Final Report on Qtia

8/12/2019 Final Report on Qtia

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Variables:

A quantity which changes its value time to time, place to place and person to person is called variableand if the corresponding probabilities are attached with the values of variable then it is called arandom variable.

For example

If we say x ! or x " or x #$ then x is a variable but if a variable appears in the following way then itis %nown as a random variable.

x &'x(

! ).*

* ).+

+ ).!

).

Population:

A large count or the whole count of the ob-ect related things is called population. here are two types

of population it may be finite or infinite. If the population elements are countable then it is %nownas finite population but if the population elements are uncountable then it is called an infinite

population.

For example:

Population of MBA students at IUGC (Finite Population)


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Population of the University tea hers in Pa!istan (Finite Population)

Population of trees (Infinite Population)

Population of sea life (Infinite Population)

"he population is also ate#ori$ed in t%o %ays&

'& omo#eneous population

& etero#eneous population

Homogeneous Population:

If all the population elements have the same properties then the population is !no%n as homo#eneous population&

For example: Population of shops* Population of houses* Population of +oys* Population of ri e in a +ox et &

Heterogeneous Population:

If all the population elements do not have the same properties then the population is !no%n as homo#eneous population&

For example: Population of MBA students (Male and Female)* Population of plants* et &

Parameter: A onstant omputed from the population or a population hara teristi is !no%n as parameter&

For ,xample:

Population Mean -* Population standard deviation .* oeffi ient of s!e%ness and !urtosis for the population&

Statistic:

A onstant omputed from the /ample or a sample hara teristi is !no%n as parameter&

For ,xample:

/ample mean * sample standard deviation s* oeffi ient of s!e%ness and !urtosis for the sample&

Estimator:

A /ample statisti used to estimate the population parameter is !no%n as estimator&

Quantitative Techniques in Analysis Page 2


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For ,xample:

/ample mean is used to estimate the population mean& /o sample mean is also alled an estimator of populationmean&

/ample variance is used to estimate the population variance. /o sample variance is also called anestimator of population variance.

Hypothesis:

An assumption a+out the population parameter tested on the +asis of sample information is alled hypothesis orhypothesis testin#&

"hese assumptions are esta+lished in the %ay that %e #enerate t%o alternative statements say 0null and alternativehypothesis1 in su h a manner if one statement is found %ron# automati ally other one is sele ted as orre t

statement&

Types of Hypothesis:

1) Null Hypothesis:

A /tatement or the first thin! a+out the parameter value is alled a null hypothesis& But statisti ally %e an say thata null hypothesis is a statement should onsist e2uality si#n su h as:

3: - 4 - 3

3: - 5 - 3

3: - 6 - 3

As it is lear from a+ove statements there are t%o types of null hypothesis&

'7 /imple null hypothesis

7 Composite null hypothesis

1-Simple Null Hypothesis:

If a null hypothesis is +ased on the sin#le value (or it onsist of only e2ual si#n) then the null hypothesis is alled a simple null hypothesis

For ,xample 3: - 4 - 3

Phrases

Avera#e rain fall in United /tates of Ameri a durin# '888 %as 33 mm&



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"he avera#e on entrations of t%o su+stan es are same

"he I9 level of MBA and BBA students are same&

I9 level is independent from edu ation level&

2-Composite Null Hypothesis:

If a null hypothesis is +ased on the interval of the parameter value (or it onsist of less then or #reater then si#n%ith e2ual si#n) then the null hypothesis is alled a Composite null hypothesis

For ,xample 3: - 5 - 3

3: - 6 - 3

Phrases

"he mean hei#ht of BBA students are at most 3 in hes

"he performan e of P ; students is at most same as MBA students

- 3 * ' : - ? - 3 )

3: - 5 - 3 ( ' :- = - 3 * ' :- > - 3 * ' : - ? - 3 )

3: - 6 - 3 ( ' :- = - 3 * ' :- > - 3 * ' : - ? - 3 )

It is lear from the a+ove stated alternatives that there are t%o different types of alternatives&

'7 @ne tailed or @ne sided alternative hypothesis

7 "%o tailed or t%o sided alternative hypothesis

1-!ne taile" Alternati e Hypothesis:

If an alternative is +ased on either the #reater then (>) or a less then (?) si#n in the statement then the alternativehypothesis is !no%n as the one tailed hypothesis&

For ,xample: ' :- > - 3 * @r ' : - ? - 3

Phrases



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Avera#e rain fall in Pa!istan is more then from avera#e rain fall in a!arta&

In$amam is more onsistent player then /haid Afridi&

aseem A!ram is a +etter +o%ler then M Grath&

Gold pri es are dependent on oil pri es&

2-T#o taile" Alternati e Hypothesis:

If an alternative is +ased on only an une2ual (=) si#n in the statement then the alternative hypothesis is !no%n asthe t%o tailed hypothesis&

For ,xample: ' : - = - 3 *

Phrases

"he Con entration of t%o su+stan es is not same&

"here is a si#nifi ant differen e +et%een the %heat produ tion of /ind and Pun a+&

"he onsisten y of D/, and //, is not same&

In this type of alternatives the total han e of type I error remain in only one side of the normal urve

In this type of alternatives the total han e of type I error is divided in t%o sides of the normal urve

Pro$a$ilities Associate" #ith %ecisions:

Ho is True Ho is &alse

Accept HoCorre t ;e ision

1-'

False ;e ision

"ype II ,rror

'

(e ect Ho

False ;e ision

"ype I ,rror

*

Corre t ;e ision

1-+



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It is lear from the a+ove fi#ures that +oth the errors an not +e minimi$ed at the same time& An in rease iso+served in the type II error %hen type I error is minimi$ed&

P- ,alue:

It is the minimum value of alpha 0E1 %hi h is needed to re e t a true null hypothesis& As it is the value of 0E1 so itan +e explain as the minimum value of type I error %hi h is asso iated %ith a hypothesis %hile it is testin#&

"herefore* it is used in t%o %ays* one in de ision ma!in# and the other to determine the pro+a+ility of type I errorasso iated %ith the testin#&

%ecision (ule on the $asis of p - alue:

e e t o if p value ? 3&3H

A ept o if p value 6 3&3H

For example:

If the p- alue for any test appears . 1& It is indi atin# that our null hypothesis is to +e re e ted and there is only' han e of re e tin# a true null hypothesis& "hat further an explain as %e are 88 onfident in re e tion of thenull hypothesis& @r %e an say that %e an re e t our this null hypothesis up to E 4 ' or 88 onfiden e level

If the p- alue for any test appears .21& It is indi atin# that our null hypothesis is to +e a epted and there is 'han e of re e tin# a true null hypothesis& "hat further an explain as %e are 8 onfident in our de ision and

re e tion of the null hypothesis& @r %e an say that our this true null hypothesis may +e re e ted at E 4 ' &


True Population

Other Population


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T-test:-

A t-test is a statisti al hypothesis test in %hi h the test statisti has a /tudentJs t distri+ution if thenull hypothesis is true& It is applied %hen the population is assumed to +e normally distri+uted

+ut the sample si$es are small enou#h that the statisti on %hi h inferen e is +ased is notnormally distri+uted +e ause it relies on an un ertain estimate of standard deviation rather thanon a pre isely !no%n value&

/ses of T-test:-

Amon# the most fre2uently used t tests are:

A test of %hether the mean of a normally distri+uted population has a value spe ified in anull hypothesis&

A test of the null hypothesis that the means of t%o normally distri+uted populations aree2ual& Given t%o data sets* ea h hara teri$ed +y its mean * standard deviation andnum+er of data points& e an use some !ind of t7test to determine %hether the meansare distin t* provided that the underlyin# distri+utions an +e assumed to +e normal&"here are different versions of the t- test dependin# on %hether the t%o samples are

o Unpaired* independent of ea h other (e&* individuals randomly assi#ned intot%o #roups* measured after an intervention and ompared %ith the other #roup)*or

o Paired* so that ea h mem+er of one sample has a uni2ue relationship %ith a parti ular mem+er of the other sample (e&* the same people measured +efore

and after an intervention &

0nterpretation of the results:-

If the al ulated p7value is +elo% the threshold hosen for statisti al si#nifi an e (usually the 3&'3* the 3&3H* or 3&3' level)* then the null hypothesis %hi h usually statesthat the t%o #roups do not differ is re e ted in favor of an alternative hypothesis* %hi htypi ally states that the #roups do differ&

