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Inferential Statistics

Chapter ElevenDr Nek Kamal Yeop YunusFaculty of business & economics

Sultan Idris Education niversity

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Inferential StatisticsChapter Eleven

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!hat are Inferential Statistics"

#efer to certain procedures that allo$ researchers tomake inferences about a population based on dataobtained from a sample%

btainin' a random sample is desirable since it

ensures that this sample is representative of a lar'erpopulation%

(he better a sample represents a population) themore researchers $ill be able to make inferences%

*akin' inferences about populations is $hatInferential Statistics are all about%

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($o Samples from ($o Distinct+opulations ,Fi'ure --%-.

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Samplin' Error

It is reasonable to assume that each sample$ill 'ive you a fairly accurate picture of itspopulation%

/o$ever) samples are not likely to be identicalto their parent populations%

(his difference bet$een a sample and itspopulation is kno$n as Samplin' Error% ,seeFi'ure --%0.

Furthermore) no t$o samples $ill be identicalin all their characteristics%

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Samplin' Error ,Fi'ure --%0.

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Distribution of Sample *eans

(here are times $here lar'e collections of randomsamples do pattern themselves in $ays that $illallo$ researchers to predict accurately somecharacteristics of the population from $hich the

sample $as taken%

1 samplin' distribution of means is a fre2uencydistribution resultin' from plottin' the means of avery lar'e number of samples from the same

population

Refer to Figure 11.3

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1 Samplin' Distribution of *eans ,Fi'ure--%3.

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Distribution of Sample *eans,Fi'ure --%4.

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Standard Error of the *ean

(he standard deviation of a samplin' distribution ofmeans is called the Standard Error of the *ean,SE*.%

If you can accurately estimate the mean and thestandard deviation of the samplin' distribution)you can determine $hether it is likely or not that aparticular sample mean could be obtained from thepopulation%

(o estimate the SE*) divide the SD of the sampleby the s2uare root of the sample si5e minus one%

Refer to Figure 11.4

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Confidence Intervals

1 Confidence Interval is a re'ion e6tendin' bothabove and belo$ a sample statistic $ithin $hich apopulation parameter may be said to fall $ith aspecified probability of bein' $ron'%

SE*7s can be used to determine boundaries orlimits) $ithin $hich the population mean lies%

If a confidence interval is 89:) there $ould be a;probability7 that 9 out of -

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(he 89 percent Confidence Interval,Fi'ure --%9.

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(he 88 percent Confidence Interval,Fi'ure --%=.

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!e Can >e 88 percent Confident,Fi'ure --%?.

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Does a Sample Difference #eflect a

+opulation Difference" ,Fi'ure --%@.

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Distribution of the Difference >et$eenSample *eans ,Fi'ure --%8.

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Confidence Intervals ,Fi'ure --%-

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/ypothesis (estin' /ypothesis testin' is a $ay of determinin' the

probability that an obtained sample statistic $ill occur)'iven a hypothetical population parameter%

(he #esearch /ypothesis specifies the predicted

outcome of a study% (he Null /ypothesis typically specifies that there is no

relationship in the population%

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#esearch and Null /ypotheses,Fi'ure --%--.

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/ypothesis (estin'B 1 #evie$ State the research hypothesis

State the null hypothesis

Determine the sample statistics pertinent to thehypothesis

Determine the probability of obtainin' the sampleresults

If the probability is small) reAect the null hypothesisand affirm the research hypothesis

If the probability is lar'e) do not reAect the nullhypothesis and do not affirm the research hypothesis

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+ractical vs% Statistical Si'nificance

(he terms si'nificance level or level ofsi'nificance refers to the probability of a samplestatistic occurrin' as a result of samplin' error%

Si'nificance levels most commonly used ineducational research are the %

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ne and ($otailed (ests

1 onetailed test is $hen the researcher obtainsa positive difference bet$een the sample mean$hich $ill support the hypothesis) $hen usin'only the positive tail of the samplin' distribution%,Fi'ure --%-3.

1 t$otailed test involves the use of probabilitiesbased on both sides of a samplin' distributionbecause the research hypothesis is a non

directional hypothesis%

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Si'nificance 1rea for anetailed (est ,Fi'ure --%-3.

