217054454 Hypothesis Testing

8/19/2019 217054454 Hypothesis Testing

1/43

HYPOTHESIS TESTING

INTRODUCTION

Definition

Hypothesis tests are procedures for making rational decisions about thereality of effects.

Rational Decisions

Most decisions require that an individual select a single alternative from anumber of possible alternatives. The decision is made without knowingwhether or not it is correct; that is, it is based on incomplete information.

For example, a person either takes or does not take an umbrella to school based upon both the weather report and observation of outside conditions.If it is not currentl raining, this decision must be made with incompleteinformation.

! rational decision is characteri"ed b the use of a procedure whichinsures the likelihood or probabilit that success is incorporated into thedecision#making process. The procedure must be stated in such a fashionthat another individual, using the same information, would make the

same decision.$ne is reminded of a %T!& T&'( episode. )aptain (irk, for one reasonor another, is stranded on a planet without his communicator and isunable to get back to the 'nterprise. %pock has assumed command and is being attacked b (lingons *who else+. %pock asks for and receivesinformation about the location of the enem , but is unable to act becausehe does not have complete information. )aptain (irk arrives at the lastmoment and saves the da because he can act on incomplete information.

This stor goes against the concept of rational man. %pock, being theultimate rational man, would not be immobili"ed b indecision. Instead,he would have selected the alternative which reali"ed the greatestexpected benefit given the information available. If complete informationwere required to make decisions, few decisions would be made brational men and women. This is obviousl not the case. The script writermisunderstood %pock and rational man.


2/43

Effects

-hen a change in one thing is associated with a change in another, wehave aneffect . The changes ma be either quantitative or qualitative, with

the h pothesis testing procedure selected based upon the t pe of changeobserved. For example, if changes in salt intake in a diet are associatedwith activit level in children, we sa an effect occurred. In another case,if the distribution of political part preference *&epublicans, emocrats,or Independents+ differs for sex *Male or Female+, then an effect is present. Much of the behavioral science is directed toward discoveringand understanding effects.

The effects discussed in the remainder of this text appear as variousstatistics including/ differences between means, contingenc tables, andcorrelation coefficients.

GENERAL PRINCIPLES

!ll h pothesis tests conform to similar principles and proceed with thesame sequence of events.

• ! model of the world is created in which there are no effects. Theexperiment is then repeated an infinite number of times.

•

The results of the experiment are compared with the model of stepone. If, given the model, the results are unlikel , then the model isre0ected and the effects are accepted as real. If, the results could beexplained b the model, the model must be retained. In the lattercase no decision can be made about the realit of effects.

1 pothesis testing is equivalent to the geometrical concept of h pothesisnegation. That is, if one wishes to prove that ! *the h pothesis+ is true,one first assumes that it isn2t true. If it is shown that this assumption islogicall impossible, then the original h pothesis is proven. In the case of h pothesis testing the h pothesis ma never be proven; rather, it isdecided that the model of no effects is unlikel enough that the oppositeh pothesis, that of real effects, must be true.

!n analogous situation exists with respect to h pothesis testing instatistics. In h pothesis testing one wishes to show real effects of anexperiment. 3 showing that the experimental results were unlikel ,given that there were no effects, one madecide that the effects are, infact, real. The h pothesis that there were no effects is called theNULLHYPOTHESIS . The s mbol 14 is used to abbreviate the 5ull

6


3/43

1 pothesis in statistics. 5ote that, unlike geometr , wecannot prove theeffects are real, rather wemay decide the effects are real.

For example, suppose the following probabilit model *distribution+

described the state of the world. In this case the decision would be thatthere were no effects; the null h pothesis is true.

'vent ! might be considered fairl likel ,given the above model was correct. !s a result the model would be

retained, along with the 5788 19:$T1'%I%. 'vent 3 on the other handis unlikel , given the model. 1ere the model would be re0ected, alongwith the 5788 19:$T1'%I%.

The Model

TheSAMPLING DISTRIBUTION is adistribution of a sample statistic . It is used as a model of what would happen if

.+ the null h pothesis were true *there reall were no effects+, and

6.+ the experiment was repeated an infinite number of times.

3ecause of its importance in h pothesis testing, the sampling distributionwill be discussed in a separate chapter.

P o!a!ilit"

P o!a!ilit" is a theory of uncertainty . It is a necessar concept becausethe world according to the scientist is unknowable in its entiret .1owever, prediction and decisions are obviousl possible. !s such, probabilit theor is a rational means of dealing with an uncertain world.

:robabilities are numbers associated with events that range from "ero toone *4# +. ! probabilit of "ero means that the event is impossible. Forexample, if I were to flip a coin, the probabilit of a leg is "ero, due to thefact that a coin ma have a head or tail, but not a leg. iven a probabilitof one, however, the event is certain. For example, if I flip a coin the probabilit of heads, tails, or an edge is one, because the coin must takeone of these possibilities.


4/43

In real life, most events have probabilities between these two extremes.For instance, the probabilit of rain tonight is .=4; tomorrow night the probabilit is . 4. Thus it can be said that rain is more likel tonight thantomorrow.

The meaning of the term probabilit depends upon one2s philosophicalorientation. In the )8!%%I)!8 approach, probabilities refer to therelative frequenc of an event, given the experiment was repeated aninfinite number of times. For example, the .=4 probabilit of rain tonightmeans that if the exact conditions of this evening were repeated aninfinite number of times, it would rain =4> of the time.

In the %ub0ective approach, however, the term probabilit refers to a?degree of belief.? That is, the individual assigning the number .=4 to the probabilit of rain tonight believes that, on a scale from 4 to , thelikelihood of rain is .=4. This leads to a branch of statistics called?3!9'%I!5 %T!TI%TI)%.? -hile man statisticians take this approach,it is not usuall taught at the introductor level. !t this point in time allthe introductor student needs to know is that a person calling themselvesa ?3a esian %tatistician? is not ignorant of statistics. Most likel , he orshe is simpl involved in the theor of statistics.

5o matter what theoretical position is taken, all probabilities must

conform to certain rules. %ome of the rules are concerned with how probabilities combine with one another to form new probabilities. Forexample, when events are independent, that is, one doesn2t effect theother, the probabilities ma be multiplied together to find the probabilitof the 0oint event. The probabilit of rain toda !5 the probabilit ofgetting a head when flipping a coin is the product of the two individual probabilities.

! deck of cards illustrates other principles of probabilit theor . In

bridge, poker, rumm , etc., the probabilit of a heart can be found bdividing thirteen, the number of hearts, b fift #two, the number of cards,assuming each card is equall likel to be drawn. The probabilit of aqueen is four *the number of queens+ divided b the number of cards. The probabilit of a queen $& a heart is sixteen divided b fift #two. Thisfigure is computed b adding the probabilit of hearts to the probabilitof a queen, and then subtracting the probabilit of a queen !5 a heartwhich equals @A6.