A test of %hether the slope of a re#ression line differs si#nifi antly from 3&

Statistical Analysis of the t-test:-

"he formula for the t7test is a ratio& "he top part of the ratio is ust the differen e+et%een the t%o means or avera#es& "he +ottom part is a measure of the varia+ility ordispersion of the s ores& "his formula is essentially another example of the si#nal7to7noise metaphor in resear h: the differen e +et%een the means is the si#nal that* in this

ase* %e thin! our pro#ram or treatment introdu ed into the dataK the +ottom part of the formula is a measure of varia+ility that is essentially noise that may ma!e it harder to see

http://en.wikipedia.org/wiki/Statistical_hypothesis_testinghttp://en.wikipedia.org/wiki/Student's_t-distributionhttp://en.wikipedia.org/wiki/Student's_t-distributionhttp://en.wikipedia.org/wiki/Null_hypothesishttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Sample_sizehttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Null_hypothesishttp://en.wikipedia.org/wiki/Meanhttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Meanhttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Statistical_independencehttp://en.wikipedia.org/wiki/P-valuehttp://en.wikipedia.org/wiki/P-valuehttp://en.wikipedia.org/wiki/Statistical_significancehttp://en.wikipedia.org/wiki/Linear_regressionhttp://en.wikipedia.org/wiki/Statistical_significancehttp://en.wikipedia.org/wiki/Statistical_significancehttp://www.socialresearchmethods.net/kb/expclass.phphttp://www.socialresearchmethods.net/kb/expclass.phphttp://en.wikipedia.org/wiki/Statistical_hypothesis_testinghttp://en.wikipedia.org/wiki/Student's_t-distributionhttp://en.wikipedia.org/wiki/Null_hypothesishttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Sample_sizehttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Null_hypothesishttp://en.wikipedia.org/wiki/Meanhttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Meanhttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Statistical_independencehttp://en.wikipedia.org/wiki/P-valuehttp://en.wikipedia.org/wiki/Statistical_significancehttp://en.wikipedia.org/wiki/Linear_regressionhttp://en.wikipedia.org/wiki/Statistical_significancehttp://www.socialresearchmethods.net/kb/expclass.phphttp://www.socialresearchmethods.net/kb/expclass.php


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the #roup differen e& Fi#ure sho%s the formula for the t7test and ho% the numerator anddenominator are related to the distri+utions&

"he top part of the formula is easy to ompute 77 ust find the differen e +et%een themeans& "he +ottom part is alled the standard error of the differen e& "o ompute it* %eta!e the varian e for ea h #roup and divide it +y the num+er of people in that #roup& eadd these t%o values and then ta!e their s2uare root

"he t7value %ill +e positive if the first mean is lar#er than the se ond and ne#ative if it is smaller& @n e you ompute the t7value %e have to loo! it up in a ta+le of si#nifi an e totest %hether the ratio is lar#e enou#h to say that the differen e +et%een the #roups is notli!ely to have +een a han e findin#& "o test the si#nifi an e* %e need to set a ris! level( alled the alpha level )& In most so ial resear h* the Lrule of thum+L is to set the alphalevel at &3H& "his means that five times out of a hundred %e %ould find a statisti ally

si#nifi ant differen e +et%een the means even if there %as none (i&e&* +y L han eL)& ealso need to determine the de#rees of freedom (df) for the test& In the t7test* the de#rees of

freedom is the sum of the persons in +oth #roups minus & Given the alpha level* the df*and the t7value* %e an loo! the t7value up in a standard ta+le of si#nifi an e (availa+le

as an appendix in the +a ! of most statisti s texts) to determine %hether the t7value islar#e enou#h to +e si#nifi ant& If it is* %e an on lude that the differen e +et%een themeans for the t%o #roups is different (even #iven the varia+ility&

Calculations:-

a) 0n"epen"ent one-sample t -test

http://www.socialresearchmethods.net/kb/statdesc.htm#Dispersionhttp://www.socialresearchmethods.net/kb/statdesc.htm#Dispersionhttp://www.socialresearchmethods.net/kb/power.phphttp://www.socialresearchmethods.net/kb/statdesc.htm#Dispersionhttp://www.socialresearchmethods.net/kb/power.php


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In testin# the null hypothesis that the population means is e2ual to a spe ified value 3 * one usesthe statisti &

here 0 s is the sample standard deviation of the sample and 0 n is the sample si$e& "hede#rees of freedom used in this test is 0 n ' 1&

$) 0n"epen"ent t#o-sample t -test:-

A) ,2ual sample si$es* e2ual varian e

"his test is only used %hen +oth:

the t%o sample si$es (that is* the n or num+er of parti ipants of ea h #roup) aree2ualK

It an +e assumed that the t%o distri+utions have the same varian e&


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"his test is used only %hen the t%o sample si$es are une2ual and the varian e is assumed to +edifferent& /ee also el hJs t test & "he t statisti to test %hether the means are different an +e

al ulated as follo%s:

Where n 1 4 num+er of parti ipants of #roup 0'1 and n2 is num+er of parti ipants #roup t%o&

In this ase* varian e is not a pooled varian e& For use in si#nifi an e testin#* the distri+utionof the test statisti is approximated as +ein# an ordinary /tudentJs t distri+ution %ith the de#reesof freedom al ulated usin#

"his is alled the el h7/atterth%aite e2uation & Oote that the true distri+ution of the test statisti a tually depends (sli#htly) on the t%o un!no%n varian es&

"his test an +e used as either a one7tailed or t%o7tailed test&

c) %epen"ent t-test for paire" samples:-

"his test is used %hen the samples are dependentK that is* %hen there is only one sample that has

+een tested t%i e (repeated measures) or %hen there are t%o samples that have +een mat hed orLpairedL&

For this e2uation* the differen es +et%een all pairs must +e al ulated& "he pairs are either one personJs pre7test and post7test s ores or +et%een pairs of persons mat hed into meanin#ful #roups (for instan e dra%n from the same family or a#e #roup: see ta+le)& "he avera#e ( XD ) and standard deviation ( sD ) of those differen es are used in the e2uation& "he onstant 3 is non7$eroif you %ant to test %hether the avera#e of the differen e is si#nifi antly different than 3& "hede#ree of freedom used is 0 N '1&

E ample 1

http://en.wikipedia.org/wiki/Welch's_t_testhttp://en.wikipedia.org/wiki/Student's_thttp://en.wikipedia.org/wiki/Welch-Satterthwaite_equationhttp://en.wikipedia.org/wiki/Welch's_t_testhttp://en.wikipedia.org/wiki/Student's_thttp://en.wikipedia.org/wiki/Welch-Satterthwaite_equation


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Analysis through SPSS:-

A) !ne-sample t-test:-

/P// need:7

') "he data should +e in the form of numeri al (i&e the numeri al varia+le)) A test value %hi h is our hypotheti al value to %hi h %e are #oin# to test&

"o analy$e the 0one7sample t7test1 I have use the employees salaries of an or#ani$ation& For this purpose* I have sele t the sample of 0Q Q1 employees of the ompany&

"he hypotheses are:

a) "he null hypothesis states that the avera#e salary of the employee is e2ual to 0R3*3331&

3 : R3*333

+) "he alternative hypothesis states that the avera#e salary of the employee is not e2ual to0R3*3331&

A: R3*333

Method:

,nter the data in the data editor and the varia+le is la+eled as employeeJs urrent salary& Oo% li ! on Analyze

%hi h %ill produ e a drop do%n menu* hoose Compare means from that and li ! on one-samples t test * adialo#ue +ox appears* in %hi h all the input varia+les appear in the left7hand side of that +ox& From this +ox %ehave to sele t a varia+le* %hi h is to +e omputed& "he varia+le omputed in our ase is Current salaries of theemployees& "he varia+les an +e sele ted for analysis +y transferrin# them to the test variable +ox& Oext* han#ethe value in the test value +ox* %hi h ori#inally appears as 3* to the one a#ainst %hi h you are testin# the samplemean& In this ase* this value %ould +e RH333& Oo% li ! on OK to run the analysis&

Pi torial epresentation

Analy3e Compare 4eans !ne-Sample T Test %rag Test ,aria$le5Scale) 6i e Test ,alue !7



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/P// output:7

!ne-Sample Statistics

N Mean Std. Deviation Std. ErrorMean

Current Salary 474 !4"4#$.%7 #7"&7%.''# 7(4.!##

Interpretation:7



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In a+ove ta+le 0O1 sho%s the total num+er of o+servation& "he avera#e salary of totalemployees is 0RQ*Q'8&H 1& "he standard deviation of the data is 0' *3 H&SS'1and the standarderror of the mean is 0 TQ&R''1&

!ne-Sample Test

)est *alue + !&&&&

t D, Si . /-tailed0 MeanDi,,eren1e

$%2 Con,iden1e 3ntervalo, t e Di,,eren1e

5o6er pper

Current Salary %.'!% 47! .&&& 4"4#$.%'( /"(7(.4& %"$'&.7!

