Testing of Hypotheses I

Transcription

1 84 Research Methodology 9 Testng of Hypotheses I (Parametrc or Standard Tests of Hypotheses) Hypothess s usually consdered as the prncpal nstrument n research. Its man functon s to suggest new experments and observatons. In fact, many experments are carred out wth the delberate object of testng hypotheses. Decson-makers often face stuatons wheren they are nterested n testng hypotheses on the bass of avalable nformaton and then take decsons on the bass of such testng. In socal scence, where drect knowledge of populaton parameter(s) s rare, hypothess testng s the often used strategy for decdng whether a sample data offer such support for a hypothess that generalsaton can be made. Thus hypothess testng enables us to make probablty statements about populaton parameter(s). The hypothess may not be proved absolutely, but n practce t s accepted f t has wthstood a crtcal testng. Before we explan how hypotheses are tested through dfferent tests meant for the purpose, t wll be approprate to explan clearly the meanng of a hypothess and the related concepts for better understandng of the hypothess testng technques. WHAT IS A HYPOTHESIS? Ordnarly, when one talks about hypothess, one smply means a mere assumpton or some supposton to be proved or dsproved. But for a researcher hypothess s a formal queston that he ntends to resolve. Thus a hypothess may be defned as a proposton or a set of proposton set forth as an explanaton for the occurrence of some specfed group of phenomena ether asserted merely as a provsonal conjecture to gude some nvestgaton or accepted as hghly probable n the lght of establshed facts. Qute often a research hypothess s a predctve statement, capable of beng tested by scentfc methods, that relates an ndependent varable to some dependent varable. For example, consder statements lke the followng ones: Students who receve counsellng wll show a greater ncrease n creatvty than students not recevng counsellng Or the automoble A s performng as well as automoble B. These are hypotheses capable of beng objectvely verfed and tested. Thus, we may conclude that a hypothess states what we are lookng for and t s a proposton whch can be put to a test to determne ts valdty.

2 Testng of Hypotheses I 85 Characterstcs of hypothess: Hypothess must possess the followng characterstcs: () Hypothess should be clear and precse. If the hypothess s not clear and precse, the nferences drawn on ts bass cannot be taken as relable. () Hypothess should be capable of beng tested. In a swamp of untestable hypotheses, many a tme the research programmes have bogged down. Some pror study may be done by researcher n order to make hypothess a testable one. A hypothess s testable f other deductons can be made from t whch, n turn, can be confrmed or dsproved by observaton. () Hypothess should state relatonshp between varables, f t happens to be a relatonal hypothess. (v) Hypothess should be lmted n scope and must be specfc. A researcher must remember that narrower hypotheses are generally more testable and he should develop such hypotheses. (v) Hypothess should be stated as far as possble n most smple terms so that the same s easly understandable by all concerned. But one must remember that smplcty of hypothess has nothng to do wth ts sgnfcance. (v) Hypothess should be consstent wth most known facts.e., t must be consstent wth a substantal body of establshed facts. In other words, t should be one whch judges accept as beng the most lkely. (v) Hypothess should be amenable to testng wthn a reasonable tme. One should not use even an excellent hypothess, f the same cannot be tested n reasonable tme for one cannot spend a lfe-tme collectng data to test t. (v) Hypothess must explan the facts that gave rse to the need for explanaton. Ths means that by usng the hypothess plus other known and accepted generalzatons, one should be able to deduce the orgnal problem condton. Thus hypothess must actually explan what t clams to explan; t should have emprcal reference. BASIC CONCEPTS CONCERNING TESTING OF HYPOTHESES Basc concepts n the context of testng of hypotheses need to be explaned. (a) Null hypothess and alternatve hypothess: In the context of statstcal analyss, we often talk about null hypothess and alternatve hypothess. If we are to compare method A wth method B about ts superorty and f we proceed on the assumpton that both methods are equally good, then ths assumpton s termed as the null hypothess. As aganst ths, we may thnk that the method A s superor or the method B s nferor, we are then statng what s termed as alternatve hypothess. The null hypothess s generally symbolzed as H 0 and the alternatve hypothess as H a. Suppose we want to test the hypothess that the populaton mean bg µ s equal to the hypothessed mean µ H0 00 d. Then we would say that the null hypothess s that the populaton mean s equal to the hypothessed mean 00 and symbolcally we can express as: H 0 : µ µ H 0 00 C. Wllam Emory, Busness Research Methods, p. 33.

3 86 Research Methodology If our sample results do not support ths null hypothess, we should conclude that somethng else s true. What we conclude rejectng the null hypothess s known as alternatve hypothess. In other words, the set of alternatves to the null hypothess s referred to as the alternatve hypothess. If we accept H 0, then we are rejectng H a and f we reject H 0, then we are acceptng H a. For H 0 : µ µ H 0 00, we may consder three possble alternatve hypotheses as follows * : Table 9. Alternatve hypothess To be read as follows H a H a H a : µ µ : µ > µ : µ < µ H H H 0 (The alternatve hypothess s that the populaton mean s not equal to 00.e., t may be more or less than 00) 0 (The alternatve hypothess s that the populaton mean s greater than 00) 0 (The alternatve hypothess s that the populaton mean s less than 00) The null hypothess and the alternatve hypothess are chosen before the sample s drawn (the researcher must avod the error of dervng hypotheses from the data that he collects and then testng the hypotheses from the same data). In the choce of null hypothess, the followng consderatons are usually kept n vew: (a) Alternatve hypothess s usually the one whch one wshes to prove and the null hypothess s the one whch one wshes to dsprove. Thus, a null hypothess represents the hypothess we are tryng to reject, and alternatve hypothess represents all other possbltes. (b) If the rejecton of a certan hypothess when t s actually true nvolves great rsk, t s taken as null hypothess because then the probablty of rejectng t when t s true s α (the level of sgnfcance) whch s chosen very small. (c) Null hypothess should always be specfc hypothess.e., t should not state about or approxmately a certan value. Generally, n hypothess testng we proceed on the bass of null hypothess, keepng the alternatve hypothess n vew. Why so? The answer s that on the assumpton that null hypothess s true, one can assgn the probabltes to dfferent possble sample results, but ths cannot be done f we proceed wth the alternatve hypothess. Hence the use of null hypothess (at tmes also known as statstcal hypothess) s qute frequent. (b) The level of sgnfcance: Ths s a very mportant concept n the context of hypothess testng. It s always some percentage (usually 5%) whch should be chosen wt great care, thought and reason. In case we take the sgnfcance level at 5 per cent, then ths mples that H 0 wll be rejected * If a hypothess s of the type µ µ H0, then we call such a hypothess as smple (or specfc) hypothess but f t s of the type µ µ or µ > µ or µ < µ, then we call t a composte (or nonspecfc) hypothess. H0 H0 H0

