+Summary of 50 samples


Verica=onC.I is

verifiedC.I is

verifiedC.I is

verifiedC.I is

verified


C.I lies in that

C.I lies in that

C.I lies in that

C.I


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+Q#4

n Hypothesis testing for significant difference between avg. marks of the of subjects.

n Method:

n Hypothesis n Ho: 1 - 2 = 0n Ha: 1 - 2 0

n Level of significancen = 0.05 n /2 = 0.025


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+Method; cont

n Critical Region

z > z /2 , z < - z /2 t>t /2 ( n1+n2 - 2) , t z 0.025 , z < - z 0.025 t > t 0.025(48) , t < - t 0.025(48) z > 1.96 , z < - 1.96 t > 2.011 , t < - 2.011

n Computationn Z = ( ! x 1 - ! x 2 ) ( 1 - 2 ) t = ( ! x 1 - ! x 2 ) ( 1 - 2 )

( S 1 /n 1 + S 2 /n 2 ) Sp. ( 1/n 1 + 1/n 2 )


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+Testing of hypothesis; for the

significance of 25 sample size

ECO-BMS ECO-SI ECO-FIN BMS-SI BMS-FIN SI-FIN H 1=2 1=2 1=2 1=2 1=2 1=2 H 12 12 12 12 12 12 = 0.05 = 0.05 = 0.05 = 0.05 = 0.05 = 0.05 /2 0.025 0.025 0.025 0.025 0.025 0.025 (1- /2) 0.975 0.975 0.975 0.975 0.975 0.975 Degree of Freedom n1+n2-2 n1+n2-2 n1+n2-2 n1+n2-2 n1+n2-2 n1+n2-2

T-Table t /2&-t /2(n1+n2-2)

t /2&-t /2(n1+n2-2)

t /2&-t /2(n1+n2-2)

t /2&-t /2(n1+n2-2)

t /2&-t /2(n1+n2-2)

t /2&-t /2(n1+n2-2)

C.Region Two-Sided Two-Sided Two-Sided Two-Sided Two-Sided Two-Sided t>t /2(n1+n2-2) t0.025(48) t0.025(48) t0.025(48) t0.025(48) t0.025(48) t0.025(48) T-Tab Value 2.3139 2.3139 2.3139 2.3139 2.3139 2.3139 t


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+ Testing of hypothesis; for the significance of 50sample size

H 1=2 1=2 1=2 1=2 1=2 1=2 H 12 12 12 12 12 12 = 0.05 = 0.05 = 0.05 = 0.05 = 0.05 = 0.05 /2 0.025 0.025 0.025 0.025 0.025 0.025 (1- /2) 0.975 0.975 0.975 0.975 0.975 0.975

Degree of Freedom n1+n2-2 n1+n2-2 n1+n2-2 n1+n2-2 n1+n2-2 n1+n2-2

T-Table

t /2&-t /2

(n1+n2-2)

t /2&-t /2

(n1+n2-2)

t /2&-t /2

(n1+n2-2)

t /2&-t /2

(n1+n2-2)

t /2&-t /2

(n1+n2-2)

t /2&-t /2

(n1+n2-2)

C.Region Two-Sided Two-Sided Two-Sided Two-Sided Two-Sided Two-Sided t>t /2(n1+n2-2) t0.025(48) t0.025(48) t0.025(48) t0.025(48) t0.025(48) t0.025(48) T-Tab Value 2.2764 2.2764 2.2764 2.2764 2.2764 2.2764 t


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Economics and Business Maths and Stats

Q5: Correlation Coefficient

H =0 H 0 = 0.05 /2 0.025 Degree of Freedom (n-2), (50-2)

C.Region Two-Sided T-Value (+/-) (+/-) 2.011

Formula r (n-2)/ (1-r^2)


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+Testing of Correlation

ECO-BMS ECO-SI ECO-FIN BMS-SI BMS-FIN SI-FIN 0 0 0 0 0 0 H =0 =0 =0 =0 =0 =0 H 0 0 0 0 0 0 = 0.05 = 0.05 = 0.05 = 0.05 = 0.05 = 0.05

/2 0.025 0.025 0.025 0.025 0.025 0.025 Degree

of Freedo

m

(n-2), (50-2) (n-2), (50-2) (n-2), (50-2) (n-2), (50-2) (n-2), (50-2) (n-2), (50-2)

C.Region

Two-Sided Two-Sided Two-Sided Two-Sided Two-Sided Two-Sided

T-Value(+/-) (+/-) 2.011 (+/-) 2.011 (+/-) 2.011 (+/-) 2.011 (+/-) 2.011 (+/-) 2.011

Formula r(n-2)/ (1-r^2)

r(n-2)/ (1-r^2)

r(n-2)/ (1-r^2)

r(n-2)/ (1-r^2)

r(n-2)/ (1-r^2)

r(n-2)/ (1-r^2)


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Summary

Courses Result Outcome

E and BMS Reject Significance Diff.E and SI Accept No Significance

Diff.