!n introductor mathematical probabilit and statistics course usuall begins with the principles of probabilit and proceeds to the applicationsof these principles. $ne problem a student might encounter concerns

=


5/43

unsorted socks in a sock drawer. %uppose one has twent #five pairs ofunsorted socks in a sock drawer. -hat is the probabilit of drawing outtwo socks at random and getting a pairB -hat is the probabilit of gettinga match to one of the first two when drawing out a third sockB 1ow man

socks on the average would need to be drawn before one could expect tofind a pairB This problem is rather difficult and will not be solved here, but is used to illustrate the t pe of problem found in mathematicalstatistics.

HYPOTHESIS TESTING PROCESS

1 pothesis testing is a s stematic method used to evaluate data and aid

the decision#making process. Following is a t pical series of stepsinvolved in h pothesis testing/. %tate the h potheses of interest

6. etermine the appropriate test statistic


6/43

average age of entering college freshmen is 6 ears.1 4The average age of entering college freshman E 6 earsIf the data one collects and anal "es indicates that the average age ofentering college freshmen is greater than or less than 6 ears, the nullh pothesis is re0ected. In this case the alternative h pothesis could bestated in the following three wa s/ * + the average age of entering collegefreshman is not 6 ears *the average age of entering college freshmen H6 +; *6+ the average age of entering college freshman is less than 6 ears*the average age of entering college freshmen 6 +; or *


7/43

longer perform better on tests. %pecificall , the research suggests thatstudents who spend four hours stud ing for an exam will get a betterscore than those who stud two hours. In this case the h potheses might be/1 4The average test scores of students who stud = hours for the test Ethe average test scores of those who stud 6 hours.1 The average test score of students who stud = hours for the test theaverage test scores of those who stud 6 hours.!s a result of the statistical anal sis, the null h pothesis can be rejected or not rejected. !s a principle of rigorous scientific method,this subtle but important point means that the null h pothesis cannot beaccepted. If the null is re0ected, the alternative h pothesis can be

accepted; however, if the null is not re0ected, we can2t conclude that thenull h pothesis is true. The rationale is that evidence that supports ah pothesis is not conclusive, but evidence that negates a h pothesis isample to discredit a h pothesis. The anal sis of stud time and test scores provides an example. If the results of one stud indicate that the testscores of students who stud = hours are significantl better than the testscores of students who stud two hours, the null h pothesis can bere0ected because the researcher has found one case when the null is nottrue. 1owever, if the results of the stud indicate that the test scores ofthose who stud = hours are not significantl better than those who stud6 hours, the null h pothesis cannot be re0ected. $ne also cannot concludethat the null h pothesis is accepted because these results are onl one setof score comparisons. Nust because the null h pothesis is true in onesituation does not mean it is alwa s true.


8/43

DETERMINING THE APPROPRIATE TEST STATISTIC#

The appropriate test statistic *the statistic to be used in statisticalh pothesis testing+ is based on various characteristics of the sample population of interest, includingsample si"e and distribution. The teststatistic can assume man numerical values. %ince the value of the teststatistic has a significant effect on the decision, one must use theappropriate statistic in order to obtain meaningful results. Most teststatistics follow this general pattern/For example, the appropriate statistic to use when testing a h pothesisabout a population means is/In this formula O E test statistic, E mean of the sample, P E mean of theΧ

population, Q E standard deviation of the sample, and R E number in thesample.

SPECI$YING THE STATISTICAL SIGNI$ICANCE LE%EL#

!s previousl noted, one can re0ect a null h pothesis or fail to re0ect anull h pothesis. ! null h pothesis that is re0ected ma , in realit , be trueor false. !dditionall , a null h pothesis that fails to be re0ected ma , inrealit , be true or false. The outcome that a researcher desires is to re0ect afalse null h pothesis or to fail to re0ect a true null h pothesis. 1owever,there alwa s is the possibilit of re0ecting a true h pothesis or failing tore0ect a false h pothesis.&e0ecting a null h pothesis that is true is called a T pe I error and failingto re0ect a false null h pothesis is called a T pe II error. The probabilitof committing a T pe I error is termed S and the probabilit ofcommitting a T pe II error is termed . !s the value of S increases, the probabilit of committing a T pe I error increases. !s the value of increases, the probabilit of committing a T pe II error increases. -hileone would like to decrease the probabilit of committing of both t pes oferrors, the reduction of S results in the increase of and vice versa. The best wa to reduce the probabilit of decreasing both t pes of error is toincrease sample si"e.The probabilit of committing a T pe I error, S, is called the level ofsignificance. 3efore data is collected one must specif a level ofsignificance, or the probabilit of committing a T pe I error *re0ecting atrue null h pothesis+. There is an inverse relationship between a

researcher2s desire to avoid making a T pe I error and the selected valueof S; if not making the error is particularl important, a low probabilit of

U

http://www.referenceforbusiness.com/knowledge/Sample_size_determination.htmlhttp://www.referenceforbusiness.com/knowledge/Sample_size_determination.html


9/43

making the error is sought. The greater the desire is to not re0ect a truenull h pothesis, the lower the selected value of S. In theor , the value of Scan be an value between 4 and . 1owever, the most common valuesused in social science research are .4A, .4 , and .44 , which respectivelcorrespond to the levels of VA percent, VV percent, and VV.V percentlikelihood that a T pe I error is not being made. Thetradeoff for choosinga higher level of certaint *significance+ is that it will take much strongerstatistical evidence to ever re0ect the null h pothesis.

'T'&MI5I5 T1' ')I%I$5 &78'.3efore data are collected and anal "ed it is necessar to determine underwhat circumstances the null h pothesis will be re0ected or fail to be

re0ected. The decision rule can be stated in terms of the computed teststatistic, or in probabilistic terms. The same decision will be reachedregardless of which method is chosen.

)$88')TI5 T1' !T! !5 :'&F$&MI5 T1')!8)78!TI$5%.

The method of data collection is determined earl in the research process.$nce a research question is determined, one must make decisionsregarding what t pe of data is needed and how the data will be collected.This decision establishes the bases for how the data will be anal "ed. $neshould use onl approved research methods for collecting and anal "ingdata.

')I I5 -1'T1'& T$ &'N')T T1' 5788 19:$T1'%I%.This step involves the application of the decision rule. The decision ruleallows one to re0ect or fail to re0ect the null h pothesis. If one re0ects thenull h pothesis, the alternative h pothesis can be accepted. 1owever, as

discussed earlier, if one fails to re0ect the null he or she can onl suggestthat the null ma be true.