4 Testng of Hypotheses I 87 when the samplng result (.e., observed evdence) has a less than 0.05 probablty of occurrng f H 0 s true. In other words, the 5 per cent level of sgnfcance means that researcher s wllng to take as much as a 5 per cent rsk of rejectng the null hypothess when t (H 0 ) happens to be true. Thus the sgnfcance level s the maxmum value of the probablty of rejectng H 0 when t s true and s usually determned n advance before testng the hypothess. (c) Decson rule or test of hypothess: Gven a hypothess H 0 and an alternatve hypothess H a, we make a rule whch s known as decson rule accordng to whch we accept H 0 (.e., reject H a ) or reject H 0 (.e., accept H a ). For nstance, f (H 0 s that a certan lot s good (there are very few defectve tems n t) aganst H a ) that the lot s not good (there are too many defectve tems n t), then we must decde the number of tems to be tested and the crteron for acceptng or rejectng the hypothess. We mght test 0 tems n the lot and plan our decson sayng that f there are none or only defectve tem among the 0, we wll accept H 0 otherwse we wll reject H 0 (or accept H a ). Ths sort of bass s known as decson rule. (d) Type I and Type II errors: In the context of testng of hypotheses, there are bascally two types of errors we can make. We may reject H 0 when H 0 s true and we may accept H 0 when n fact H 0 s not true. The former s known as Type I error and the latter as Type II error. In other words, Type I error means rejecton of hypothess whch should have been accepted and Type II error means acceptng the hypothess whch should have been rejected. Type I error s denoted by α (alpha) known as α error, also called the level of sgnfcance of test; and Type II error s denoted by β (beta) known as β error. In a tabular form the sad two errors can be presented as follows: Table 9. Decson Accept H 0 Reject H 0 H 0 (true) Correct Type I error decson ( α error) H 0 (false) Type II error Correct ( β error) decson The probablty of Type I error s usually determned n advance and s understood as the level of sgnfcance of testng the hypothess. If type I error s fxed at 5 per cent, t means that there are about 5 chances n 00 that we wll reject H 0 when H 0 s true. We can control Type I error just by fxng t at a lower level. For nstance, f we fx t at per cent, we wll say that the maxmum probablty of commttng Type I error would only be 0.0. But wth a fxed sample sze, n, when we try to reduce Type I error, the probablty of commttng Type II error ncreases. Both types of errors cannot be reduced smultaneously. There s a trade-off between two types of errors whch means that the probablty of makng one type of error can only be reduced f we are wllng to ncrease the probablty of makng the other type of error. To deal wth ths trade-off n busness stuatons, decson-makers decde the approprate level of Type I error by examnng the costs or penaltes attached to both types of errors. If Type I error nvolves the tme and trouble of reworkng a batch of chemcals that should have been accepted, whereas Type II error means takng a chance that an entre group of users of ths chemcal compound wll be posoned, then

5 88 Research Methodology n such a stuaton one should prefer a Type I error to a Type II error. As a result one must set very hgh level for Type I error n one s testng technque of a gven hypothess. Hence, n the testng of hypothess, one must make all possble effort to strke an adequate balance between Type I and Type II errors. (e) Two-taled and One-taled tests: In the context of hypothess testng, these two terms are qute mportant and must be clearly understood. A two-taled test rejects the null hypothess f, say, the sample mean s sgnfcantly hgher or lower than the hypothessed value of the mean of the populaton. Such a test s approprate when the null hypothess s some specfed value and the alternatve hypothess s a value not equal to the specfed value of the null hypothess. Symbolcally, the twotaled test s approprate when we have H0 : µ µ H and H 0 a : µ µ whch may mean H µ > µ H0 0 or µ < µ H0. Thus, n a two-taled test, there are two rejecton regons *, one on each tal of the curve whch can be llustrated as under: Acceptance and rejecton regons n case of a two-taled test (wth 5% sgnfcance level) Rejecton regon Acceptance regon (Accept H0 f the sample mean ( X ) falls n ths regon) Rejecton regon Lmt Lmt of area of area 0.05 of area 0.05 of area Both taken together equals 0.95 or 95% of area Z.96 H 0 Z.96 Reject H0 f the sample mean ( X ) falls n ether of these two regons Fg. 9. Rchard I. Levn, Statstcs for Management, p * Also known as crtcal regons.