Interpretation:7

"hrou#h a+ove ta+le %e an o+serve that*

i) 0"1 value is positive %hi h sho% that our estimated mean value is #reater thana tual value of mean&

ii) ;e#ree of freedom is (O ') 4 Q R&

iii) "he 0P7value1 is 03&3331 %hi h is less than 03&3H1&

iv) "he differen e +et%een the estimated a tual mean is 0Q*Q'8&HST1&

v) Confiden e interval has the lo%er upper limit *T T&Q H*8S3& R respe tively& "he

onfiden e interval limits does not ontains $ero&

;e ision:7

@n the +asis of follo%in# o+servation I re e t my 0Oull hypothesis1 and a ept the 0Alternativehypothesis1& I am almost 0'33 1 sure on my de ision&

i) "he 0P7value1 is 03&3331 %hi h is less than 03&3H1&ii) "he onfiden e interval limits does not ontains $ero&

Comments:7

"he avera#e salary of employees is not e2ual to 0R3*3331&

E ample 2

8) 0n"epen"ent t-test:-

/P// need:7



14/98

') "%o varia+le are re2uired one should +e numeri al and other should +e ate#ori al%ith t%o levels&

"o analy$e the 0independent t7test1 I have use the employees salaries of an

or#ani$ation& For this purpose* I have sele t the sample of 0Q Q1 employees of theompany ontainin# the +oth males and females& In my analysis I assi#ned males as 0m1and female as 0f1&

"he hypotheses are:

a) "he null hypothesis states that the avera#e salary of the male employee is e2ual toavera#e salary of the male employee&

3 :

i&e

+) "he alternative hypothesis states that the avera#e salary of the male employee is note2ual to avera#e salary of the male employee&

A :

i&e

Method:

,nter the data in the data editor and the varia+les are la+eled as employeeJs +e#innin# salary and employeeJsdesi#nations respe tively& Cli ! on Analyze %hi h %ill produ e a drop do%n menu* hoose Compare means fromthat and li ! on independent samples t-test * a dialo#ue +ox appears* in %hi h all the input varia+les appear inthe left7hand side of that +ox& "o perform the independent samples t7test* transfer the dependent varia+le into thetest variable +ox and transfer the varia+le that identifies the #roups into the roupin variable +ox& In this ase*the 8e innin salary of the employees is the dependent varia+le to +e analy$ed and should +e transferred into testvaria+le +ox +y li !in# on the first arro% in the middle of the t%o +oxes& 9ob 1ate ory is the varia+le %hi h %illidentify the #roups of the employees and it should +e transferred into the #roupin# varia+le +ox&

@n e the #roupin# varia+le is transferred* the de,ine roups +utton %hi h %as earlier ina tive turns a tive& Cli !on it to define the t%o #roups& In this ase roup# represents the employees +elon# to leri al ate#ory and roup/represents the employees +elon# to the ustodial ate#ory& "herefore put ' in the +ox a#ainst #roup' and in the

+ox a#ainst #roup and li ! ontinue& Oo% li ! on OK to run the analysis&


Analy3e Compare 4eans 0n"epen"ent-Samples T Test %rag Test 96rouping ,aria$le %efine 6roups !7



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/P// output:7

6roup Statistics



16/98

Interpretation:7


i) "otal num+er of male is 0 HT1 and the female is 0 'S1&ii) "he mean value of salaries of male employee is Q'*QQ'& T the female employee is

S*3R'&8 &

iii) /tandard deviation of salaries of male employee is '8*QQ8& 'Q the femaleemployee is *HHT&3 '&

iv) /tandard error of mean of salaries of male employees is '* 'R&8ST the /tandarderror of mean of salaries of female employees is H'Q& HT&

0n"epen"ent Samples Test

Interpretation:7

In a+ove ta+le %e have t%o parts (a) f7test* (+) t7test* throu#h %hi h %e an o+serve that*

i) 0F1 value is 0''8&SS81 %ith si#nifi ant value of 03&331 %hi h is less than 03&3H1&ii) @n the +asis of P7value of F7test part %e assume that that the varian e of the t%o

populations is not e2ual&

iii) 0"1 value is positive %hi h sho% that the mean value of salaries of male employeesis #reater than the mean value of salaries of female employees

iv) ;e#ree of freedom is 0RQQ& S 1&


:ender N Mean Std. Deviation Std. Error Mean

Current Salary Male

;emale

/%(

/#'

4#"44#.7(

/'"&!#.$/

#$"4$$./#4

7"%%(.&/#

#"/#!.$'(

%#4./%(

Current /alary

VeveneJs "est for ,2uality of


17/98

v) "he 0P7value1 is 03&3331 %hi h is less than 03&3H1&

vi) "he differen e +et%een the t%o population mean is 0'H*Q38&TS 1&

vii) "he standard error differen e +et%een the t%o population mean is 0'*R'T&Q331&

viii) Confiden e interval has the lo%er upper limit 0' *T'S& T1 0'T*33 &88S1respe tively& "he onfiden e interval limits does not ontains $ero&

;e ision:7

@n the +asis of follo%in# o+servation I re e t my 0Oull hypothesis1 and a ept the 0Alternativehypothesis1& I am almost 0'33 1 sure on my de ision&

i) "he 0P7value1 is 03&3331 %hi h is less than 03&3H1&ii) "he onfiden e interval limits does not ontains $ero&

Comments:7

"he avera#e salaries of male female employees are not e2ual&

E ample

C) Paire" t-test:-

/P// need:7

') "%o numeri al varia+les are re2uired %hi h should +e e2ual in num+ers&

"o analy$e the 0paired t7test1 I used the +e##in# endin# salaries of the employees ofan or#ani$ation& For this purpose* I have sele t the sample of 0Q Q1 employees of theor#ani$ation&

"he hypotheses are:

a) "he null hypothesis states that the avera#e salary of the male employee is e2ual toavera#e salary of the male employee&

3 :

i&e 3



18/98

+) "he alternative hypothesis states that the avera#e salary of the male employee is note2ual to avera#e salary of the male employee&

A :

i&e 3

Method:

,nter the data in the data editor and the varia+les are la+eled as employeeJs urrent and +e#innin# salaryrespe tively& Cli ! on Analyze %hi h %ill produ e a drop do%n menu* hoose Compare means from that and li !on


19/98

/P// output:7

Paire" Samples Statistics

Mean O /td& ;eviation/td& ,rror

Mean Pair ' Current /alary WRQ*Q'8&H Q Q W' *3 H&SS' W TQ&R''

Be#innin# /alary W' *3'S&38 Q Q W *T 3&SRT WRS'&H'3

Interpretation:7


i) "he mean vale of urrent +e#innin# salary is 0RQ*Q'8&H 1 0' *3'S&381respe tively&

ii) "otal num+er of +oth #roups is 0Q Q1 individually&

iii) "he standard deviation of urrent +e#innin# salary is 0' *3 H&SS'1 0 *T 3&SRT1 respe tively&

iv) "he standard error mean of urrent +e#innin# salary is 0 TQ&RR'1 0RS'&H'31respe tively&

Paire" Samples Correlations

O Correlation /i#& Pair ' Current /alary

Be#innin# /alary Q Q &TT3 &333

Interpretation:7


An


20/98


i) "he total num+er of pair is 0Q Q1&ii) 03&TT1 sho% that the +oth values of #roup are hi#hly o7related* %hi h indi ate that

the employees %ho has #reater +e##in# salary has also #reater urrent salary&iii) "he P7value is 03&331 %hi h is less than 03&3H1&

Paire" Samples Test

t df /i#& ( 7tailed Mean /td& ;eviation

/td& ,rror Mean

8H Confiden e Interval of the ;ifferen e

Vo%er Upper

Pair ' Current /alary7 Be#innin# /alary W' *Q3R&QT' W'3*T'Q&S 3 WQ8S& R W'S*Q &Q3 W'T*R 8&HHH RH&3RS Q R

Interpretation:7

In a+ove ta+le %e have t%o parts (a) f7test* (+) t7test* throu#h %hi h %e an o+serve that*

i) "he mean value of pair is 0' *Q3R&QT'1&ii) "he standard deviation of pair is 0'3*T'Q&S 31&

iii) "he standard error mean of pair is 0Q8S& R 1&

iv) Confiden e interval has the lo%er upper limit 0'S*Q &Q3 1 0'T*R 8&HHH1respe tively& "he onfiden e interval limits does not ontains $ero&

v) "7


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iv) "he onfiden e interval limits does not ontains $ero&

Comments:7

"he mean differen e of the t%o paired varia+les i&e& urrent and +e#innin# salary is si#nifi ant or not same&

!ne-;ay AN!,A:

AO@


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Analy$e Compare Means @ne7 ay AO@


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!utput:

"he a+ove ta+le #ives the test results for the analysis of one7%ay AO@


24/98

In t#o-;ay Analysis< %e have t%o independent varia+les or !no%n fa tors and %e are interested in !no%in# theireffe t on the same dependent varia+le&

;ata /our e:C:X/P//,age: 3? : - i 4 - A? : - i = - Y for all i

/P// Oeed:/P// need t%o types of varia+les for analy$in# t%o7%ay AO@


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!utput:



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Between-Subjects Factors

" 9" 6

C" 7;2< 7

l(ry 7i**ell

1.002.003.00

=a >a)ede*i)n

1.002.003.00

randna'e

Value Label ?

"his ta+le sho%s the value la+el under ea h ate#ory and the fre2uen y of ea h value la+el& e have totaled S valuela+els under pa !a#e desi#n and +rand name&

Tests of Between-Subjects Effects

Dependent Variable: =re eren e

a

!37.231 2 26 .616 16. 3 .00036.10 2 1 .0!4 1.13! .3!1

206. 33 13 1!.91037! .000 22

S(ur e

pa >a)ebrand&rr(r

(tal

ype Su'( S8uare* d +ean S8uare Si).

< S8uared @ .763 # dAu*ted < S8uared @ .617%a.