'W!M:8'.W9O )orporation is a compan that is focused on a stable workforce thathas ver little turnover. W9O has been in business for A4 ears and hasmore than 4,444 emplo ees. The compan has alwa s promoted the ideathat its emplo ees sta with them for a ver long time, and it has used thefollowing line in itsrecruitment brochures/ ?The average tenure of ouremplo ees is 64 ears.? %ince W9O isn2t quite sure if that statement is still

V

http://www.referenceforbusiness.com/knowledge/Trade_off.htmlhttp://www.referenceforbusiness.com/knowledge/Recruitment.htmlhttp://www.referenceforbusiness.com/knowledge/Trade_off.htmlhttp://www.referenceforbusiness.com/knowledge/Recruitment.html


10/43

true, a random sample of 44 emplo ees is taken and the average ageturns out to be V ears with a standard deviation of 6 ears. )an W9Ocontinue to make its claim, or does it need to make a changeB

. %tate the h potheses.1 4 E 64 ears1 H 64 ears

6. etermine the test statistic. %ince we are testing a population meanthat is normall distributed, the appropriate test statistic is/


11/43

H"&othesis testin'#

O!(ecti)es• ealing with paired parametric data

• )omparing confidence intervals and p values

In covering these ob0ectives the following terms will be introduced/• :arametric and non#parametric anal sis

• :aired " test

• :aired t test

-e have shown previousl that statistical inference enables generalconclusions to be drawn from specific data. For example estimating a population2s mean from a sample mean. !t first glance this ma notappear important. In practice however the abilit to make theseestimations is fundamental to most medical investigations. These tend toconcentrate on dealing with one or more of the following questions/

1ave the observations changed with time and@or interventionB

o two or more groups of observations differ from each otherB

Is there an association between different observationsB

To answer these questions man different t pes of statistical inferencetests have been developed to deal with var ing sample si"es and differentt pes of data. Though the tests differ the have the common aim ofassessing whether the null h pothesis is likel to be correct *box +. Theare known collectivel as Ztests of significance[. Bo* + The null h pothesisThere is no difference between the groups with respect to themeasurement made.

The significance test chosen is dependent upon the t pe of data we aredealing with, whether it has a normal distribution and the t pe of question being asked.6 $nce the distribution of the data is known, ou can tell ifthe null h pothesis should be tested using parametric or non#parametricmethods.

http://emj.bmj.com/content/18/2/124.full#ref-2http://emj.bmj.com/content/18/2/124.full#ref-2


12/43

Pa a,et ic and non-&a a,et ic anal"sis

PARAMETRIC ANALYSIS

! normal distribution is a regular shape. !s such it is possible to draw thecurve exactl b simpl knowing the mean, standard deviation andvariance of the data. -hen considering a normal distribution of a population these features are known as parameters. :arametric anal sisrelies on the data being normall *or nearl + distributed so that anestimation of the underl ing population2s parameters can be made.< Thesecan then be used to test the null h pothesis. !s onl quantitative data canhave a normal distribution, it follows that parametric anal sis can onl beused on quantitative data .

(e point!ll parametric tests use quantitative data but not all quantitative data haveto be anal sed using parametric tests.

NON-PARAMETRIC ANALYSIS

These tests of the null h pothesis do not assume an particulardistribution for the data. Instead the look at the categor or rank order of the values and ignore the absolute difference between them.)onsequentl non#parametric anal sis is used on nominal and ordinaldata as well as quantitative data that are not normall *or nearl normall +distributed *table +.

If a difference exists between the stud groups, it is more likel to befound using parametric tests. It is therefore important to know for certain

if the data are normall distributed. 9ou can sometimes determine this bchecking the distribution curve of the plotted data. ! more formal wa isto use a computer to show how precisel the data fit with a normaldistribution. This will be described in greater detail in the next article.-hen data are not normall distributed attempts are often made totransform it so that parametric anal sis can be carried out. Thecommonest method used is logarithmic transformation.6 This has theadded advantage of allowing geometric means and confidence intervals to be calculated that have the same units as the original data.

(e points

6

http://emj.bmj.com/content/18/2/124.full#ref-3http://emj.bmj.com/content/18/2/124.full#ref-2http://emj.bmj.com/content/18/2/124.full#ref-3http://emj.bmj.com/content/18/2/124.full#ref-2


13/43

• 5on#parametric anal sis can be used on an data but parametricanal sis can onl be used when the data are normall distributed.

• :rovided the are appropriatel used, parametric tests derive more

information about the whole population than non#parametric ones.1aving determined that parametric anal sis is appropriate ou then needto select the best statistical test. This depends upon the si"e of the samplesand the t pe of question being asked. In all cases however the followings stematic approach is used *box 6+.

Bo* . % stem for statistical comparison of two groups• %tate the null h pothesis and the alternative h pothesis of the stud

• %elect the level of significance

• 'stablish the critical values• %elect a sample and calculate its mean and standard error of the

mean *%'M+\• )alculate the test statistic

• )ompare the calculated test statistic with the critical values

• 'xpress the chances of obtaining results at least this extreme if thenull h pothesis is true

Pai ed o inde&endent &a a,et ic anal"sisTwo t pes of parametric statistical tests can be used to compare themeans of two stud groups. The choice depends upon whether the data

ou are dealing with are independent or paired.ata can be considered to be paired when two related observations are

taken with anal sis concentrating on the difference between the pairedscores. 'xamples of these t pe of data include/

• Z3efore[ and Zafter[ studies carried out on the same sub0ects• $bservations made on individuall matched pairs where onl the

factor under investigation is different Independent data are when the

sub0ects for the two groups are picked at random such that selection forone group will not effect the sub0ects chosen for the other. The tests used


14/43

to anal se these data will be discussed in the next article. For now we willconcentrate on paired data.

•

PAIRED TESTS3oth " andt tests can be used to investigate the difference between themeans of two sets of paired observations. The former is, however, onlvalid if the samples are sufficientl large.=-hen this is not the case wecan use thet statistic provided that the population of the mean differencesin scores between the pairs is approximatel normall distributed.= !sthis is often the case the pairedt test, rather than its " counterpart, is morecommonl seen in the medical literature. To show how these tests areapplied consider the following examples.

PAIRED / TESTr 'gbert 'verard continues to work in the 'mergenc epartment ofeathstar eneral. 1is consultant, r )anute, asks him to find out if

5everwhee"e, the new bronchodilator for asthmatics, significantlchanges patient2s peak flow rate *:F&+. To do this he follows thes stematic approach described in box 6.

+ State the n0ll h"&othesis and alte nati)e h"&othesis of the st0d"1aving considered the problem, 'gbert writes down the null h pothesisas/

ZThere is no difference in asthmatic patient2s :F& before and afterreceiving 5everwhee"e[.

This can be summarised to/

Mean difference in :F& E 4

The alternative h pothesis is the logical opposite of this, that is/

ZThere is a difference in asthmatic patient2s :F& before and afterreceiving 5everwhee"e[.