6 Testng of Hypotheses I 89 Mathematcally we can state: Acceptance Regon A: Z < 96. Rejecton Regon R : Z > 96. If the sgnfcance level s 5 per cent and the two-taled test s to be appled, the probablty of the rejecton area wll be 0.05 (equally spltted on both tals of the curve as 0.05) and that of the acceptance regon wll be 0.95 as shown n the above curve. If we take µ 00 and f our sample mean devates sgnfcantly from 00 n ether drecton, then we shall reject the null hypothess; but f the sample mean does not devate sgnfcantly from µ, n that case we shall accept the null hypothess. But there are stuatons when only one-taled test s consdered approprate. A one-taled test would be used when we are to test, say, whether the populaton mean s ether lower than or hgher than some hypothessed value. For nstance, f our H0 : µ µ H and H 0 a : µ < µ H, then we are 0 nterested n what s known as left-taled test (wheren there s one rejecton regon only on the left tal) whch can be llustrated as below: Acceptance and rejecton regons n case of one taled test (left-tal) wth 5% sgnfcance Rejecton regon Acceptance regon (Accept H 0 f the sample mean falls n ths regon) Lmt 0.45 of area 0.50 of area 0.05 of area Both taken together equals 0.95 or 95% of area Z.645 H 0 Reject H0 f the sample mean ( X ) falls n ths regon Fg. 9. Mathematcally we can state: Acceptance Regon A: Z > 645. Rejecton Regon R : Z < 645.

7 90 Research Methodology If our µ 00 and f our sample mean devates sgnfcantly from00 n the lower drecton, we shall reject H 0, otherwse we shall accept H 0 at a certan level of sgnfcance. If the sgnfcance level n the gven case s kept at 5%, then the rejecton regon wll be equal to 0.05 of area n the left tal as has been shown n the above curve. In case our H0 : µ µ H and H 0 a : µ > µ H, we are then nterested n what s known as onetaled test (rght tal) and the rejecton regon wll be on the rght tal of the curve as shown 0 below: Acceptance and rejecton regons n case of one-taled test (rght tal) wth 5% sgnfcance level Acceptance regon (Accept H 0 f the sample mean falls n ths regon) Rejecton regon Lmt 0.05 of area 0.45 of area Both taken together equals 0.95 or 95% of area 0.05 of area H 0 Z.645 Reject H 0 f the sample mean falls n ths regon Fg. 9.3 Mathematcally we can state: Acceptance Regon A: Z < 645. Rejecton Regon A: Z > 645. If our µ 00 and f our sample mean devates sgnfcantly from 00 n the upward drecton, we shall reject H 0, otherwse we shall accept the same. If n the gven case the sgnfcance level s kept at 5%, then the rejecton regon wll be equal to 0.05 of area n the rght-tal as has been shown n the above curve. It should always be remembered that acceptng H 0 on the bass of sample nformaton does not consttute the proof that H 0 s true. We only mean that there s no statstcal evdence to reject t, but we are certanly not sayng that H 0 s true (although we behave as f H 0 s true).

8 Testng of Hypotheses I 9 PROCEDURE FOR HYPOTHESIS TESTING To test a hypothess means to tell (on the bass of the data the researcher has collected) whether or not the hypothess seems to be vald. In hypothess testng the man queston s: whether to accept the null hypothess or not to accept the null hypothess? Procedure for hypothess testng refers to all those steps that we undertake for makng a choce between the two actons.e., rejecton and acceptance of a null hypothess. The varous steps nvolved n hypothess testng are stated below: () Makng a formal statement: The step conssts n makng a formal statement of the null hypothess (H 0 ) and also of the alternatve hypothess (H a ). Ths means that hypotheses should be clearly stated, consderng the nature of the research problem. For nstance, Mr. Mohan of the Cvl Engneerng Department wants to test the load bearng capacty of an old brdge whch must be more than 0 tons, n that case he can state hs hypotheses as under: Null hypothess H 0 : µ0 tons Alternatve Hypothess H a : µ > 0 tons Take another example. The average score n an apttude test admnstered at the natonal level s 80. To evaluate a state s educaton system, the average score of 00 of the state s students selected on random bass was 75. The state wants to know f there s a sgnfcant dfference between the local scores and the natonal scores. In such a stuaton the hypotheses may be stated as under: Null hypothess H 0 : µ 80 Alternatve Hypothess H a : µ 80 The formulaton of hypotheses s an mportant step whch must be accomplshed wth due care n accordance wth the object and nature of the problem under consderaton. It also ndcates whether we should use a one-taled test or a two-taled test. If H a s of the type greater than (or of the type lesser than), we use a one-taled test, but when H a s of the type whether greater or smaller then we use a two-taled test. () Selectng a sgnfcance level: The hypotheses are tested on a pre-determned level of sgnfcance and as such the same should be specfed. Generally, n practce, ether 5% level or % level s adopted for the purpose. The factors that affect the level of sgnfcance are: (a) the magntude of the dfference between sample means; (b) the sze of the samples; (c) the varablty of measurements wthn samples; and (d) whether the hypothess s drectonal or non-drectonal (A drectonal hypothess s one whch predcts the drecton of the dfference between, say, means). In bref, the level of sgnfcance must be adequate n the context of the purpose and nature of enqury. () Decdng the dstrbuton to use: After decdng the level of sgnfcance, the next step n hypothess testng s to determne the approprate samplng dstrbuton. The choce generally remans between normal dstrbuton and the t-dstrbuton. The rules for selectng the correct dstrbuton are smlar to those whch we have stated earler n the context of estmaton. (v) Selectng a random sample and computng an approprate value: Another step s to select a random sample(s) and compute an approprate value from the sample data concernng the test statstc utlzng the relevant dstrbuton. In other words, draw a sample to furnsh emprcal data. (v) Calculaton of the probablty: One has then to calculate the probablty that the sample result would dverge as wdely as t has from expectatons, f the null hypothess were n fact true.