"he a+ove ta+le #ives the test results for the analysis of t%o7%ay AO@


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Pac>age "esign

As our null hypothesis for pa !a#e desi#n is re e ted* so multiple omparisons are used to assess that %hi h #roup

mean is different from the others& "he a+ove ta+le #ives the results for multiple omparisons +et%een ea h valuela+el under pa !a#e desi#n ate#ory&

"he Post7 o test presents the result of the omparison +et%een all the possi+le pairs& /in e %e have three #roups*a total of six pairs %ill +e possi+le in %hi h three %ill +e mirror ima#es& "he results are sho%n in three ro%s& "he p7value for A@ = 8@ and A@ = C@ omparison is sho%n as 3&333* %hereas it is 3&R for 8@ = C@ omparison& "hismeans that the avera#e preferen e for pa !a#e desi#n +et%een AZ and BZ as %ell as AZ and CZ are si#nifi antlydifferent* %hereas the same is not si#nifi antly different +et%een BZ CZ&

Conclusion: As our null hypothesis for pa !a#e desi#n is re e ted and %e on lude that all mean preferen es for pa !a#e desi#n are not same& "o identify the mean %hi h is different from other %e used V/; test and on lude that AO@a)e de*i)n"

C" "C"

""

# % =a,>a)e de*i)n "

"

C"

+eanDi eren,e

# -$% Std. &rr(r Si). L(/er (und pper (und9! C(n iden,e nter al

a*ed (n (b*er ed 'ean*.

5e 'ean di eren,e i* *i)ni i,ant at t5e .0! le el.".


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Chi-S uare goo"ness of fit test is used %hen the distri+ution is non7normal and the sample si$e is less than R3* sothe hi7s2uare #oodness of fit test determines %hether the distri+ution follo%s uniform distri+ution or not&

;ata /our e:C:X/P//,


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,xplanation of Graph

From the a+ove #raph %e see that our numeri al varia+le (pri e) is on x7axis and its fre2uen y on the y7axis& "he

mean and standard deviation of o+servations are &33 and 3&T T respe tively&"he a+ove #raph learly sho%s that the sele ted numeri al varia+le i&e& pri e does not follo% a normal distri+ution*

so %e use hi7s2uare #oodness of fit test to determine if the sample under investi#ation has +een dra%n from a population* %hi h follo%s some spe ified distri+ution&

Method:

First of all enter the data in the data editor and the varia+les are la+eled as pri1e & Cli ! onAnalyze %hi h %ill produ e a drop do%n menu* hoose non-parametri1 test from that and li ! on C i-sBuare test * a dialo#ue +oxappears* in %hi h all the input varia+les appear in the left7hand side of that +ox& /ele t the varia+le you %ant toanaly$e& hen you sele t the test varia+le* the arro% +et%een the t%o +oxes %ill no% a tive and you an transferthe varia+le on the +ox la+eled test variable list +y li !in# on the arro%& In this ase our test varia+le is pri1e andit should +e transferred to the test varia+le +ox& \ou also li ! on the options +utton* if you are interested to !no%the des riptive statisti s of the tested varia+le& Oo% li ! on OK to run the analysis&


Analy3e Non-parametric test Chi-s uare %efine test ,aria$le list

!7



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!utput

Price

7.3 .76 7.3 -1.3

7.3 .722

$1.19 $1.39$1.!9

(tal

Bb*er ed ? & pe ted ?


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Cross ta+ulation is used to examine the variation in the ate#ori al data* it is a ross measurin# analysis& A+ove %eare ross examine the #ender and desi#nation of the employees&

e ta!e desi#nation of the employees in the olumn and #ender of the employees in ro%* and %e have totaled Q Qo+servations&

"he results are #iven in t%o ro%sK the first ro% sho%s the num+er of female employees in ea h employmentate#ory&

"he se ond ro% sho%s the num+er of male employees in ea h employment ate#ory&

C&i-S uare Tests

79.277 a 2 .000474

=ear*(n C5i-S8uare? ( Valid Ca*e*

Value d *y'p. Si).

#2-*ided%

0 ell* #.0 % 5a e e pe ted (unt le** t5an !. 5e

'ini'u' e pe ted (unt i* 12.30.

a.

"he a+ove ta+le #ives the test results for the hi7s2uare test for independen e& "he first ro% la+eled Pearson Chi- S uare sho%s that the value of ] is 8& %ith de#ree of freedom& "he t%o7tailed p7value is sho%n as 3&333*%hi h is less than 3&3H* so %e an re e t our null hypothesis and on lude that the ;esi#nation is not independent of/ex&

Conclusion: "he test results are statisti ally si#nifi ant at H level of si#nifi an e and the data provide suffi ienteviden e to on lude that the desi#nation of the employees is not independent of the sex* and %e are almost '33

onfident on our de ision and the re e tion of the null hypothesis&

/e ond Approa h

Consider a ase in %hi h the data is not availa+le and only the ta+le la+eled as 6en"er@Employment CategoryCrossta$ulation in the a+ove output* is #iven& @n the +asis of the output ta+le you an easily find out the sameresult as a+ove +y usin# /P// %ei#ht ases options& Belo% %e +riefly explain that ho% to enter the data on the +asisof the ta+le and to find out the desired results&

Method

First of all* in the varia+le vie% of /P// define three varia+les and la+eled them as :ender *EmploymentCate ory and *alue & Oo% on the data vie% of the /P//* enter the data in a different manner& e see that the ta+le

ontains t%o ro%s and R olumns& In ro% %e have t%o ate#ories i&e& Female and MaleK similarly in olumns %ehave three ate#ories i&e& Cleri al* Custodial* and Mana#er& Oo% the female and male employees* +oth are fall inthe three employment ate#ory&

/o in the data vie% %e simply define the ro% data i&e& Gender and its opposite define the olumn data i&e& ,mployment ate#ory and its orrespondin# fre2uen ies in the


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&

After definin# the data ust li ! on Data * %hi h %ill produ e a drop do%n menu* hoose 6ei t 1ases from that* adialo#ue +ox appears in %hi h all the varia+les are on the left hand side of that +ox& "i ! 6ei t 1ases by and dra#value in the +ox la+eled ;reBuen1y *ariable +y li !in# on the arro% +et%een the t%o +oxes& Oo% li ! OK toreturn to the previous %indo%&

"he Further pro ess is same as des ri+ed a+ove& ust define Gender in ro% and ,mployment ate#ory in Column&"i ! C i-sBuare +y li !in# on the Statisti1s +utton& Oo% li !OK to run the analysis& hen the output appears*

you %ill see that /P// %ill #ive you the same result as %e find out earlier throu#h data&

Regression Analysis Regression is the relationship between selected values of independent variable andobserved values of dependent variable , from which the most probable value of dependentvariable can be predicted for any value of independent variable.

The use of regression to make quantitative predictions of one variable from the values ofanother variable is called regression analysis . There are following several types ofregression, which may be used by the researcher.

Linear regression Multiple linear regression

Quadratic / urvilinear regression Logistic / !inary logistic regression Multivariate logistic regression

Linear Regression

"hen one dependent variable depends on single independent variable then theirdependency called linear regression and its model is given by



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y = a + bx

"here, y is a depending variable x is a independent variable

a is called the regression constant b is called the regression coefficient

Regression Coefficient

#egression coefficient is a measure of how strongly the independent variable predicts thedependent variable. There are two types of regression coefficient.

$n%standardi&ed coefficients 'tandardi&ed coefficients commonly known as Beta .

The un-standardized coefficients can be used in the equation as coefficients of differentindependent variables along with the constant term to predict the value of dependentvariable.

The standardized coefficient is, however, measured, in standard deviations. The betavalue of ( associated with a particular independent variable indicates that a change of )standard deviation in that particular independent variable will result in change of ( standarddeviations in the dependent variable.

Data Source: *+' ''- L+-mployee 0ata

Variables: 1ere we are interested to analy&e two numerical variables i.e.

urrent salary 23umerical4 !eginning salary 23umerical4

Hypothesis:

1 5* #egression coefficient is &ero.1 * #egression coefficient is not &ero.

SPSS Need: ' '' need two numerical variables and both should be scaled.



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Method:

The given data is entered in the data editor and the variables are labeled as current salaryand beginning salary . lick on Analyze which will produce a drop down menu, chooseRegression from that and click on Linear , a dialogue bo6 appears, in which all the inputvariables appear in the left%hand side of that bo6. Transfer the dependent variable into theright%hand side bo6 labeled Dependent . Transfer the independent variable into the bo6labeled Independent(s) . 7n our case, current salary is a dependent variable and beginningsalary is an independent variable. 3e6t we have to select the method for analysis in the bo6labeled Method . ' '' gives five options here* Enter , Stepwise , Remo e , !orward , and"ac#ward . 7n the absence of a strong theoretical reason for using a particular method,Enter should be used. The bo6 labeled Selection ariable is used if we want to restrict theanalysis to cases satisfying particular selection criteria. The bo6 labeled $ase labels is usedfor designating a variable to identify points on plots.

fter making the appropriate selections click on Statistics button. This will produce adialogue bo6 labeled Linear regression% Statistics . Tick against the statistics you want inthe output. The Estimates option gives the estimate of regression coefficient. The Model&it option gives the fit indices for the overall model. !esides these the R'S uared change option is used to get the incremental #%square value when the models change. 8theroptions are not commonly used. lick on the $ontinue button to return to the maindialogue bo6.

The lots button in the main dialogue bo6 may be used for producing histograms andnormal probability plots of residual. The Sa e button can be used to save statistics like

predicted values, residuals, and distances. The options button can be used to specify thecriteria for stepwise regression.

3ow click on *+ in the main dialogue bo6 to run the analysis.

Pictorial Representation

Analyze Regression Linear Define DV & V !lots "ic# $istogra% & or%al !robability !lot '(



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OUTPUT



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Coefficients a

192 .206 .6 0 2.170 .0311.909 .047 . 0 40.276 .000

#C(n*tant%e)innin) Salary

+(del1

Std. &rr(r

n*tandardi edC(e i ient*

eta

Standardi edC(e i ient*

t Si).