E and F Reject Significance Diff

BMS and SI Reject Significance Diff

BMS and FI Reject Significance Diff SI and FI Reject Significance Diff

Q5: Correlation Coefficient


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+Q#6

n Regression Line and testing of their significance asindependence

n Beta is calculated from excel slope formula.

n Alpha is calculated from forecast formula where x=0


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+Regression Line


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+Testing of Significance

ECO-BMS ECO-SI ECO-FIN BMS-SI BMS-FIN SI-FIN 0 0 0 0 0 0

H =0 =0 =0 =0 =0 =0 H 0 0 0 0 0 0 = 0.05 = 0.05 = 0.05 = 0.05 = 0.05 = 0.05

/2 0.025 0.025 0.025 0.025 0.025 0.025

Degree of Freedom

(n-2), (50-2) (n-2), (50-2) (n-2), (50-2) (n-2), (50-2) (n-2), (50-2) (n-2), (50-2)

C.Region Two-Sided Two-Sided Two-Sided Two-Sided Two-Sided Two-Sided

T-Value(+/-)

(+/-) 2.011 (+/-) 2.011 (+/-) 2.011 (+/-) 2.011 (+/-) 2.011 (+/-) 2.011

Sb 0.154 0.507 0.171 1.040 0.066 0.049 Formula (b-)/Sb (b-)/Sb (b-)/Sb (b-)/Sb (b-)/Sb (b-)/Sb

(0.72-0)/0.15 (0.69-0)/0.51 (0.47-0)/0.17 (0.28-0)/1.04 (0.4-0)/0.07 (0.11-0)/0.05 Compute

d Value

4.68 1.36 2.74 0.27 6.15 2.28

Result T(comp)>T

(tab) value

T(comp)T(tab)

value

T(comp)T

(tab) value

T(comp)>T(tab)

value

Reject Nullhypothesis

Accept NullHypotheses


Accept NullHypotheses



Conclusion

is not equalsto 0

is equals to 0 is not equalsto 0

is equals to 0 is not equalsto 0

is not equalsto 0

I n or d

er

t o t e s t t h e s i gni f i c an

c ew

em

u s t d ev

el o

p

H y p o t h e s i s i nwh i c h

t h e p o p b e t awi l l b

e 0


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+Sb and S.Y.X

Ex 4204.5 4204.5 4204.5 4159 4159 3691 EY 4159 3691 4178 3691 4178 4178 Ex2 355627.25 355627.25 355627.25 349391 349391 299023 Ey2 349391 299023 352458 299023 352458 352458 Exy 351224.5 311808 352297 309703 348898 311360

Alpha 22.51 15.68 44.21 8.98 50.44 75.39 Beta 0.72 0.69 0.47 0.28 0.40 0.11 SYX 7.02 23.08 7.76 61.05 3.86 7.93 Sb 0.154 0.507 0.171 1.040 0.066 0.049


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+Q#7

n Confidence Interval for first 25 sample size for onepopulation variance marks at 95%.

Dataalpha 0.05

n size 25 degree of free V Tab-value /2 0.025 24 39.36101- /2 0.975 24 12.4010


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+Confidence interval at 95%significance level

C.I for 25

@95% Eco BMS SI Fin

Lower 21.85 33.61 193.56 45.97 Upper 69.35 106.68 614.36 145.91


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+Confidence interval at 99%significance level

C.I for 25

@99%Eco BMS SI Fin

Lower 18.88 29.04 167.23 39.72Upper 86.99 133.82 770.63 183.03


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+Q#8

n Testing of Hypothesis for one population variance at theExtreme Values,

n This testing includes developing two Hypothesis for each

coursen The purpose is to verify that pop mean lies in the confidence

interval

n so One alternate hyp would be >lower & other would be


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+ At 95% significance level

n Testing of hypothesis for the Confidence interval of 25samples at 0.05 level of significance:


Verica=onC.I is

verifiedC.I is

verifiedC.I is

verifiedC.I is

verified

Remarks lies in

that C.I lies in

that C.I lies in

that C.I lies in

that C.I


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+ At 99% significance level

n Testing of hypothesis for the Confidence interval of 25samples at 0.05 level of significance:

Summary ECO BMS SI FINLower: Ho Reject Reject Reject RejectUpper: Ho Reject Reject Reject Reject

Verica=on C.I is verified C.I is verified C.I is verified C.I is verified


C.I lies in that

C.I lies in that

C.I lies in that

C.I


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+Q#9

n Test the hypothesis that is there any significancedifference between two-population variance of the marksin;n Eco-BMS,n Eco-SI,n ECO-FIN,n BMS-SI,n BMS-FIN,n SI-FIN.

n At 0.05 level of significance (using full first 25 as samplesize)?