9 9 Research Methodology (v) Comparng the probablty: Yet another step conssts n comparng the probablty thus calculated wth the specfed value for α, the sgnfcance level. If the calculated probablty s equal to or smaller than the α value n case of one-taled test (and α / n case of two-taled test), then reject the null hypothess (.e., accept the alternatve hypothess), but f the calculated probablty s greater, then accept the null hypothess. In case we reject H 0, we run a rsk of (at most the level of sgnfcance) commttng an error of Type I, but f we accept H 0, then we run some rsk (the sze of whch cannot be specfed as long as the H 0 happens to be vague rather than specfc) of commttng an error of Type II. FLOW DIAGRAM FOR HYPOTHESIS TESTING The above stated general procedure for hypothess testng can also be depcted n the from of a flowchart for better understandng as shown n Fg. 9.4: 3 FLOW DIAGRAM FOR HYPOTHESIS TESTING State H as well as H 0 a Specfy the level of sgnfcance (or the value) Decde the correct samplng dstrbuton Sample a random sample(s) and workout an approprate value from sample data Calculate the probablty that sample result would dverge as wdely as t has from expectatons, f were true H 0 Is ths probablty equal to or smaller than value n case of one-taled test and / n case of two-taled test Yes No Reject H 0 Accept H 0 thereby run the rsk of commttng Type I error thereby run some rsk of commttng Type II error Fg Based on the flow dagram n Wllam A. Chance s Statstcal Methods for Decson Makng, Rchard D. Irwn INC., Illnos, 969, p.48.

10 Testng of Hypotheses I 93 MEASURING THE POWER OF A HYPOTHESIS TEST As stated above we may commt Type I and Type II errors whle testng a hypothess. The probablty of Type I error s denoted as α (the sgnfcance level of the test) and the probablty of Type II error s referred to as β. Usually the sgnfcance level of a test s assgned n advance and once we decde t, there s nothng else we can do about α. But what can we say about β? We all know that hypothess test cannot be foolproof; sometmes the test does not reject H 0 when t happens to be a false one and ths way a Type II error s made. But we would certanly lke that β (the probablty of acceptng H 0 when H 0 s not true) to be as small as possble. Alternatvely, we would lke that β (the probablty of rejectng H 0 when H 0 s not true) to be as large as possble. If β s very much nearer to unty (.e., nearer to.0), we can nfer that the test s workng qute well, meanng thereby that the test s rejectng H 0 when t s not true and f β s very much nearer to 0.0, then we nfer that the test s poorly workng, meanng thereby that t s not rejectng H 0 when H 0 s not true. Accordngly β value s the measure of how well the test s workng or what s techncally descrbed as the power of the test. In case we plot the values of β for each possble value of the populaton parameter (say µ, the true populaton mean) for whch the H 0 s not true (alternatvely the H a s true), the resultng curve s known as the power curve assocated wth the gven test. Thus power curve of a hypothess test s the curve that shows the condtonal probablty of rejectng H 0 as a functon of the populaton parameter and sze of the sample. The functon defnng ths curve s known as the power functon. In other words, the power functon of a test s that functon defned for all values of the parameter(s) whch yelds the probablty that H 0 s rejected and the value of the power functon at a specfc parameter pont s called the power of the test at that pont. As the populaton parameter gets closer and closer to hypothessed value of the populaton parameter, the power of the test (.e., β ) must get closer and closer to the probablty of rejectng H 0 when the populaton parameter s exactly equal to hypothessed value of the parameter. We know that ths probablty s smply the sgnfcance level of the test, and as such the power curve of a test termnates at a pont that les at a heght of α (the sgnfcance level) drectly over the populaton parameter. Closely related to the power functon, there s another functon whch s known as the operatng characterstc functon whch shows the condtonal probablty of acceptng H 0 for all values of populaton parameter(s) for a gven sample sze, whether or not the decson happens to be a correct one. If power functon s represented as H and operatng characterstc functon as L, then we have L H. However, one needs only one of these two functons for any decson rule n the context of testng hypotheses. How to compute the power of a test (.e., β ) can be explaned through examples. Illustraton A certan chemcal process s sad to have produced 5 or less pounds of waste materal for every 60 lbs. batch wth a correspondng standard devaton of 5 lbs. A random sample of 00 batches gves an average of 6 lbs. of waste per batch. Test at 0 per cent level whether the average quantty of waste per batch has ncreased. Compute the power of the test for µ 6 lbs. If we rase the level of sgnfcance to 0 per cent, then how the power of the test for µ 6 lbs. would be affected?

11 94 Research Methodology Soluton: As we want to test the hypothess that the average quantty of waste per batch of 60 lbs. s 5 or less pounds aganst the hypothess that the waste quantty s more than 5 lbs., we can wrte as under: H 0 : µ< 5 lbs. H a : µ >5 lbs. As H a s one-sded, we shall use the one-taled test (n the rght tal because H a s of more than type) at 0% level for fndng the value of standard devate (z), correspondng to.4000 area of normal curve whch comes to.8 as per normal curve area table. * From ths we can fnd the lmt of µ for acceptng H 0 as under: Accept H0f X < ( α p/ n) or X < e5/ 00j or X < at 0% level of sgnfcance otherwse accept H a. But the sample average s 6 lbs. whch does not come n the acceptance regon as above. We, therefore, reject H 0 and conclude that average quantty of waste per batch has ncreased. For fndng the power of the test, we frst calculate β and then subtract t from one. Snce β s a condtonal probablty whch depends on the value of µ, we take t as 6 as gven n the queston. We can now wrte β p (Accept H 0 : µ < 5 µ 6). Snce we have already worked out that H 0 s accepted f X < 564. (at 0% level of sgnfcance), therefore β p ( X < µ 6) whch can be depcted as follows: Acceptance regon Rejecton regon b g X Fg. 9.5 * Table No.. gven n appendx at the end of the book.