This can be summarised to/

Mean difference in :F& H 4

. Select the le)el of si'nificance

=

http://emj.bmj.com/content/18/2/124.full#ref-4http://emj.bmj.com/content/18/2/124.full#ref-4http://emj.bmj.com/content/18/2/124.full#ref-4http://emj.bmj.com/content/18/2/124.full#ref-4


15/43

If the null h pothesis is correct the :F& before and after 5everwhee"eshould be the same. 1owever, even if this were true it would be verunlikel the would be exactl the same because of random variation between patients. 9ou would expect some :F& to increase and others tofall. $verall however the mean difference between the two groups should be "ero if the null h pothesis is valid.

-hen groups are widel separated it is likel that the null h pothesis isnot valid. For example, if 44 people died in our department one daand none the next then its highl unlikel that a difference this big would be attributable to chance. In contrast ou would be less confident to ruleout the effect of chance if the difference in death rate was onl . Thequestion therefore is how big does the difference need to be before the

null h pothesis can be re0ectedB&ather than guessing, it is better to consider what values are possible. Ifwe measured the mean :F& difference in groups of asthmatic patientsselected randoml we would find the sample means form a normaldistribution around the population2s mean difference. !s it is a normaldistribution it is possible to convert this to a standard normaldistribution.A The probabilit of getting a particular sample mean can then be read from the table of " statistics where/

" E Ksample mean difference Ypopulation mean difference *P+L@standarderror of the mean *%'M+

-here/

%'M E population standard deviation *Q+@]number in the sample *n+

In this case we do not know the value of Q. 5evertheless, provided thesample si"e is large enough *that is, greater than or equal to 44+ the "statistic can still be used.= This relies on the fact that a valid estimation of

the population2s standard deviation can be derived from the sample data*s+.=

(e point:rovided the sample is 44/

%'M E '%'M E s@]n

3 convention, the outer 4.46A probabilities *that is, the tips of the twotails representing 6.A> of the area under the curve+ are considered to besufficientl awa from the population mean as to represent values that

A

http://emj.bmj.com/content/18/2/124.full#ref-5http://emj.bmj.com/content/18/2/124.full#ref-4http://emj.bmj.com/content/18/2/124.full#ref-4http://emj.bmj.com/content/18/2/124.full#ref-5http://emj.bmj.com/content/18/2/124.full#ref-4http://emj.bmj.com/content/18/2/124.full#ref-4


16/43

cannot be simpl attributed to chance variation *fig +. )onsequentl , ifthe sample mean is found to lie in either of these two tails then the nullh pothesis is re0ected. )onversel , if the sample mean lies within thesetwo extremes then the null h pothesis will be accepted *fig +. In doingthis we are accepting that normal samples that fall into these two tails will be incorrectl labelled as being abnormal. Therefore a total of A> of all possible sample means from the normal population will incorrectl re0ectthe null h pothesis.

Figure 1 Random sampling distribution of mean differences for a hypothetical population. μ hyp = mean of the hypothetical population. Z crit = criticalvalue of z separating the areas of acceptance and rejection of the nullhypothesis.Following convention, 'gbert picks a significance level of 4.4A for hisstud . 1e now needs to determine the :F& that demarcates this level of

probabilit . These are known as the critical values.3 Establish the critical values7sing the " table 'gbert finds that the critical value *")&IT+ demarcatingthe middle VA> of the distribution curve is " E X@Y .VC *fig +. In otherwords a " value of X@Y .VC separates the middle VA> area of acceptanceof the null h pothesis from two 6.A> areas of re0ection.-ith the null and alternative h potheses defined, and the critical valuesestablished *")&IT+, the patients for the stud can now be selected. The "statistic derived from the sample *")!8) + can then be determined.

C

http://emj.bmj.com/content/18/2/124.full#ref-1http://emj.bmj.com/content/18/2/124/F1.large.jpghttp://emj.bmj.com/content/18/2/124.full#ref-1


17/43

(e pointThe " test is when ou compare ")&IT and ")!8)

4 Study a sample and calculate its mean

-ith the critical values known, 'dgar now gathers a sample of 44 patients and measures the :F&. Following 5everwhee"e he finds themean :F& increases b A l@minute and s is 64 l@minute.

5 Calculate the test statistic!s explained before the " statistic is equal to/

" E Ksample mean difference Y population mean difference *P+L@'%'M

-here '%'M equals s@]n

Therefore/ '%'M E 64@ 4 E 6

!ccording to the null h pothesis the mean difference for the population is"ero, consequentl /

" statistic E AY4@6 E 6.A

In other words the mean difference before and after using 5everwhee"elies 6.A '%'M above the population2s mean difference of "ero.

6 Compare the calculated test statistic with the critical valuesThe calculated value of X6.A lies above the larger critical value of .VC. Ittherefore falls into the area of re0ecting the null h pothesis.

7 E press the chances o! obtaining results at least this e treme i! thenull hypothesis is trueThe p value is the probabilit of getting a mean difference equal to orgreater than that found in the experiment, if the null h pothesis wascorrect. !s the " value can be negative or positive, there are two wa s ofgetting a difference with a magnitude of 6.A. )onsequentl the p value isrepresented b the area demarcated b Y6.A to the tip of the left tail plusthe area demarcated b X6.A to the tip of the right tail *fig +.From the " statistic table 'gbert finds the probabilit of getting adifference equal to, or greater than, X6.A is 4.AY4.=V+ that adifference of 6.A '%'M could be produce if the null h pothesis wascorrect.



18/43

)onsequentl 'gbert can tell r )anute that ZThe h pothesis that there isno difference in the :F& before and after 5everwhee"e is re0ected. Themean difference E A l@minute, p E 4.4 6[.

(e points• The " distribution tables can be used to convert the " statistic into a p value.

• The p value represents the chances of getting an experimentalresult this big, or greater, if the null h pothesis applied.

In da s gone b the anal sis would stop at this point. 5owada s it isusual practice to also consider the confidence interval of the resultswhenever possible. 3efore carr ing out these calculations it is pertinent toconsider wh the confidence intervals are considered so useful.Confidence inte )als and & )al0es!s demonstrated in the previous example statistical inference can be usedto produce a p value for the mean difference. The latter is known as the point *or sample+ estimate along with a p value. In contrast, a confidenceinterval *)I+ around the point estimate provides a range within which thevalue of the particular parameter would lie if the whole population wasconsidered.=For example, a VA> confidence interval around a stud 2smean difference is the range of values the population2s mean differencecould be expected to be found VA> of the time. This is a similar part tothat pla ed b the standard error of the mean *%'M+.A In these cases theVA> confidence interval is equal to the sample mean X@Y .VC %'M.To help understand the importance of this, consider a trial of twoantih pertensives, ! and 3. This stud found that the group taking drug !had a mean s stolic blood pressure that was =4 mm 1g less than group 3*p E 4.444 +. The low p value indicates the result is statisticallsignificant and the large point estimate implies the finding is clinicallrelevant. 1owever, if ou repeated this stud using similar, but differentgroups of patients, then the magnitude of the blood pressure changewould var . The )I allows ou to work out how wide this variation islikel to be. In this stud the VA>)I was of the whole population of similar h pertensives lies between


19/43

It is important to bear in mind that although we are VA> confident thatthe true value lies within the range provided, it does not mean it has anequal chance of l ing an where along the )I. In actual fact the probabilit varies, with the most likel value being that calculatedoriginall . Therefore, using the example above, the most likel reductionin blood pressure for all similar h pertensives is =4 mm 1g but in VA> of cases it could var between )I is often chosen, the actual level is up to that whichou consider the most appropriate. For example, ou can use )I of VV.V>

when the treatment is potentiall harmful or ver expensive. The effect isto widen the range of values covering the point estimate to ensure thewidest range of possible differences is identified.