Dependent Variable: Current Salarya.

The above table gives the regression constant and coefficient and their significance. Theseregression coefficient and constant can be used to construct an ordinary least squares28L'4 equation and also to test the hypothesis of the independent variable. $sing theregression coefficient and the constant term given under the column labeled !? one canconstruct the 8L' equation for predicting the current salary i.e.

urrent salary = ./012034 + 5.2/3/6 5Beginning salary6

3ow we test our hypothesis, we see that the p%value for regression coefficient of beginningsalary is given by 5.555, which is less that 5.5:, so we can re


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and ) respectively, which shows that the fitted model is best and the chances of error isminimum.

or%al ! 7! !lot of Regression )tandardized Residual

1.00.0.60.40.20.0

Obser)e# Cum Prob

1.0

0.

0.6

0.4

0.2

0.0

E - p e c

t e # C u m

P r o

b

+epen#ent Variable, Current Salar!

The above 3ormal probability plot of regression standardi&ed residual shows the regressionline which touches ma6imum number of points presents in the model and it also shows theaccuracy of the fitted model.

)catter !lot

7.!!.02.!0.0-2.!-!.0

(e%ression Stu#enti*e# +elete# .Press/ (esi#ual

$140,000

$120,000

$100,000

$ 0,000

$60,000

$40,000

$20,000

$0

C u

r r e n

t S a

l a r !

+epen#ent Variable, Current Salar!



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The above scatter plot also shows the adequacy of the fitted model as we can see that thedata is scattered and it does not follow any particular pattern, so we can say that the fittedmodel has minimum chances of error.

Multiple Regression Hierarchical Method!

Multiple regression is the most commonly used technique to assess the relationshipbetween one dependent variable and several independent variables. There are three ma


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The further procedure and uses of advance options for e6tra results are discussed earlier inthe case of linear regression. 3ow after making appropriate selections of options for betterresults click on *+ to run the analysis.

OUTPUT

Variables Entere#'(emo)e# b

e)innin) Salary a . &nter &du ati(nal Le el #year*%a . &nter +(nt5* *in e Eire a . &nter

+(del123

Variable* &nteredVariable*


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dependent variable. "e can see the #%square change value in row three is 5.55>. Thismeans that the inclusion of month since hire after beginning salary and educational levelhelps in e6plaining the additional 5.>; variance in the current salary of the employees. The

p%value for all three models shows that our value falls in the critical region, so we can re.(5= B 2).A5A4 2!'49'D:L 0 ' @ %9>5>.9)C B 2).=9D4 2!'4 B 2)5(5.DA54 2-L49'D:L ; ' @ )AA>=.:5 B 2).=>A4 2!'4 B 2A==.)594 2-L4 B 2)::.95)4 2M'14

3ow we test our hypothesis, we see that the p%value for regression coefficient in all threemodels is less that 5.5:, so we can re


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Charts:

This model also produces three diagrams for the standardi&ed residual i.e. -istogram ,.ormal robability lot , and Scatter lot . The charts and its interpretation are almostsame as we discuss under the case of linear regression. 'o we are not describing thesecharts and its interpretations again.

Cur"ilinear # $uadratic Regression

The relationship between variables when the regression equation is nonlinear i.e. quadraticor higher order then their dependency called curvilinear or quadratic regression. There maybe more than one dependent variable depending on one independent variable.

Data Source: *+' ''- L+-mployee 0ata

Variables: 1ere we are interested to analy&e three numerical variables i.e.

urrent salary 23umerical4 !eginning salary 23umerical4 -ducational level 23umerical4

Hypothesis:

1 5* #egression coefficient is &ero.1 * #egression coefficient is not &ero.

SPSS Need: ' '' need two numerical variables and both should be scaled.

Method:

The given data is entered in the data editor and the variables are labeled as currentsalary , beginning salary and Educational le el . lick on Analyze which will produce adrop down menu, choose Regression from that and click on $ur e Estimation , a dialoguebo6 appears, in which all the input variables appear in the left%hand side of that bo6.Transfer the dependent variable into the right%hand side bo6 labeled Dependent(s) .Transfer the independent variable into the bo6 labeled Independent . 7n our case, current



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salary and !eginning salary are dependent variables and -ducational level is anindependent variable.

3ow choose an appropriate model you want by ticking its bo6 appearing below the window

labeled $ur e Estimation . 7n this case we choose /uadratic model by ticking itscorresponding bo6.

The Sa e button can be used to save statistics like predicted values, residuals, and predicted intervals. 3ow click on *+ in the main dialogue bo6 to run the analysis.


Analyze Regression urve :sti%ation Define DVs and V "ic# uadratic '(



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OUTPUT

Mo#el +escription

+BDF2Current Salary

e)innin) Salary

Guadrati&du ati(nal Le el #year*%

n luded

n*pe i ied

.0001

+(del ?a'e12

Dependent Variable

1&8uati(nndependent Variable

C(n*tantVariable 5(*e Value* Label Bb*er ati(n* in=l(t*

(leran e (r &nterin) er'* in &8uati(n*

The above table gives the description of the model. 7n this case we have two dependentvariables i.e. urrent salary and !eginning salary along with one independent variable i.e.

-ducational level 2years4.



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Case Processin% Summar!

474000

(tal Ca*e*& luded Ca*e* a

(re a*ted Ca*e*?e/ly Created Ca*e*

?

Ca*e* /it5 a 'i**in) alue in anyariable are e luded r(' t5e analy*i*.

a.

The above table shows the number of cases fall in the selected model. 7n our case the totalnumber of cases is C9C, with no e6cluded or missing cases.

The above table gives the test results for the quadratic regression. #%value shows thecorrelation between the observed and e6pected values of the dependent variables. 7n thiscase the E%value is given by DD9.(C=, with level of significance equals 5.555 which is lessthat 5.5:. This means that our value falls in the critical region, so we can re


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Scatter Plots

$140,000

$120,000

$100,000

$ 0,000

$60,000

$40,000

$20,000

$0

2220116141210

E#ucational 0e)el ears

GuadratiBb*er ed

Current Salar!

$ 0,000

$60,000

$40,000

$20,000

$0

2220116141210

E#ucational 0e)el .!ears/

Guadrati

Bb*er ed

Be%innin% Salar!



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The above charts for residuals of dependent variables clearly show that the residual valuesare not scattered and it follows a particular pattern, this means that the fitted model is notgood.

Lin%er &ype Scaling

Lin#er type scaling is a method used for nominations on categorical data in order to makethe categorical data meaningful, when we have to apply some statistical test on the data.Through this scaling approach the ranks assign can be treated as numerical values and itsstatement at once.

'uppose we have to collect the data about the awareness, preferences, usage, likeness, anddislikeness as well as the agreement with any statement that should return in qualitativeform and we have to record the responses, which can be analy&e statistically. 'o in thiscondition we use lin#er type scaling .

Data Source: #$3 ++temp+temp+ li #a&a+Mateen.sav

Hypothesis:

1 5* FMale @ F Eemale1 * FMale F Eemale

Variables:

1ere we are interested to analy&e two categorical variables i.e.

Gender. 2 ategorical4 reference of cellular service with respect to network coverage.

2 ategorical but treated as 3umerical4

1ere we consider the preference of cellular service as a numerical variable and statisticallytest the hypothesis that 9ean preference of %ale and fe%ale over cellular service>it8 respect to net>or# coverage is sa%e . The method, we use to test the abovehypothesis is ndependent sa%ples t-test .

Method:

-nter the data in the data editor and the variables are labeled as 0ender and pre&erence .lick on Analyze which will produce a drop down menu, choose $ompare means from

that and click on independent samples t'test , a dialogue bo6 appears, in which all theinput variables appear in the left%hand side of that bo6. To perform the independentsamples t%test, transfer the dependent variable into the test ariable bo6 and transfer thevariable that identifies the groups into the grouping ariable bo6. 7n this case, the

re&erence is the dependent variable to be analy&ed and should be transferred into test

http://smb//temp/Teacher's%20%20temorary%20data/ali%20raza/RUN%20/temp/temp/Alihttp://smb//temp/Teacher's%20%20temorary%20data/ali%20raza/RUN%20/temp/temp/Ali


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variable bo6. 0ender is the variable which will identify the groups and it should betransferred into the grouping variable bo6.

8nce the grouping variable is transferred, the de&ine groups button which was earlierinactive turns active. lick on it to define the two groups. 7n this case group1 represents

Male and group2 represents female. Therefore put ) in the bo6 against group) and ( in thebo6 against group( and click continue. 3ow click on *+ to run the analysis.

Pictorial Representation naly&e ompare Means 7ndependent%'amples T Test 0rag Test H Grouping ariable 0efine Groups 8I



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OUTPUT

"roup Sta tistics

100 4.12 . 91 .0 9

100 4.32 .61 .062

ender (ntain (t5'ale and e'ale+ale

e'ale

ide net/(r> ( era)e'(ti ate t5e indu idualt( pre er a parti ular

ellulare *er i e

? +ean Std. De iati(nStd. &rr(r

+ean

This table contains the descriptive statistics for both groups. "e have taken (55observations for the independent samples t%Test in which )55 belongs to male category and)55 to female category. The column labeled 9ean shows that the mean preferences ofcellular service with respect to network coverage for both groups are appro6imately ? . Thismeans that both groups are Agree that wide network coverage motivate the individual to

prefer a particular cellular service.