12 Testng of Hypotheses I 95 We can fnd out the probablty of the area that les between 5.64 and 6 n the above curve frst by fndng z and then usng the area table for the purpose. In the gven case z ( X µ )/ ( σ/ n) ( ) /( 5/ 00) 0. 7 correspondng to whch the area s Hence, β and the power of the test ( β ) (.358) for µ 6. In case the sgnfcance level s rased to 0%, then we shall have the followng crtera: b ge Accept H0 f X < or X < 5. 4, otherwse accept H a d β p X < 54. µ 6 or β. 30,, usng normal curve area table as explaned above. b g b g Hence, β TESTS OF HYPOTHESES j As has been stated above that hypothess testng determnes the valdty of the assumpton (techncally descrbed as null hypothess) wth a vew to choose between two conflctng hypotheses about the value of a populaton parameter. Hypothess testng helps to decde on the bass of a sample data, whether a hypothess about the populaton s lkely to be true or false. Statstcans have developed several tests of hypotheses (also known as the tests of sgnfcance) for the purpose of testng of hypotheses whch can be classfed as: (a) Parametrc tests or standard tests of hypotheses; and (b) Non-parametrc tests or dstrbuton-free test of hypotheses. Parametrc tests usually assume certan propertes of the parent populaton from whch we draw samples. Assumptons lke observatons come from a normal populaton, sample sze s large, assumptons about the populaton parameters lke mean, varance, etc., must hold good before parametrc tests can be used. But there are stuatons when the researcher cannot or does not want to make such assumptons. In such stuatons we use statstcal methods for testng hypotheses whch are called non-parametrc tests because such tests do not depend on any assumpton about the parameters of the parent populaton. Besdes, most non-parametrc tests assume only nomnal or ordnal data, whereas parametrc tests requre measurement equvalent to at least an nterval scale. As a result, non-parametrc tests need more observatons than parametrc tests to acheve the same sze of Type I and Type II errors. 4 We take up n the present chapter some of the mportant parametrc tests, whereas non-parametrc tests wll be dealt wth n a separate chapter later n the book. IMPORTANT PARAMETRIC TESTS The mportant parametrc tests are: () z-test; () t-test; ( * 3) χ -test, and (4) F-test. All these tests are based on the assumpton of normalty.e., the source of data s consdered to be normally dstrbuted. 4 Donald L. Harnett and James L. Murphy, Introductory Statstcal Analyss, p * χ - test s also used as a test of goodness of ft and also as a test of ndependence n whch case t s a non-parametrc test. Ths has been made clear n Chapter 0 enttled χ -test.

13 96 Research Methodology In some cases the populaton may not be normally dstrbuted, yet the tests wll be applcable on account of the fact that we mostly deal wth samples and the samplng dstrbutons closely approach normal dstrbutons. z-test s based on the normal probablty dstrbuton and s used for judgng the sgnfcance of several statstcal measures, partcularly the mean. The relevant test statstc *, z, s worked out and compared wth ts probable value (to be read from table showng area under normal curve) at a specfed level of sgnfcance for judgng the sgnfcance of the measure concerned. Ths s a most frequently used test n research studes. Ths test s used even when bnomal dstrbuton or t-dstrbuton s applcable on the presumpton that such a dstrbuton tends to approxmate normal dstrbuton as n becomes larger. z-test s generally used for comparng the mean of a sample to some hypothessed mean for the populaton n case of large sample, or when populaton varance s known. z-test s also used for judgng he sgnfcance of dfference between means of two ndependent samples n case of large samples, or when populaton varance s known. z-test s also used for comparng the sample proporton to a theoretcal value of populaton proporton or for judgng the dfference n proportons of two ndependent samples when n happens to be large. Besdes, ths test may be used for judgng the sgnfcance of medan, mode, coeffcent of correlaton and several other measures. t-test s based on t-dstrbuton and s consdered an approprate test for judgng the sgnfcance of a sample mean or for judgng the sgnfcance of dfference between the means of two samples n case of small sample(s) when populaton varance s not known (n whch case we use varance of the sample as an estmate of the populaton varance). In case two samples are related, we use pared t-test (or what s known as dfference test) for judgng the sgnfcance of the mean of dfference between the two related samples. It can also be used for judgng the sgnfcance of the coeffcents of smple and partal correlatons. The relevant test statstc, t, s calculated from the sample data and then compared wth ts probable value based on t-dstrbuton (to be read from the table that gves probable values of t for dfferent levels of sgnfcance for dfferent degrees of freedom) at a specfed level of sgnfcance for concernng degrees of freedom for acceptng or rejectng the null hypothess. It may be noted that t-test apples only n case of small sample(s) when populaton varance s unknown. χ -test s based on ch-square dstrbuton and as a parametrc test s used for comparng a sample varance to a theoretcal populaton varance. F-test s based on F-dstrbuton and s used to compare the varance of the two-ndependent samples. Ths test s also used n the context of analyss of varance (ANOVA) for judgng the sgnfcance of more than two sample means at one and the same tme. It s also used for judgng the sgnfcance of multple correlaton coeffcents. Test statstc, F, s calculated and compared wth ts probable value (to be seen n the F-rato tables for dfferent degrees of freedom for greater and smaller varances at specfed level of sgnfcance) for acceptng or rejectng the null hypothess. The table on pages 98 0 summarses the mportant parametrc tests along wth test statstcs and test stuatons for testng hypotheses relatng to mportant parameters (often used n research studes) n the context of one sample and also n the context of two samples. We can now explan and llustrate the use of the above stated test statstcs n testng of hypotheses. * The test statstc s the value obtaned from the sample data that corresponds to the parameter under nvestgaton.

14 Testng of Hypotheses I 97 HYPOTHESIS TESTING OF MEANS Mean of the populaton can be tested presumng dfferent stuatons such as the populaton may be normal or other than normal, t may be fnte or nfnte, sample sze may be large or small, varance of the populaton may be known or unknown and the alternatve hypothess may be two-sded or onesded. Our testng technque wll dffer n dfferent stuatons. We may consder some of the mportant stuatons.. Populaton normal, populaton nfnte, sample sze may be large or small but varance of the populaton s known, H a may be one-sded or two-sded: In such a stuaton z-test s used for testng hypothess of mean and the test statstc z s worked our as under: X µ H z σ n p. Populaton normal, populaton fnte, sample sze may be large or small but varance of the populaton s known, H a may be one-sded or two-sded: In such a stuaton z-test s used and the test statstc z s worked out as under (usng fnte populaton multpler): z X µ H0 eσ p nj bn ngbn g 3. Populaton normal, populaton nfnte, sample sze small and varance of the populaton unknown, H a may be one-sded or two-sded: In such a stuaton t-test s used and the test statstc t s worked out as under: X µ H t σ / n X X and σ s n s d b 0 0 wth d.f. (n ) g 4. Populaton normal, populaton fnte, sample sze small and varance of the populaton unknown, and H a may be one-sded or two-sded: In such a stuaton t-test s used and the test statstc t s worked out as under (usng fnte populaton multpler): t X µ H0 eσs/ nj bn ng/ bn g wth d.f. (n )