(e pointThe choice of )I is a balance between ensuring the population mean isincluded while minimising parts of the scale where it is unlikel to be.

The confidence interval also provides information on the precision of thestud Dthat is, the abilit to determine the true value for the whole population. If the VA> )I in a similar stud , involving two h potensives) and , was YA.4 to A4 mm 1g then it could be that roup ) did worsethan *that is, the difference E YA.4 mm 1g+ or did ver much better*that is, the difference E A4 mm 1g+. 'quall the true difference could liean where between these two extremes. -ide intervals around the pointestimate indicate the stud lacks precision. 7suall this is due to there

being too few sub0ects in the experiment.(e point

enerall , confidence intervals decrease with increases in sample si"e

The above example also shows that a confidence interval can include 4.-hen this occurs it means there is a chance there is no difference between the stud groups. This is the same as having a p value greaterthan 4.4A *or our chosen significant level+ and not re0ecting the nullh pothesis. )onsequentl the )I can provide all the information available

V


20/43

from a p value. In addition it tells ou the suitabilit of re0ecting oraccepting the null h pothesis.

(e point

! negative result *that is, accept the null h pothesis+ occurs when theconfidence interval includes "ero or onl clinicall irrelevant difference between the groups

In summar therefore, confidence intervals provide information on/

• The magnitude of the difference• The precision of the stud

• The statistical significance

(e point!s p values impl little about the magnitude and precision of thedifferences between the groups, )I should be reported instead.

)onfidence intervals are worked out on a computer using a set ofmathematical rules. 1owever, the method used to calculate them needs totake into account the t pe of data and the stud design. !dvice istherefore recommended in choosing the most appropriate t pe of )Icalculation. Furthermore, although confidence intervals are an excellentwa of summarising information, the cannot control for other errors instud design such as improper patient selection and poor experimentalmethodolog . For example, a small )I obtained from a biased stud isless likel to include the true population value than one that is unbiased.)onsequentl the narrow )I gives a false impression of precision.

In view of the importance of confidence intervals, 'gbert now wants todetermine the VA> )I for the mean difference.

!s described in the previous article,= the VA> )I is/

-here " o is the " statistic appropriate to the required )I. )onsequentl theVA> confidence interval of the difference is/

!s this range does not include "ero, 'gbert concludes that data are notcompatible with the null h pothesis being correct. 1owever, the range of:F& covers small values. Therefore rather than simpl presenting a p

value 'gbert will be able to provide more information if he uses the VA>confidence interval when discussing the clinical relevance of these data.

64



21/43

! lot of information has been presented over the last few paragraphs. It istherefore useful to take a moment to re#read the basic s stem forcomparing two groups statisticall *box 6+. This is used, with slightvariations, in the ma0orit of situations ou will come across because itapplies equall to comparing means, proportions, slopes of lines andman other common statistical anal ses.

Chi-s10a ed tests use the same calculations and the same probabilitdistribution for different applications/

• )hi#squared tests for variance are used to determine whether anormal population has a specified variance. The null h pothesis is thatit does.

•

)hi#squared tests of independence are used for deciding whethertwo variables are associated or are independent. The variables arecategorical rather than numeric. It can be used to decide whetherleft#handedness is correlated withlibertarian politics *or not+. The nullh pothesis is that the variables are independent. The numbers used inthe calculation are the observed and expected frequencies ofoccurrence *fromcontingenc tables+.

• )hi#squared goodness of fit tests are used to determine theadequac of curves fit to data. The null h pothesis is that the curve fit

is adequate. It is common to determine curve shapes to minimi"e themean square error, so it is appropriate that the goodness#of#fitcalculation sums the squared errors.

Pai ed t testIf the sample si"e in the example above was smaller than 44 then a pairedt test would have to be used. To demonstrate this, consider the

following case.'gbert informs r 'ndora 8onel about his findings regarding 5everwhee"e. %he is surprised because the use a lot of it in the'mergenc epartment at %t 1eartsinc where she works as a %p&. %hetherefore decides to repeat the stud using 6A patients attending herdepartment.

+ State the n0ll h"&othesis and alte nati)e h"&othesis of the st0d"These remain the same. )onsequentl /

The null h pothesis can be summarised to/

6

https://en.wikipedia.org/wiki/Chi-squared_testhttps://en.wikipedia.org/wiki/Left-handednesshttps://en.wikipedia.org/wiki/Left-handednesshttps://en.wikipedia.org/wiki/Libertarianismhttps://en.wikipedia.org/wiki/Contingency_tablehttps://en.wikipedia.org/wiki/Chi-squared_testhttps://en.wikipedia.org/wiki/Left-handednesshttps://en.wikipedia.org/wiki/Left-handednesshttps://en.wikipedia.org/wiki/Libertarianismhttps://en.wikipedia.org/wiki/Contingency_table


22/43

Mean difference in :F& E 4

!nd the alternative is/

Mean difference in :F& H 4

. Select the le)el of si'nificanceFollowing convention, 'ndora picks a significance level of 4.4A for herstud .

2 Esta!lish the c itical )al0esIt is not possible to calculate the %'M when the standard deviation of the population is not known, or the sample si"e is less than 44. In thesecases thet statistic has to be used instead of the " statistic.=

!s described in the previous article, thet tables use degrees of freedomrather than the number in the group.= This is equal to one less than thegroup si"e. )onsequentl 'ndora looks up the value for t with asignificance of 4.4A and 6= degrees of freedom *fig 6+. This is the criticalvalue *t)&IT+ and in this case is equal to 6.4C=. In other words, for asample si"e of 6A, at value of X@Y 6.4C= separates the middle VA> area of acceptance of the null h pothesis from two 6.A> areas of re0ection.