The above table contains the test statistics for independent samples t%test.

Levene@s "est< The table contains two sets of analysis, the first one assuming equalvariances in the two groups and the second one assuming unequal variances. The LeveneJstest tells us which statistic to consider analy&ing the equality of means. The p%value forLeveneJs test is given by 32.3 , which is greater than 323 . Therefore, the statisticassociated with equal variances assumed should be used for the t%test for equality of meansof two independent populations.

!-Value< shows that the value of our test statistic does not fall in the critical region i.e.3234 K 323 so we can accept our 3ull 1ypothesis i.e. C 9ale = C ,e%ale

onclusion< The test results are statistically significant at A:; confidence level and thedata provide sufficient evidence to conclude that the mean preference of cellular service


1n#epen#ent Sa mples Test

2.730 .100 -1. 4! 19 .067 -.200 .10 -.414 .014

-1. 4! 176.31 .067 -.200 .10 -.414 .014

&8ual arian e*a**u'ed&8ual arian e*n(t a**u'ed

ide net/(r> ( era)e'(ti ate t5e indu idualt( pre er a parti ular

ellulare *er i e

Si).

Le eneH*e*t (r

&8uality ( Varian e*

t d Si).

#2-tailed%+ean

Di eren eStd. &rr(r Di eren e L(/er pper

9!C(n iden enter al ( t5eDi eren e

t-te*t (r &8uality ( +ean*


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with respect to network coverage for male and female is same and there is only =.9;chance of re


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hoose appropriate Model by clicking on that bo6, here we choose Alpha as model. 3owclick on Statistics button, a dialogue bo6 appears. Tick the corresponding bo6 which youwant to analy&e in the output. 3ow click $ontinue to return to the main dialogue bo6. lick

on *+ to run the analysis.

Pictorial Representation naly&e 'cale #eliability nalysis0rag 7tems hoose Model Give 'tatistics 8I



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OUTPUT


2440 100.00 .0

2440 100.0

Valid& ludeda

(tal

Ca*e*?

Li*t/i*e deleti(n ba*ed (n allariable* in t5e pr( edure.

a.

The above table shows the total number of cases fall in the data set. "e have (CC5observations with no missing and e6cluded cases.

(eliabilit! Statistics

.!76 !

Cr(nba 5H* lp5a ? ( te'*

The above table shows the test results for the reliability analysis. The value of ronbachJs lpha is given by 32 4 ? the number of items in the data set is . The value associatedwith lpha is said to be !oor and the conclusions draw from this data is not reliable tounderstand and forecast.

1tem-Total Statistics

1641!1.7603 !!3319661 .6 .4 0

140212.214 4646009 01 .!0! .43

132761.3423 !160 1!037 .314 .!331064!4.2! 7 192 !23141 .!6! .477

1 1191.6111 6 01!3713 -.032 .61!

pprai*ed Land Value pprai*ed Value (

'pr( e'ent*(tal pprai*ed Value

Sale =ri e


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item is deleted from the analysis and retests the reliability of the entire data, the value ofronbachJs lpha becomes 324. . 'o, in order to improve the value of lpha to make our

data set more reliable we delete the last item and retest the value of our ronbachJs lpha.

(eliabilit! Statistics

.61! 4

Cr(nba 5H* lp5a ? ( te'*

1ere we retest our data after deletion of one item and our new value of lpha is given by324. . 3ow the total number of items in the entire data set is ? . The value associated with

lpha in this set of reliability statistics is said to be Acceptable and the conclusions drawfrom this data is reliable to understand and forecast.

1tem-Total Statistics

1641!0.!7 !!3319 039 .6 .!40

140211.03 464600 210 .!0! .493

132760.16 !160 13 63 .314 .!991064!3.07 192 !3033! .!6! .!36

pprai*ed Land Value pprai*ed Value (

'pr( e'ent*(tal pprai*ed Value

Sale =ri e

S ale +ean i te' Deleted

S aleVarian e i te' Deleted

C(rre tedte'- (tal

C(rrelati(n

Cr(nba 5H* lp5a i te'

Deleted

This table shows that if we delete any other item from the data set and retest the reliability,then our value of lpha becomes !oor . !ecause all the values associated with theremaining four items in last column of the above table is less than the current value of our

ronbachJs lpha i.e. 324. . 'o we donNt need to further retest the reliability of the dataset, which means the data is reliable at the current value of our ronbachJs lpha.

Correlation Analysis

orrelation refers to the degree of relation between two numerical variables. 7t is denotedby r , which is typically known as orrelation oefficient .



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Correlation Coefficient

The orrelation coefficient gives the mathematical value for measuring the strength ofthe linear relation between two variables. Mathematically the value of r always lay

between %) and ) with*2a4 B) representing absolute positive linear relationship 2as O increases, P

increases4.2b4 5 representing no linear relationship 2O and P have no pattern4.2c4 %) representing absolute inverse relationship 2as O increases, P, 0ecreases4.

'i"ariate Correlation

Bivariate correlation tests the strength of relationship between two variables withoutgiving any consideration to the interference some other variables might cause to therelationship between the two variables being tested. Eor e6ample, while testing thecorrelation between the urrent and Beginning salary of the employees, bivariatecorrelation will not consider the impact of some other variables like :ducational Level and!revious :xperience of the employees. 7n such cases, a bivariate analysis may show us astrong relationship between urrent and !eginning salary? but in reality, this strongrelationship could be the result of some other e6traneous factors like -ducational Level and

revious -6perience etc.

Data Source:

*+' ''- L+-mployee data

Hypothesis:

1 5* There is no orrelation between ariables 5r =361 * There is some orrelation between ariables 5r 36

Variables: 1ere we are interested to analy&e three numerical variables i.e.

urrent salary 23umerical4 !eginning salary 23umerical4

-ducational Level 2years4 23umerical4

Technically correlation analysis can be run with any kind of data, but the output will be ofno use if a correlation is run on a categorical variable with more than two categories. Eore6ample, in a data set, if the respondents are categori&ed according to nationalities andreligions, correlation between these variables is meaningless.



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OUTPUT

Correlations

1 . 0"" .661"".000 .000

474 474 474. 0"" 1 .633"".000 .000474 474 474

.661"" .633"" 1

.000 .000474 474 474

=ear*(n C(rrelati(nSi). #2-tailed%?=ear*(n C(rrelati(nSi). #2-tailed%?=ear*(n C(rrelati(nSi). #2-tailed%?

Current Salary

e)innin) Salary

&du ati(nal Le el #year*%

Current Salarye)innin)Salary

&du ati(nalLe el #year*%

C(rrelati(n i* *i)ni i ant at t5e 0.01 le el #2-tailed%."".

The above table gives the correlation for all pairs of variables and each correlation is produced twice in the matri6. 'o here we get following D correlations for the given data.

urrent salary and !eginning salary urrent salary and -ducational level !eginning salary and -ducational level

The value of correlation coefficient is . in the cells where ' '' compare two same variables2 urrent salary and urrent salary and so on4. This means that there is a perfect

positive correlation between the variables.

7n each cell of the correlation matri6, we get earsonJs correlation coefficient, p%value fortwo%tailed test of significance and the sample si&e. Erom the output we can see that thecorrelation coefficient between urrent salary and Beginning salary is 3211 and the p%value for two%tailed test of significance is less than 323 . Erom these figures we canconclude that there is a strong positive correlation between urrent salary and beginningsalary and that this correlation is significant at the significance level of 5.5).

'imilarly, the correlation coefficient for urrent salary and :ducational level is 3244. .'o there is a moderate positive correlation between these variables.

The correlation coefficient for Beginning salary and :ducational level is 324;; and its p%

value is given by 5.555, so we can re


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Partial Correlation

!artial correlation allows us to e6amine the correlation between two variables whilecontrolling for the effects of one or more of the additional variables without throwing out

any of the data.7n other words, it is the degree of relation between the dependent variable and one of theindependent variable by controlling the effect of other independent variables, because weknow that, in a multiple regression model, one dependent variable depends on two or moreindependent variables.

Data Source:

*+' ''- L+-mployee data

Hypothesis:

1 5* There is no orrelation between ariables 5r =361 * There is some orrelation between ariables 5r 36

Variables: 1ere we are interested to analy&e two numerical variables, while controlling one additionalvariable.

urrent salary !eginning salary -ducational level 2 ontrol variable4

SPSS Need:

' '' need two or more numerical variables to perform partial orrelation.

Method:

-nter the data in the data editor and labeled them. lick on Analyze which will produce adrop down menu, choose $orrelate from that and click on artial , a dialogue bo6 appears,in which all the input variables appear in the left%hand side of that bo6. To perform the

partial correlation, transfer the variables for which you want to know the correlationbetween them in the bo6 labeled 4ariables , while controlling the effect of one or more

additional variables by transferring them to the bo6 labeled $ontrolling &or .

7n our case, we want to find the correlation between $urrent salary and beginning salaryof the employees, so these variables should be transferred to the bo6 labeled 4ariables ,while controlling for the effect of Educational le el and re ious e3perience of theemployees by transferring them to the bo6 labeled $ontrolling &or . 3ow click on *+ to runthe analysis.