15 Table 9.3: Names of Some Parametrc Tests along wth Test Stuatons and Test Statstcs used n Context of Hypothess Testng Unknown Test stuaton (Populaton Name of the test and the test statstc to be used parameter characterstcs and other condtons. Random One sample Two samples samplng s assumed n all stuatons along wth Independent Related nfnte populaton Mean ( µ ) Populaton(s) normal or z-test and the z-test for dfference n means and the test Sample sze large (.e., test statstc statstc n > 30) or populaton varance(s) known X µ H X X 0 z z σ p n F I σ p + HG n n KJ In case σ p s not s used when two samples are drawn from the known, we use same populaton. In case σ p s not known, we use σ s n ts place σ s n ts place calculatng calculatng σ s Σd X X n σ s where D X X s e j e s j n σ + D + n σ + D d d D X X X n X n n + n X + n + n Contd. 98 Research Methodology

16 3 4 5 z σ X n OR p X σ + n p s used when two samples are drawn from dfferent populatons. In case σ p and σ p are not known. We use σs and σ s respectvely n ther places calculatng Σd σ s and X X n Σd σ s X X n Mean ( µ ) Populatons(s) normal t-test and the t-test for dfference n means and the test statstc Pared t-test or and test statstc dfference test and sample sze small (.e., X X t + the test statstc n < 30 ) X µ n n and H0 Σ X X + X X t d Σd D 0 t populaton varance(s) σ s n n + n Σ D D, n unknown (but the n wth wth d.f. (n n + n ) populaton varances d.f. (n ) wth d.f (n ) assumed equal n case of test on dfference between where where n number of means) Contd. Testng of Hypotheses I 99

17 3 4 5 ΣdX X σ s pars n two samples. n Alternatvely, t can be worked out as under: R S T X X bn gσ s + bn gσ n + n s + n n wth d. f. n + n b Proporton Repeated ndependent z-test and the z-test for dfference n proportons of two (p) trals, sample sze test statstc samples and the test statstc large (presumng normal approxmaton of bnomal p$ p p$ p$ z z dstrbuton) p q/ n p$ q$ p$ q$ + n n If p and q are not known, then we use p and q n ther places g U V W D dfferences.e., D X Y s used n case of heterogenous populatons. But when populatons are smlar wth respect to a gven attrbute, we work out the best estmate of the populaton proporton as under: p 0 n p$ + n p$ n + n b g Contd. 00 Research Methodology

18 3 4 5 z p q 0 0 p$ p$ F HG + n n I KJ and q p n whch case 0 0 we calculate test statstc varance Populaton(s) χ -test and the test F-test and the test statstc normal, observatons statstc eσ p j X X n s F are ndependent σ d / σs X X / n In the table the varous symbols stand as under: χ σ s n σ p wth d.f. (n ) b g where σ s d s treated > σ s wth d.f. v (n ) for greater varance and d.f. v (n ) for smaller varance X mean of the sample, X mean of sample one, X mean of sample two, n No. of tems n a sample, n No. of tems n sample one, n No. of tems n sample two, µ H0 Hypothessed mean for populaton, σ p standard devaton of populaton, σ s standard devaton of sample, p populaton proporton, q p, p$ sample proporton, q$ p$. Testng of Hypotheses I 0

19 0 Research Methodology X X and σ s n d b 5. Populaton may not be normal but sample sze s large, varance of the populaton may be known or unknown, and H a may be one-sded or two-sded: In such a stuaton we use z-test and work out the test statstc z as under: X µ H z σ / n p (Ths apples n case of nfnte populaton when varance of the populaton s known but when varance s not known, we use σ s n place of σ p n ths formula.) z OR X µ H0 eσp/ nj bn ng/ bn g (Ths apples n case of fnte populaton when varance of the populaton s known but when varance s not known, we use σ s n place of σ p n ths formula.) Illustraton A sample of 400 male students s found to have a mean heght nches. Can t be reasonably regarded as a sample from a large populaton wth mean heght nches and standard devaton.30 nches? Test at 5% level of sgnfcance. Soluton: Takng the null hypothess that the mean heght of the populaton s equal to nches, we can wrte: H0 : µ H " 0 : µ " H a H and the gven nformaton as X ", σ p 30. ", n 400. Assumng the populaton to be normal, we can work out the test statstc z as under: X µ H z 3. σ / n 30. / p As H a s two-sded n the gven queston, we shall be applyng a two-taled test for determnng the rejecton regons at 5% level of sgnfcance whch comes to as under, usng normal curve area table: R : z >.96 The observed value of z s.3 whch s n the acceptance regon snce R : z >.96 and thus H 0 s accepted. We may conclude that the gven sample (wth mean heght 67.47") can be regarded 0 g