Figure "

66

http://emj.bmj.com/content/18/2/124.full#ref-4http://emj.bmj.com/content/18/2/124.full#ref-4http://emj.bmj.com/content/18/2/124/F2.large.jpghttp://emj.bmj.com/content/18/2/124.full#ref-4http://emj.bmj.com/content/18/2/124.full#ref-4


23/43

E tract of the table of the t statistic values. !he first column lists thedegrees of freedom "df#. !he headings of the other columns give

probabilities for t to lie $ithin the t$o tails of the distribution.!s with the " test, thet statistic derived from the sample *t)!8) + can now be determined.(e pointThe t test is when ou compare t)&IT and t)!8)

3 St0d" a ando, sa,&le and calc0late its ,ean-ith the critical values known, 'ndora can now carr out her stud .Following 5everwhee"e she finds the mean :F& increases b U l@minuteand s is +. There is therefore onl a 4.44 ^4.4 chance that a differenceof < '%'M could be produce if the null h pothesis was correct.)onsequentl 'ndora can claim that ZThe h pothesis that there is no

difference in the :F& before and after 3reathee"e is re0ected,t E


24/43

(e pointThe t distribution table can be used to convert the t statistic into a p value

'ndora now wants to determine the VA> )I for the mean difference.

!s described previousl ,= the VA> )I is/-here t o is thet statistic appropriate to the required )I. For a sample si"eof 6A *df E 6=+ this is 6.4C=. )onsequentl the VA> confidence interval of the difference is/!s this range does not include "ero, she concludes that data are notcompatible with the null h pothesis being correct. 1owever, the range of:F& again includes small values. The VA> confidence interval istherefore helpful when discussing the clinical relevance of these data.

H"&othesis Testin' - Anal"sis of %a iance 8ANO%A9

Int od0ction

This module will continue the discussion of h pothesis testing, where aspecific statement or h pothesis is generated about a population parameter, and sample statistics are used to assess the likelihood that the

h pothesis is true. The h pothesis is based on available information andthe investigator2s belief about the population parameters. The specific testconsidered here is called anal sis of variance *!5$_!+ and is a test ofh pothesis that is appropriate to compare means of a continuous variablein two or more independent comparison groups. For example, in someclinical trials there are more than two comparison groups. In a clinicaltrial to evaluate a new medication for asthma, investigators mightcompare an experimental medication to a placebo and to a standardtreatment *i.e., a medication currentl being used+. In an observationalstud such as the Framingham 1eart %tud , it might be of interest tocompare mean blood pressure or mean cholesterol levels in persons whoare underweight, normal weight, overweight and obese.

The technique to test for a difference in more than two independentmeans is an extension of the two independent samples procedurediscussed previousl which applies when there are exactl twoindependent comparison groups. The !5$_! technique applies whenthere are two or more than two independent groups. The !5$_! procedure is used to compare the means of the comparison groups and isconducted using the same five step approach used in the scenariosdiscussed in previous sections. 3ecause there are more than two groups,

6=



25/43

however, the computation of the test statistic is more involved. The teststatistic must take into account the sample si"es, sample means andsample standard deviations in each of the comparison groups.

O!(ecti)es. :erform anal sis of variance b hand6. !ppropriatel interpret results of anal sis of variance tests


26/43

If the variabilit in the k comparison groups is not similar, thenalternative techniques must be used.

The F statistic is computed b taking the ratio of what is called the

?between treatment? variabilit to the ?residual or error? variabilit . Thisis where the name of the procedure originates. In anal sis of variance weare testing for a difference in means *14/ means are all equal versus 1/means are not all equal+ b evaluating variabilit in the data. Thenumerator captures between treatment variabilit *i.e., differences amongthe sample means+ and the denominator contains an estimate of thevariabilit in the outcome. The test statistic is a measure that allows us toassess whether the differences among the sample means *numerator+ aremore than would be expected b chance if the null h pothesis is true.&ecall in the two independent sample test, the test statistic was computed b taking the ratio of the difference in sample means *numerator+ to thevariabilit in the outcome *estimated b %p+.

The decision rule for the F test in !5$_! is set up in a similar wa todecision rules we established for t tests. The decision rule again dependson the level of significance and the degrees of freedom. The F statistichas two degrees of freedom. These are denoted df and df 6, and called thenumerator and denominator degrees of freedom, respectivel . Thedegrees of freedom are defined as follows/

df E k# and df 6E5#k,

where k is the number of comparison groups and 5 is the total number ofobservations in the anal sis. If the null h pothesis is true, the betweentreatment variation *numerator+ will not exceed the residual or errorvariation *denominator+ and the F statistic will small. If the nullh pothesis is false, then the F statistic will be large. The re0ection regionfor the F test is alwa s in the upper *right#hand+ tail of the distribution as

shown below.Re(ection Re'ion fo $ Test 6ith a ;


27/43

For the scenario depicted here, the decision rule is/ &e0ect 14 if F J 6.U .

ERRORS IN HYPOTHESIS TESTING

superintendent in a medium si"e school has a problem. The mathematicalscores on nationall standardi"ed achievement tests such as the %!T and!)T of the students attending her school are lower than the nationalaverage. The school board members, who don2t care whether the footballor basketball teams win or not, is greatl concerned about this deficienc .The superintendent fears that if it is not corrected, she will loose her 0ob before long.

!s the superintendent was sitting in her office wondering what to do, asalesperson approached with a briefcase and a sales pitch. The

salesperson had heard about the problem of the mathematics scores andwas prepared to offer the superintendent a ?deal she couldn2t refuse.? Thedeal was teaching machines to teach mathematics, guaranteed to increasethe mathematics scores of the students. In addition, the machines nevertake breaks or demand a pa increase.

The superintendent agreed that the machines might work, but wasconcerned about the cost. The salesperson finall wrote some figures.%ince there were about 444 students in the school and one machine was

needed for ever ten students, the school would need about one hundredmachines. !t a cost of ` 4,444 per machine, the total cost to the school

6


28/43

would be about ` ,444,444. !s the superintendent picked herself up offthe floor, she said she would consider the offer, but didn2t think the school board would go for such a big expenditure without prior evidence that themachines actuall worked. 3esides, how did she know that the compan

that manufactures the machines might not go bankrupt in the next ear,meaning the school would be stuck with a million dollar2s worth ofuseless electronic 0unk.

The salesperson was prepared, because an offer to lease ten machines fortesting purposes to the school for one ear at a cost of `A44 each wasmade. !t the end of a ear the superintendent would make a decisionabout the effectiveness of the machines. If the worked, she would pitchthem to the school board; if not, then she would return the machines withno further obligation.

!n experimental design was agreed upon. $ne hundred students would be randoml selected from the student population and taught using themachines for one ear. !t the end of the ear, the mean mathematicsscores of those students would be compared to the mean scores of thestudents who did not use the machine. If the means were differentenough, the machines would be purchased. The astute student willrecogni"e this as a nested t#test.

In order to help decide how different the two means would have to be inorder to bu the machines, the superintendent did a theoretical anal sis of the decision process. This anal sis is presented in the following decision box.

?&eal -orld?

')I%I$5The machines don2twork.

The machineswork.

3u the machines.

ecide the machines work.