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Pictorial Representation naly&e orrelate artial

0rag ariables 0rag ontrolling ariables 8I



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OUTPUT

Correlations

1.000 .79!. .000

0 471.79! 1.000.000 .471 0

C(rrelati(nSi)ni i an e #2-tailed%

d C(rrelati(nSi)ni i an e #2-tailed%d

Current Salary

e)innin) Salary

C(ntr(l Variable*&du ati(nal Le el #year*%

Current Salarye)innin)Salary

The above table shows the test results for the partial correlation between urrent salary and beginning salary of the employees. The variable we are controlling for in the analysisis :ducational level , and it is shown in the left%hand side of the table.

"e can see that the correlation coefficient between urrent salary and Beginning salary is 32 / , which is considerably smaller as compared to 3211 in case of Bivariate . Thismeans that both the variables are still have positive correlation, but the value of correlationcoefficient decreased if we control for the :ducational level of the employees and thevariables are no longer strongly correlated with each other.

onclusion< The test results are significant at :; level of significance and the data providesufficient evidence to conclude that there is some correlation present between the urrentsalary and !eginning salary of the employees, but it is considerably smaller in the case of

partial correlation than in case of bivariate correlation.



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Logistic Regression

Logistic regression starts in )955. 7f a categorical variable depends on any numerical orcategorical variable then their dependency may called the logistic regression. 7t is used to

predict a discrete outcome based on variables may be discrete, continuous, or mi6ed. Thuswhen the dependent variable is categorical with two or more than two discrete outcomes,logistic regression is a commonly used technique. 7t has the following two types*

!inary logistic regression / Logit Multinomial logistic regression

Coefficient of Logistic Regression

Logistic regression computes the log odds for a particular outcome. The odds of anoutcome is given by the ratio of the probability of it happening and not happening as E! F5.-!6G , where ! is the probability of an event. There are some mathematical problems inreporting these odds, so natural logarithms of these odds are calculated. positive valueindicates that odds are in favor of the event and the event is likely to occur while a negativevalue indicates that odds are against the events and the event is not likely to occur. Theformula to do so may be written either

'inary Logistic Regression

7f a categorical variable having only two levels -.g. Male and Eemale, Pes or 3o, Good and!ad etc, and it is depending on different categorical or numerical independent variablesthen their relation can be referred as Binary logistic regression2 The e6pression forbinary logistic regression may be given as

H = b 3 + b . I . + b 0 I 0 + JJJ22b # I #

Data Source: *+' ''- L+ ML 'urvival

Hypothesis:

1 5* #egression coefficients are &ero1 * #egression coefficients are not &ero



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Variables:1ere we are interested to analy&e three 0ifferent variables i.e.

'tatus 2 ategorical4 Time 23umerical4 hemotherapy 2 ategorical4

1ere )tatus is our dependent variable depending on "i%e and 8e%ot8erapy . s in thiscase our dependent variable is categorical having only two levels i.e. ensored andRelapsed , so we use binary logistic regression to analy&e the dependency between thevariables.

SPSS Need:

' '' need one dependent variable and it must be ategorical, while the independentvariables can be categorical as well as numerical.

Method:

Eirstly the data is entered in the data editor and the variables are labeled as Status , 6ime ,and $hemotherapy . lick on Analyze which will produce a drop down menu, chooseRegression from that and click on "inary Logistics , a dialogue bo6 appears, in which allthe input variables appear in the left%hand side of that bo6. To perform the "inaryLogistics Regression , transfer the dependent variable in the bo6 labeled Dependent andthe independent variables in the bo6 labeled $o ariates .

7n our case, Status is an only dependent variable and should be transfer to the bo6 labeledDependent . 6ime and $hemotherapy are independent variables and should be transferto the bo6 labeled $o ariates .

3e6t we have to select the method of for analysis in the bo6 labeled Method . ' '' givesseven options, of which the Enter method is most commonly used. Eor common purposeone does not need to use the Sa e and *ptions buttons. dvance users may e6perimentwith these. The Sa e button can be used to save statistics like predicted values, residuals,and distances. The options button can be used to specify the criteria for stepwiseregression.

fter making appropriate selections, click on *+ to run the analysis.



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Pictorial Representation naly&e #egression !inary Logistic 0rag 0ependent 0rag ovariates 8I



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OUTPUT


23 100.00 .0

23 100.00 .0

23 100.0

n/ei)5ted Ca*e* a

n luded in naly*i*+i**in) Ca*e*

(tal

Sele ted Ca*e*

n*ele ted Ca*e*(tal

? =er ent

/ei)5t i* in e e t, *ee la**i i ati(n table (r t5e t(talnu'ber ( a*e*.

a.

The above table gives the description of cases selected for the analysis. "e have totaled (Dcases included in the analysis with no missing and unselected cases.

+epen#ent Variable Enco#in%

01

Bri)inal ValueCen*(red


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The above table shows the observed or actual number of cases fall in each category of thedependent variable. The last column labeled !ercentage orrect shows that our modelcan predict 5; status of the censored patients and )55; status for the relapsed patients.8verall, our model can predict 9>.D; status of the patients.

Block 1: Method = Enter

Omnibus Tests of Mo#el Coefficients

4.609 2 .1004.609 2 .1004.609 2 .100

Stepl( >

+(del

Step 1C5i-*8uare d Si).

The above table reports significance levels by the traditional chi%square method. 7t tests ifthe model with the predictors is significantly different from the model. The omnibus testmay be interpreted as a test of the capability of all predictors in the model eli5((d

C( I Snell< S8uare

?a)el>er>e< S8uare

&*ti'ati(n ter'inated at iterati(n nu'ber ! be au*epara'eter e*ti'ate* 5an)ed by le** t5an .001.

a.

The above table gives the ox & )nell R-)*uare value, which gives an appro6imationabout how much variance in the dependent variable can be e6plained with the hypothesi&edmodel. 7n this case "i%e and 8e%ot8erapy can e6plain .120K of the patientJs current)tatus .

Classification Table a

1 4 20.00 1 100.0

2.6

Bb*er edCen*(red


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The above lassification table summari&es the results of our predictions about patientJs'tatus based on Time and hemotherapy. "e can see that our model can correctly predict(5; status of censored patients and )55; status of the relapsed patients. 8verall, ourmodel predicts >(.=; status of the patients.

Variables in t&e E uation

-1.49 1.262 1.409 1 .23! .224-.024 .024 1.0!! 1 .304 .9762.962 1.207 6.02! 1 .014 19.332

5e'(ti'eC(n*tant

Step1

a

S.&. ald d Si). & p# %

Variable#*% entered (n *tep 1: 5e'(, ti'e.a.

The above table gives the !eta coefficients for the independent variables along with theirsignificance. 3egative beta coefficients for time and chemotherapy mean that withincreasing chemotherapy and time of the treatment, it chances for the patient of having arelapsed status. 'ame as Multiple linear regression models, we can construct an 8L'equation for the status of the patient by the help of above regression constant andcoefficients. The e6pression for status of the patient is given by*

)tatus = 02/40 + 5-.2?/16 5 8e%ot8erapy6 + 5-3230?6 5"i%e6

The last column labeled :xp5B6 takes a value of more than one, if the beta coefficients are positive and less than one, if it is negative . 7n our case, the beta coefficients for

hemotherapy and time are negative, so coefficients are having the values of less than onein column labeled :xp5B62 value of 32/ 4 for Time indicates that for . >ee# increase inthe treatment, the odds of a patient having a relapsed status increases by a factor of32/ 4 . These values can also use to construct an equation for the odds of a patient, and itis given by*

@ )

) B e -5./2;;0 + 3200? + 32/ 4 "6

Non(Para)etric &ests

on-!ara%etric tests are used to test the hypothesis regarding the population parameters of non%normal data with small sample si&e 2less than D54. These tests aresometimes also referred as Distribution-,ree tests



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'ino)ial &est

Bino%ial tests are used to test the hypothesis regarding the population proportion. 7t runson a categorical variable having two levels only.

Data Source: *+' ''- L+ arpet

Hypothesis:

1 5* @ 5.: 1 * 5.:

Variables:

1ere we are interested to analy&e a categorical variable i.e. $ouse #eeping )eal . 7n ourcase a superstore owner claims that :5; of their customers got house keeping seal on the

purchase of the product.

SPSS Need: ' '' need one categorical variable 2( levels only4.

Method:

Eirstly the data is entered in the data editor and the variable is labeled as -ouse #eepingseal . lick on Analyze which will produce a drop down menu, choose .on' arametric6ests from that and click on "inomial , a dialogue bo6 appears, in which all the inputvariables appear in the left%hand side of that bo6. To perform the "inomial test , transferthe test variable in the bo6 labeled 6est ariable list . 7n our case -ouse #eeping seal isa test variable, so it should be transferred to the bo6 labeled Test variable list by clicking onthe arrow between the two bo6es. 3ow give the test value in the bo6 below labeled as 6est

roportion . 7n our case the test value is 32 3 .



naly&e 3on% arametric tests !inomial 0rag test ariable Give test roportion 8I



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OUTPUT

NPar Tests

Binomial Test

Je* .36 .!0 .2 6?( 14 .64

22 1.00

r(up 1r(up 2(tal

((d E(u*e>eepin) *ealCate)(ry ?