20 Testng of Hypotheses I 03 to have been taken from a populaton wth mean heght 67.39" and standard devaton.30" at 5% level of sgnfcance. Illustraton 3 Suppose we are nterested n a populaton of 0 ndustral unts of the same sze, all of whch are experencng excessve labour turnover problems. The past records show that the mean of the dstrbuton of annual turnover s 30 employees, wth a standard devaton of 75 employees. A sample of 5 of these ndustral unts s taken at random whch gves a mean of annual turnover as 300 employees. Is the sample mean consstent wth the populaton mean? Test at 5% level. Soluton: Takng the null hypothess that the populaton mean s 30 employees, we can wrte: H0 : µ H 30 employees 0 H a : µ H0 30 employees and the gven nformaton as under: X 300 employees, σ p 75 employees n 5; N 0 Assumng the populaton to be normal, we can work out the test statstc z as under: z * σ p X µ / n N n / N H 0 b gb g / / b g b g b gb g 0.67 As H a s two-sded n the gven queston, we shall apply a two-taled test for determnng the rejecton regons at 5% level of sgnfcance whch comes to as under, usng normal curve area table: R : z >.96 The observed value of z s 0.67 whch s n the acceptance regon snce R : z >.96 and thus, H 0 s accepted and we may conclude that the sample mean s consstent wth populaton mean.e., the populaton mean 30 s supported by sample results. Illustraton 4 The mean of a certan producton process s known to be 50 wth a standard devaton of.5. The producton manager may welcome any change s mean value towards hgher sde but would lke to safeguard aganst decreasng values of mean. He takes a sample of tems that gves a mean value of What nference should the manager take for the producton process on the bass of sample results? Use 5 per cent level of sgnfcance for the purpose. Soluton: Takng the mean value of the populaton to be 50, we may wrte: H : µ 50 * Beng a case of fnte populaton. H 0 0

21 04 Research Methodology H a :µ 0 < 50 (Snce the manager wants to safeguard aganst decreasng values of mean.) H and the gven nformaton as X we can work out the test statstc z as under: 48., 5 σ. 5 and n. Assumng the populaton to be normal, p X µ H z σ / n 5./. 5 / p b gb As H a s one-sded n the gven queston, we shall determne the rejecton regon applyng onetaled test (n the left tal because H a s of less than type) at 5 per cent level of sgnfcance and t comes to as under, usng normal curve area table: R : z <.645 The observed value of z s.0784 whch s n the rejecton regon and thus, H 0 s rejected at 5 per cent level of sgnfcance. We can conclude that the producton process s showng mean whch s sgnfcantly less than the populaton mean and ths calls for some correctve acton concernng the sad process. Illustraton 5 The specmen of copper wres drawn form a large lot have the followng breakng strength (n kg. weght): 578, 57, 570, 568, 57, 578, 570, 57, 596, 544 Test (usng Student s t-statstc)whether the mean breakng strength of the lot may be taken to be 578 kg. weght (Test at 5 per cent level of sgnfcance). Verfy the nference so drawn by usng Sandler s A-statstc as well. Soluton: Takng the null hypothess that the populaton mean s equal to hypothessed mean of 578 kg., we can wrte: H0 : µ µ H 578 kg. 0 H a :µ µ H 0 As the sample sze s mall (snce n 0) and the populaton standard devaton s not known, we shall use t-test assumng normal populaton and shall work out the test statstc t as under: g t X σ µ s / H n 0 To fnd X and σ s we make the followng computatons: S. No. X X X d dx X Contd.

22 Testng of Hypotheses I 05 S. No. X X X d dx X d 456 n 0 X 570 X X X X kg. n 0 d X X and σ s n kg Hence, t / 0 Degree of freedom (n ) (0 ) 9 As H a s two-sded, we shall determne the rejecton regon applyng two-taled test at 5 per cent level of sgnfcance, and t comes to as under, usng table of t-dstrbuton * for 9 d.f.: R : t >.6 As the observed value of t (.e.,.488) s n the acceptance regon, we accept H 0 at 5 per cent level and conclude that the mean breakng strength of copper wres lot may be taken as 578 kg. weght. The same nference can be drawn usng Sandler s A-statstc as shown below: Table 9.3: Computatons for A-Statstc S. No. X Hypothessed mean D X µ H 0 m 578 kg. H 0 d D contd. * Table No. gven n appendx at the end of the book.

23 06 Research Methodology S. No. X Hypothessed mean D X µ H 0 m 578 kg. H 0 d D n 0 D 60 D 86 b g b g A D / D 86/ Null hypothess H : µ H 578 kg. 0 0 Alternate hypothess H a : µ H0 578 kg. As H a s two-sded, the crtcal value of A-statstc from the A-statstc table (Table No. 0 gven n appendx at the end of the book) for (n ).e., 0 9 d.f. at 5% level s Computed value of A (0.5044), beng greater than 0.76 shows that A-statstc s nsgnfcant n the gven case and accordngly we accept H 0 and conclude that the mean breakng strength of copper wre lot maybe taken as578 kg. weght. Thus, the nference on the bass of t-statstc stands verfed by A-statstc. Illustraton 6 Raju Restaurant near the ralway staton at Falna has been havng average sales of 500 tea cups per day. Because of the development of bus stand nearby, t expects to ncrease ts sales. Durng the frst days after the start of the bus stand, the daly sales were as under: 550, 570, 490, 65, 505, 580, 570, 460, 600, 580, 530, 56 On the bass of ths sample nformaton, can one conclude that Raju Restaurant s sales have ncreased? Use 5 per cent level of sgnfcance. Soluton: Takng the null hypothess that sales average 500 tea cups per day and they have not ncreased unless proved, we can wrte: H 0 : µ 500 cups per day H a : µ > 500 (as we want to conclude that sales have ncreased). As the sample sze s small and the populaton standard devaton s not known, we shall use t-test assumng normal populaton and shall work out the test statstc t as: t X µ σ / s n (To fnd X and σ s we make the followng computatons:)

24 Testng of Hypotheses I 07 Table 9.4 S. No. X X X d dx X d 3978 n 0 X 6576 X X X X n d X X and σ s n Hence, t / Degree of freedom n As H a s one-sded, we shall determne the rejecton regon applyng one-taled test (n the rght tal because H a s of more than type) at 5 per cent level of sgnfcance and t comes to as under, usng table of t-dstrbuton for degrees of freedom: R : t >.796 The observed value of t s whch s n the rejecton regon and thus H 0 s rejected at 5 per cent level of sgnfcance and we can conclude that the sample data ndcate that Raju restaurant s sales have ncreased. HYPOTHESIS TESTING FOR DIFFERENCES BETWEEN MEANS In many decson-stuatons, we may be nterested n knowng whether the parameters of two populatons are alke or dfferent. For nstance, we may be nterested n testng whether female workers earn less than male workers for the same job. We shall explan now the technque of