T pe I

'&&$&

probabilit E

)$&&')T

probabilit E #

?power?

o not bu the machines. )$&&')T T pe II

6U


29/43

ecide that the machines do notwork

probabilit E # '&&$&

probabilit EThe decision box has the decision that the superintendent must make onthe left hand side. For simplicit 2s sake, onl two possibilities are permitted/ either bu all the machines or bu none of the machines. Thecolumns at the top represent ?the state of the real world?. The state of thereal world can never be trul known, because if it was known whether ornot the machines worked, there would be no point in doing theexperiment. The four cells represent various places one could be,

depending upon the state of the world and the decision made. 'ach cellwill be discussed in turn.

. 3u ing the machines when the do not work.

This is called a T pe I error and in this case is ver costl *` ,444,444+.The probabilit of this t pe of error is , also called the significancelevel, and is directl controlled b the experimenter. 3efore theexperiment begins, the experimenter directl sets the value of . In thiscase the value of would be set low, lower than the usual value of .4A, perhaps as low as .444 , which means that one time out of 4,444 theexperimenter would bu the machines when the didn2t work.

6. 5ot bu ing the machines when the reall didn2t work.

This is a correct decision, made with probabilit # when in fact theteaching machines don2t work and the machines are not purchased.

The relationship between the probabilities in these two decision boxescan be illustrated using the sampling distribution when the nullh pothesis is true. The decision point is set b , the area in the tail ortails of the distribution. %etting smaller moves the decision point further into the tails of the distribution.

6V


30/43


31/43

The relationship between the si"e of and can be seen in the followingillustration combining the two previous distributions into overlappingdistributions, the top graph with E.4A and the bottom with E .4 .

1 4 true 1 true

The si"e of the effect is the difference between the center points * + of the

two distributions. If the si"e of the effect is increased, the relationship between the probabilities of the two t pes of errors is changed.


32/43

-hen the error variance of the scores are decreased, the probabilit of at pe II error is decreased if ever thing else remains constant, asillustrated below.

!n interactive exercise designed to allow exploration of the relationships between alpha, si"e of effects, si"e of sample *5+, si"e of error, and betacan now be understood. The values of alpha, si"e of effects, si"e ofsample, and si"e of error can all be ad0usted with the appropriate scroll bars. -hen one of these values is changed, the graphs will change and thevalue of beta will be re#computed. The area representing the value ofalpha on the graph is drawn in dark gra . The area representing beta is


33/43

drawn in dark blue, while the corresponding value of power isrepresented b the light blue area. 7sing this exercise the student shouldverif /

•

The si"e of beta decreases as the si"e of error decreases.• The si"e of beta decreases as the si"e of the sample increases.• The si"e of beta decreases as the si"e of alpha increases.• The si"e of beta decreases as the si"e of the effects increase.

The si"e of the increase or decrease in beta is a complex function ofchanges in all of the other values. For example, changes in the si"e of thesample ma have either small or large effects on beta depending upon theother values. If a large treatment effect and small error is present in theexperiment, then changes in the sample si"e are going to have a smalleffect.

A SECOND CHANCE

!s might be expected, in the previous situation the superintendentdecided not to purchase the teaching machines, because she hadessentiall stacked the deck against deciding that there were an effects.-hen she described the experiment and the result to the salesperson thenext ear, the salesperson listened carefull and understood the reason

wh had been set so low.The salesperson had a new offer to make, however. 3ecause of anadvance in microchip technolog , the entire teaching machine had been placed on a single integrated circuit. !s a result the price had dropped toÀ44 a machine. 5ow it would cost the superintendent a total of À4,444to purchase the machines, a sum that is quite reasonable.

The anal sis of the probabilities of the two t pes of errors revealed thatthe cost of a T pe I error, bu ing the machines when the reall don2twork *À4,444+, is small when compared to the loss encountered in aT pe II error, when the machines are not purchased when in fact the dowork, although it is difficult to put into dollars the cost of the students notlearning to their highest potential.

In an case, the superintendent would probabl set the value of to afairl large value *. 4 perhaps+ relative to the standard value of .4A. Thiswould have the effect of decreasing the value of and increasing the power * # + of the experiment. Thus the decision to bu the machineswould be made more often if in fact the machines worked. The


34/43

experiment was repeated the next ear under the same conditions as the previous ear, except the si"e of was set to . 4.

The results of the significance test indicated that the means were

significantl different, the null h pothesis was re0ected, and a decisionabout the realit of effects made. The machines were purchased, thesalesperson earned a commission, the math scores of the studentsincreased, and ever one lived happil ever after.

THE ANALYSIS GENERALI/ED TO ALL E?PERIMENTS

The anal sis of the realit of the effects of the teaching machines ma begenerali"ed to all significance tests. &ather than bu ing or not bu ing themachines, one re0ects or retains the null h pothesis. In the ?real world,?rather than the machines working or not working, the null h pothesis istrue or false. The following presents the boxes representing significancetests in general.

?&eal -orld?

')I%I$5 5788 T&7'!8T'&5!TI_'F!8%'

5o 'ffects

5788 F!8%'

!8T'&5!TI_'T&7'

&eal 'ffects&e0ect 5ull

!ccept !lternative

ecide there arereal effects.

T pe I

'&&$&

prob E

)$&&')T

prob E #

?power?

&etain 5ull

&etain !lternative

ecide that no effects werediscovered.

)$&&')T

prob E #

T pe II

'&&$&

prob E


35/43

Inte & etation of Data

Interpretation of data refers to drawing inferences b anal "ing data.Itcan be said that interpretation helps to convert the statistical data

into information. !nal sis and interpretation of data are closel .!nal sis and interpretation of data can be done simultaneousl .!nal sis and interpretation of data facilitates research findings,recommendations and conclusions.

Unde standin' and Inte & etin' Data

Figuring out what data means is 0ust as important as collecting it. 'ven ifthe data collection process is sound, data can be misinterpreted. -hen

interpreting data, the data user must not onl attempt to discern thedifferences between causalit and coincidence, but also must consider all possible factors that ma have led to a result.

!fter considering the design of a surve , consumers should look at thereported data interpretation. %uppose a report states that A6 percent of all!mericans prefer )hevrolet to other car manufacturers. The surve orswant ou to think that more than half of all !mericans prefer )hevrolet,

but is this reall the caseB :erhaps not all those surve ed were!mericans. !lso, the A6 percent comes from the sample, so it is importantto ask if the sample was large enough, unbiased, and randoml chosen.$ne also needs to be aware of margins of error and confidence intervals.If the margin of error for this surve is A percent than this means that the percentage of car owners in the 7nited %tates who prefer )hevrolet couldactuall be between = and A percent *A percent higher or lower than theA6 percent+.

%imilar questions are important to consider when we tr tounderstand&olls . uring the 6444 presidential race, the evening newsand newspapers were often filled with poll reports. For example, one pollstated A percent of !mericans preferred eorge -. 3ush, =C percent preferred !l ore, and < percent were undecided, with a margin of errorof plus or minus A percent.