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+Methodology

H : 1= 2 Ha : 1 2

level of significance=

0.05

Critical Region:F < F ( 1- /2)(V1,V2)

F > F ( /2)(V1,V2)


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ECO-BMS ECO-SI ECO-FIN BMS-SI BMS-FIN SI-FIN H 1= 2 1= 2 1= 2 1= 2 1= 2 1= 2

H 1 2 1 2 1 2 1 2 1 2 1 2 /2 0.025 0.025 0.025 0.025 0.025 0.025 (1- /2) 0.975 0.975 0.975 0.975 0.975 0.975

F table f /2&1- /2(n1-1)-(n2-1)

f /2&1- /2(n1-1)-(n2-1)

f /2&1- /2(n1-1)-(n2-1)

f /2&1- /2(n1-1)-(n2-1)

f /2&1- /2(n1-1)-(n2-1)

f /2&1- /2(n1-1)-(n2-1)

C.Region Two-Sided Two-Sided Two-Sided Two-Sided Two-Sided Two-Sided F>F /2(v1,v2) f0.025(24,24) f0.025(24,24) f0.025(24,24) f0.025(24,24) f0.025(24,24) f0.025(24,24) F-Tab Value 2.2693 2.2693 2.2693 2.2693 2.2693 2.2693

F


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+Q#10

n Construct a contingency table:

n 1): Gender and CGPA 23, 34.

n 2): CGPA 23, 34 and Each for Eco, BMS, SI, FIN

n 3): Mathematical Background or each for Eco, BMS, SIand FIN.

n Using the above contingency table test the independenceof the said variables (relationship b/w gender and CGPA,CGPA and marks and subjects and backgrounds in math)

at 0.05 level of significance?


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+Methodology

n Ho:n There is no relationship exist

n Ha:n There is a relationship

n Critical Values:n Chi Square (r-1)(c-1)

n Computation:n X= (O-E)*2/E

n Result:n X> X (r-1)(c-1); reject null hypothesisn X


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+Summary table of all theindependence test

Particular Calculated Value Critical Vale Remarks

Gender W.R.T CGPA 0.16 3.84 No relation

ECO GPA W.R.T CGPA 11.079 14.067 No relation

BMS GPA W.R.T CGPA 16.480 14.067 Relationship exist

SI GPA W.R.T CGPA 11.244 18.307 No relation

FIN GPA W.R.T CGPA 15.020 14.067 Relationship exist

Math Background WRT ECO GPA 7.343 14.067 No relation

Math Background WRT BMS GPA 16.201 14.067 Relationship exist

Math Background WRT SI GPA 13.310 18.307 No relation

Math Background WRT Fin GPA 15.020 14.067 Relationship exist

Criteria for giving remarks:1): If calculated value is greater than critical one: There is a relationship between said variables.2): If calculated value is less than critical one: There is no relationship b/w said variables in particular.


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+Q#11

n Test the hypothesis that all population means are equalby using analysis of variance (ANOVA) table:


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+Methodology

n Ho: 1 = 2 = 3 = 4

n Ha: At least two of the means are not equal

n Level of significancen = 0.05

n Critical Regionn V1 = k 1 = 3,n V2 = n k = 196n F > F (V1,V2)n F > F0.05 (3,196)n F > 2.6507


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+Computation

n ComputationF=Sb2/Sw2

n Sb2BSS/(K-1)

n Sw2WSS/(n-K)

n BSS(T.J^2/r) (T..^2/n)

n WSS: TSS-BSSn TSS

(X^2ij) (T..^2/n)


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+Data

n K 4

n r 50

n n 200

n T.. 16232.5

n T..^2 263494056.3

n T.J^2 66054266.25

n X^2ij 1356499.25


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+Execution

n Sb21205.01

n Sw2180.68

n F6.67

n Results:n Since the calculated value is greater than the tabulated value i.e

(6.67>2.6507), we may reject our null hypothesis and concludethat the population means of all the courses are hetregenous oratleast two of them are not equal


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Thank you

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