Bb*er ed=r(p. e*t =r(p.

& a t Si).#2-tailed%

The above table gives the test results for the !inomial 3on%parametric test.

The first column labeled ategory gives the two categories 2Pes or 3o4 of the test variablei.e. Good 1ouse keeping seal.

The second column labeled as gives the total number cases analy&ed, and also thenumber of cases fall in each category of our test variable. 7n this case we have selected thesample of (( persons out of which > persons say Hes they got the house keeping seal andthe remaining says o .

The third column labeled as 'bserved !roportion gives the percentage of the personssaying Pes or 3o. D=; individuals says Pes they got the house keeping seal while =C;individuals says 3o.

The last column gives the p-value for the (%tailed test and it is given by 32014 , which is

greater than 323 , so we can accept our null hypothesis and conclude that the claim of thesuperstore owner is correct, the proportion is 5.:5.



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naly&e 3on% arametric tests #uns 0rag test ariable Tick !o6 2Median4 8I



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OUTPUT

(uns Test

11.!011112213

.21

. 27

e*t Value a

Ca*e* K e*t ValueCa*e* @ e*t Value

(tal Ca*e*?u'ber (


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*ne Sa)ple +(S &est

'ne sa%ple (-) "est is used to test the goodness of fit of any specific distribution for thegiven data. This distribution is called Iolmogrov%'mirnov R commonly known as 3on%

parametric hi%square .

Data Source:

*+' ''- L+ arpet

Hypothesis:

1 5* Eit is Good 20ata follows the fitted distribution4

1 * Eit is not Good 20ata does not follow the fitted distribution4

Variables:



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1ere we are interested to analy&e a numerical variable i.e. !rice .

SPSS Need:

' '' need a numerical variable with small sample si&e.

Method:

Eirstly the data is entered in the data editor and the variable is labeled as rice . lick onAnalyze which will produce a drop down menu, choose .on' arametric 6ests from thatand click on 1'sample +'S , a dialogue bo6 appears, in which all the input variables appearin the left%hand side of that bo6. To perform the +'S test , transfer the test variable in thebo6 labeled 6est ariable list . 7n our case rice is a test variable, so it should be

transferred to the bo6 labeled Test variable list by clicking on the arrow between the twobo6es. 3ow tick the bo6 in the section labeled 6est Distribution at the bottom of thedialogue bo6. 7n our case the fitted distribution is oisson , so we tick the correspondingbo6 labeled oisson.



naly&e 3on% arametric tests #uns

0rag test ariable Tick !o6 2Median4 8I



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OUTPUT

One-Sample 3olmo%oro)-Smirno) Te st

222.0000

.143

.143

-.13!.670.760

?+ean=(i**(n =ara'eter a,b

b*(lute=(*iti e?e)ati e

+(*t & tre'eDi eren e*

;(l'()(r( -S'irn( M *y'p. Si). #2-tailed%

=ri e

e*t di*tributi(n i* =(i**(n.a.

Cal ulated r(' data.b.



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The above table gives the test results for the one sa%ple (-) test . "e have taken ((observations for the analysis. The mean of oisson distribution calculated from data is givenby 0 .

The row labeled Absolute gives the difference between e6treme values i.e. e6tremely highvalues and -6tremely Low values and it given by 32.?; .

The row labeled !ositive gives the difference between the Ma6imum and Minimum values,when we subtract Minimum value from ma6imum value and it is 32.?; .

The row labeled egative also gives the same as row labeled ositive, but here we subtractMa6imum value from the Minimum value, so the resulted value is given by 5-32.; 6 .

The (ol%ogrov-)%irnov N value is given by 324 , which calculated from the formula.

The last row gives the p-value for the analysis. 7n our case the p%value is given by 32 4 ,and it is greater than 323 . 'o we can accept our null hypothesis and conclude this test asthe Eit is Good and the data follows the oisson distribution.



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4atri of Nonparametric Statistics ^^^^^^&&^&^^^^^^^^^^^^^^&'8

Nonparametric Tests

'& @ne7 ay Chi7/2uare test and its assumptions& @+served and expe ted fre2uen ies

R& ;e#rees of freedom in a @ne7 ay Chi7/2uare test Q& "%o7 ay Chi7/2uare test H& ,xpe ted fre2uen ies and the rule of independent pro+a+ilitiesS& ;e#rees of freedom in a "%o7 ay Chi7/2uare test

& Chi7/2uare and the pro+lem of small expe ted fre2uen iesT& emedies for small expe ted fre2uen ies8& \ates s Corre tion for Continuity'3& Fisher s ,xa t Pro+a+ility "est ''& Assumptions of the "%o7 ay Chi7/2uare test ' & "he Median "est 'R& ;ealin# %ith tied ran!s in the Median "est 'Q& ;e#rees of freedom in the Median "est



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'H& Assumption of the Median "est 'S& "he Mann7 hitney U "est ' & "he normal distri+ution and the Mann7 hitney U "est %ith lar#e samples'T& Assumptions of the Mann7 hitney U "est '8& Drus!al7 allis @ne7 ay AO@


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Nonparametric Statistic:-

A group of statistical procedures that have two things in co on !

;esi#ned to +e used %ith nominal and[or ordinal varia+les

Ma!e fe% or no assumptions a+out population parameters

Statistical Po#er:-

All other thin#s +ein# e2ual* nonparametri te hni2ues are less po%erful tests of si#nifi an e than their parametriounterparts&

Assumin# that the null hypothesis is false*

Oonparametri tests have less po%er to dis over si#nifi ant relationships than parametri tests&

The follo#ing are "escri$e" some non-parametric test:-

1) !ne-;ay Chi-S uare Test:-

A test that co pares the o"served fre#uenc$ of a varia"le in a single group with what would "e thee%pected "$ chance&

'%a ple (elon$ cases are assigned at rando to four district cri inal courts& )he annual o"served

fre#uenc$ of drug cases is given "elow& *ince the nu "er of cases in each court is not the sa e is thecase assign ent s$ste not rando ,

District ourt1.2 nd 1/0 th 2 th 234th

22 234 211 103

)otal nu "er of cases = /03

5f the assign ent process is trul$ rando we would e%pect that 6/03 7 48 = 224 cases would "e assignedto each court "$ chance&

9ow uch do the o"served fre#uencies 6f o8 differ fro the e%pected fre#uencies 6f e = 2248,



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A one-wa$ chi-s#uare has df = 6? < 18& (or the court e%a ple df = 64 < 18 = . 6? = @ courts8

)he critical value of 2 for . df at = & is >&/1 &

*ince 11&4 > >&/1 the null h$pothesis is re ected and it is concluded that the assign ent s$ste is notrando &

Analysis t rou S (elon$ asesB$ Cretrial *tatus

5n ail :n Bail E:E

.2 21 1>

5f the null h$pothesis assu es e#ual nu "ers of cases in each status !

Do the o"served fre#uencies differ significantl$ fro this assu ption,

S


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Chi7s2uare 4 H&' '*

df 4 (! 7 ') 4 (R7') 4

p 4 3&3 H

Test Statistics

!.1712

.07!

C5i-S8uare a

d *y'p. Si).

=


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/0 ) e Median )est -

Fsed to deter ine whether two rando sa ples co e fro populations with the sa e edians&

Assess ent enter Eating "$ )wo )ea sG:fficer andidates Eando l$ Assigned to Assess ent )ea s

)ea A )ea B*u" ect Eating *u" ect Eating

Holdin >2 :lsen 0>esani 3> * ither >3

Critchard /> )rantha /.Birdwell 43 Hordon 30

haveI / Hraha 3:JNeal 3. Andel 3/

ohnson /4 9utton 02)ate . Caul //Bird 32 KcHuire >4Luni >> osto >.

o pton /2 Eaines 3Mewis /0 Battan 4

MitI ann 4.

Step # - Deter ine the edian rating for the two assess ent groups co "ined

Kidpoint = 6N + 18 7 2 = 62 + 18 7 2 = 1.

Ean?ing of the co "ined ratingsof the two groups

0> 3002 3//0 3>// 3/> 3./4 32/. //2 3

>> 4>3 .>4 43>. 4.>2

edian of co "ined groups



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Step / - Deter ine the nu "er of officers in either group whose ratings were e#ual to or a"ove theedian and the nu "er not a"ove the edian&

)ea A )ea B

A"ove Kdn 3 >

Below Kdn 3 3

Step ! Eun a two-wa$ chi-s#uare to deter ine whether there is an association "etweenassess ent tea and the ratings&

6'%pected fre#uencies are in parentheses8

Cosition )ea A )ea B )otal

A"ove Kdn 3 63&248 > 63&>38 1.

Below Kdn 3 6 &>38 3 63&248 12

)otal 12 1. 2

2 = 63 < 3&2482 7 3&24 + 6> < 3&>382 7 3&>3 + 63 < &>382 7 &>3 + 63 < 3&2482 7 3&24

2 = & .3>

3nterpretation -

)he critical value of 2 for 1 df at = & is .&/41&

*ince & .3> < .&/41 the null h$pothesis is accepted&

5t is concluded that there is no significant difference in the edian ratings given "$ thetwo assess ent tea s&



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!0 ) e Mann-= itney )est -

Fsed to co pare the ran?s of two independent groups co para"le to the purpose of the t test&

5t is ore powerfu

Documents

Final Report on Qtia