The news anchor then went on to report thatmost !mericans prefereorge -. 3ush. 1owever, given the data outlined above, this conclusion


36/43

is questionable. 3ecause the difference between eorge -. 3ush and !lore is the same as the margin of error, it is impossible to know which

candidate was actuall preferred. In addition, if we do not know an of

the circumstances behind the poll, we should be skeptical about itsfindings.

!s another example, consider census data that shows a radical increase inthe number of people living in Florida and !ri"ona along with a decreasein the number of people living in 5ew 9ork. $ne could easil *andfalsel + conclude that the data ?proves? that people are finding 5ew 9orkto be a less desirable place to live and therefore are moving awa .

3ut this hast conclusion could be missing the big picture. -hat if thedata also reveals that the average age of 5ew 9orkers has dropped since

VV4B Further interpretation of the data ma reveal that when 5ew9orkers grow older, the move to warmer climates to retire. Thisillustrates wh data must be thoroughl interpreted before anconclusions can be drawn.

A Data Chec>list# -hen reading an surve , listening to an

advertisement, or hearing about poll results, informed consumers shouldask questions about the soundness of the data interpretation. ! recap ofke points follows.

. -as the sample unbiased *representative of the whole population+B6. -as the sample large enough for the purpose of the surve *margin

of error of the sample+B


37/43

Phases of Inte & etation

• eductive :hase ^ the researcher applies the process ofdeduction.

• Inductive :hase ^ formulation of generali"ations or principles that masubstantiate h potheses.

Three modes in :resenting ata %efore the data could be interpreted they must be presented first

by&

!e tual mode ^ embraces the discussion the discussion and anal sis.!abular mode ^ used to present through tables , the data of stud .'raphic mode ^ presented through graphs, charts and other devices.

(lassification of )ata- !nal "ing the characteristics of a large group classified as;

. 1omogeneous ^ no breakdown into subgroups

6. ichotomous ^ twofold categories

*orting and !abulating )ata

!abulation ^ a process of transferring data from the data#gatheringinstrument to the tabular form in which the s stematicall evaluated.

Hand *orting+ Hand Recording and Hand !abulation ^ this methodrecommended in tabulation to save time and to ensure greater accurac .

!ables and ,igures- To help researcher to see the similarities and relationship of the

data.

!able ^ a s stematic method of presenting statistical data in verticalcolumns and hori"ontal rows, according to some classification of sub0ectmatter.


38/43

Rules for the Handling of !ables- ood tables relativel simple, concentrating on a limited

number of ideas- Text reference should identif tables b number - Tables should not exceed in the page si"e- The word table is centered between the page margins and t ped

in capital letters followed b the table number in capital &omannumerals or !rabic.

- The top of the table is placed three spaces below the last line ofthe title.

- 5umerical data are usuall arranged in descending order - ecimal points should alwa s be aligned in the column

,igure ^ a device that presents statistical data in graphic form. Figuresinclude graphs, charts, drawings, diagrams. Maps, photographs, blueprints and other computer print#outs

(haracteristics of 'ood ,igures

- The title should clearl describe the nature of data- %imple and enough to conve a clear ideas- 5umerical data should be presented- ata should be presented carefull and correctl- Figures should be used sparingl- Figures are numbered with !rabic- The title of the figure is place below


39/43

%T!TI%TI)!8 T')15I 7'% 7%' I5 !5!89OI5 !T!

%tatistics defined as a bod of mathematical techniques or processfor gathering, organi"ing, anal "ing and interpreting numerical data.

escriptive %tatistics#summari"es data collected for a sample population. -hether aresearch stud is a surve or an experiment, it is essential todescribe what was observed. -e need a means of summari"ingsome of the basic characteristics of the observations

Three aspects in escriptive %tatistics -. ,re uency )istribution

- It is an listing of a set of classes *test scores+ and frequenc ofdistribution in the class *no. of students who made that score+.

- ! first step in summari"ing and describing data. It removes thenames of the sub0ects and provides a wa of grouping themeasurements. -hether the class categor is a measurement ofintelligence, anxiet , or reaction time, we can create a frequencdistribution to show how man observations fall in each class.

/ays in presenting fre uency distribution

. Histogram or %ar 'raph # is a graphical representation of thedistribution of data.


40/43

6. ,re uency 0olygon or curve # ! graph made b 0oining themiddle#top points of the columns of a frequenc histogram.

The fre uency distribution can be describe as symmetric or ske$ed . In symmetric distribution , the mean and median areidentical. In ske$ed distribution , one tail of the distribution isdisproportionatel longer than the other. In positive ske$ed distribution, most of the scores are at the low end of the scale, andfew scores are at the high. Innegative ske$ed distribution, most ofthe scores are at the high end of the scale and few scores are at thelow end.

%. 1easures of (entral !endency- $ne wa of identif ing t pical, most likel and value in a group

of scores.- o not summari"e a distribution of scores completel .

. 1ode # is the value that appears most often in a set of data6. 1edian # is the numerical value separating the higher half of adatasample, a population, or a probabilit distribution, from thelower half.


41/43

1ethods in describing variability in a distribution

3. Range

- ! crude measure of variabilit .- the distance between the highest and the lowest score.

2. *tandard )eviation- %table measure of variabilit- ! distance of ever score from the mean. -hatever the value of

the standard deviation direct proportional to average distance ifthe score.

)orrelation )oefficient- !ssess the strength of a correlation *degree and direction of

relation between two variables+- ! number ranging from # , which indicates a perfect negative

correlation between the two variables, through 4, whichindicates no correlation, then, X which indicates a perfect positive correlation.

Inferential %tatisticsTo determine the relationship between the h potheses and

variables are supported b the findings of the research.

0robability- The basic tool in inferential statistics- eveloped when the researchers and mathematicians can

estimate reasonabl and accuratel the chances that particularevent will occur.

*tatistical *ignificance- !llow researchers to determine exactl how small the

probabilit is that their results have come about b chance.

LIMITATIONS AND SOURCES O$ ERROR IN THE ANALYSIS

AND INTERPRETATION O$ DATA

=


42/43

. (onfusing statements $ith facts . ! common fault is the acceptanceof statement is facts. -hat individuals report ma be a sincereexpression of what the believe to be the facts in a case, but thesestatements are not necessaril true.

6. ,ailure to recognize limitations . The ver nature of researchimplies certain restrictions or limitations about the group orsituation described ^ its si"e, its representativeness, and itsdistinctive composition. Failure to recogni"e these limitations malead to the formulation of generali"ations that are not warranted bthe data collected.


43/43

It is this abilit to see all implications in the data that producessignificant discoveries.

ANALYSIS O$ DATA

. Must anal "e his research problem carefull to see what necessarto provide a solution to it. The researcher must assume himself, and be able to satisf those to whom he reports his stud , that thismethod of attacking the problems provides a crucial approach.

6. Must see the factors that he chooses for stud will satisf theconditions of the problem.

217054454 Hypothesis Testing

Documents

Transcript of 217054454 Hypothesis Testing