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netailed (est sin' a Distribution of

Differences >et$een Sample *eans,Fi'ure

--%-4.

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Contin'ency Coefficient alues for DifferentSi5edCrossbreak (ables ,(able --%-.

Size of Table Upper limita

(No. of Cells) for C Calculated

2 by 2 .71

3 by 3 .82

4 by 4 .87

5 by 5 .89

6 by 6 .91

aThe upper limit for unequal-size tables !su"h as 2 by 3 or 3 by 4# are un$no%n but "an be estimate

from the &alues 'i&en. Thus( the upper imit for a 3 by 4 table %oul appro)imate .85

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Commonly sed Inferential (echni2ues,(able --%0.

Parametric Nonparametric

Quantitative t-test for inepenent means *ann-+hitney Utest

t-test for "orrelate means ,rus$al-+allis one-%ay analysis of &arian"e

nalysis of &arian"e !/0# i'n test

nalysis of "o&arian"e !/0# rieman t%o-%ay analysis of &arian"e

*ulti&ariate analysis of &arian"e !*/0#

t-test for r

Categorical t-test for ifferen"e in proportions hi square

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(ype I vs% (ype II Error 1 null hypothesis predicts no relationship% 1 (ype II error results $hen the researcher

fails to reAect the null hypothesis that is false%

1 (ype I error results $hen the researcherreAects the null $hen it is true% Fi'ure --%-= provides an e6ample of (ype -

and (ype II errors%

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/ypothetical E6ample of (ype I and

(ype II Errors ,Fi'ure --%-=.

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#eAectin' the Null /ypothesis,Fi'ure --%-?.

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Inference (echni2ues

(here are t$o basic types of inferencetechni2uesB

-. +arametricB makes assumptions about the

nature of the population from $hich the samplesinvolved in the research study $ere taken

0. NonparametricB makes fe$ assumptions aboutthe nature of the population from $hich thesamples are taken

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1n Illustration of +o$er nder an 1ssumed+opulation alue

,Fi'ure --%-@.

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+arametric (echni2ues for 1naly5in'Guantitative Data

(he ttest is a parametric statistical test usedto see $hether a difference bet$een the meansof t$o samples is si'nificant%

(here are t$o forms of ttestsB

-. (test for correlated means

0. (test for independent means

1nalysis of ariance ,1N1. is used todetermine if si'nificant differences e6ist

bet$een t$o or more 'roups%

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+arametric (echni2ues for1naly5in' Guantitative Data ,cont%.

1nalysis of Covariance ,1NC1. is a variation ofan 1N1 used $hen 'roups are 'iven a pretestrelated in some $ay to the dependent variable andtheir mean scores on this pretest are found to

differ% *ultivariate 1nalysis of ariance ,*1N1.incorporates t$o or more dependent variables inthe same analysis) thus permittin' a morepo$erful test of differences amon' means%

(test for r is used to see $hether a correlationcoefficient calculated on sample data is si'nificant%

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Non+arametric (echni2ues for1naly5in' Guantitative Data

(he *ann!hitney test is a nonparametricalternative to the t test used $hen a researcher$ishes to analy5e ranked data

(he Kruskal!allis one$ay analysis of variance isused $hen you have t$o or more independentvariables to compare

(he Si'n test is used $hen you $ant to analy5e t$orelated samples% #elated samples are connected insome $ay

(he Friedman t$o$ay analysis of variance is used$hen t$o or more related 'roups are involved

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(echni2ues for *easurin'Cate'orical Data

+arametric (echni2ue (test for +roportions ,findin' differences in

proportions $ithin cate'ories.

Non+arametric (echni2ue Chis2uare test is used to analy5e data that

are reported in cate'ories

(he Contin'ency Coefficient is a descriptive

statistic indicatin' the de'ree of relationshipthat e6ists bet$een t$o cate'orical variables

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+o$er of a Statistical (est

+o$er is the probability that the test $illcorrectly lead to the conclusion that there isa difference $hen) it fact) a difference

e6ists% +arametric tests are 'enerally) but not

al$ays) more po$erful than nonparametrictests%

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1 +o$er Curve ,Fi'ure --%-8.

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1ny 2uestions"

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(hank You