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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
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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+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
Reject Nullhypothesis
Accept NullHypotheses
Reject Nullhypothesis
Reject Nullhypothesis
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:
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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+ 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
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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+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
Q/A