25 08 Research Methodology hypothess testng for dfferences between means. The null hypothess for testng of dfference between means s generally stated as H 0 : µ µ, where µ s populaton mean of one populaton and µ s populaton mean of the second populaton, assumng both the populatons to be normal populatons. Alternatve hypothess may be of not equal to or less than or greater than type as stated earler and accordngly we shall determne the acceptance or rejecton regons for testng the hypotheses. There may be dfferent stuatons when we are examnng the sgnfcance of dfference between two means, but the followng may be taken as the usual stuatons:. Populaton varances are known or the samples happen to be large samples: In ths stuaton we use z-test for dfference n means and work out the test statstc z as under: z σ X p n + X σ p n In case σ p and σ p are not known, we use σ s and σ s respectvely n ther places calculatng σ s dx d X X X and σs n n. Samples happen to be large but presumed to have been drawn from the same populaton whose varance s known: In ths stuaton we use z test for dfference n means and work out the test statstc z as under: z σ X p F HG X + n n I KJ In case σ p s not known, we use σ s. (combned standard devaton of the two samples) n ts place calculatng d d where D X X. D X X. σ s. e s j e s j n σ + D + n σ + D n + n

26 Testng of Hypotheses I 09 X. nx n + nx + n 3. Samples happen to be small samples and populaton varances not known but assumed to be equal: In ths stuaton we use t-test for dfference n means and work out the test statstc t as under: t X X X + X X + n + n n n X d d wth d.f. (n + n ) Alternatvely, we can also state t X X bn gσs + bn gσs + n + n n n wth d.f. (n + n ) Illustraton 7 The mean produce of wheat of a sample of 00 felds n 00 lbs. per acre wth a standard devaton of 0 lbs. Another samples of 50 felds gves the mean of 0 lbs. wth a standard devaton of lbs. Can the two samples be consdered to have been taken from the same populaton whose standard devaton s lbs? Use 5 per cent level of sgnfcance. Soluton: Takng the null hypothess that the means of two populatons do not dffer, we can wrte H 0 : µ µ H a : µ µ and the gven nformaton as n 00; n 50; X σ 00 lbs.; X 0 lbs.; s 0 lbs.; σ lbs.; and σ p lbs. Assumng the populaton to be normal, we can work out the test statstc z as under: s z σ X p F HG X + n n I KJ 00 0 bg F HG I K J

27 0 Research Methodology As H a s two-sded, we shall apply a two-taled test for determnng the rejecton regons at 5 per cent level of sgnfcance whch come to as under, usng normal curve area table: R : z >.96 The observed value of z s 4.08 whch falls n the rejecton regon and thus we reject H 0 and conclude that the two samples cannot be consdered to have been taken at 5 per cent level of sgnfcance from the same populaton whose standard devaton s lbs. Ths means that the dfference between means of two samples s statstcally sgnfcant and not due to samplng fluctuatons. Illustraton 8 A smple random samplng survey n respect of monthly earnngs of sem-sklled workers n two ctes gves the followng statstcal nformaton: Table 9.5 Cty Mean monthly Standard devaton of Sze of earnngs (Rs) sample data of sample monthly earnngs (Rs) A B Test the hypothess at 5 per cent level that there s no dfference between monthly earnngs of workers n the two ctes. Soluton: Takng the null hypothess that there s no dfference n earnngs of workers n the two ctes, we can wrte: H 0 : µ µ H a : µ µ and the gven nformaton as Sample (Cty A) Sample (Cty B) X 695 Rs X 70 Rs σ s 40 Rs σ s 60 Rs n 00 n 75 As the sample sze s large, we shall use z-test for dfference n means assumng the populatons to be normal and shall work out the test statstc z as under: z X σ n s X s σ + n

28 Testng of Hypotheses I (Snce the populaton varances are not known, we have used the sample varances, consderng the sample varances as the estmates of populaton varances.) Hence z bg bg As H a s two-sded, we shall apply a two-taled test for determnng the rejecton regons at 5 per cent level of sgnfcance whch come to as under, usng normal curve area table: R : z >.96 The observed value of z s.809 whch falls n the rejecton regon and thus we reject H 0 at 5 per cent level and conclude that earnng of workers n the two ctes dffer sgnfcantly. Illustraton 9 Sample of sales n smlar shops n two towns are taken for a new product wth the followng results: Town Mean sales Varance Sze of sample A B Is there any evdence of dfference n sales n the two towns? Use 5 per cent level of sgnfcance for testng ths dfference between the means of two samples. Soluton: Takng the null hypothess that the means of two populatons do not dffer we can wrte: H 0 : µ µ H a : µ µ and the gven nformaton as follows: Table 9.6 Sample from town A as sample one X 57 σ s 53. n 5 Sample from town B As sample two X 6 σ s 48. n 7 Snce n the gven queston varances of the populaton are not known and the sze of samples s small, we shall use t-test for dfference n means, assumng the populatons to be normal and can work out the test statstc t as under: wth d.f. (n + n ) t X X bn gσs + bn gσs + n + n n n

29 Research Methodology b. g b. g Degrees of freedom (n + n ) As H a s two-sded, we shall apply a two-taled test for determnng the rejecton regons at 5 per cent level whch come to as under, usng table of t-dstrbuton for 0 degrees of freedom: R : t >.8 The observed value of t s whch falls n the rejecton regon and thus, we reject H 0 and conclude that the dfference n sales n the two towns s sgnfcant at 5 per cent level. Illustraton 0 A group of seven-week old chckens reared on a hgh proten det wegh, 5,, 6, 4, 4, and 6 ounces; a second group of fve chckens, smlarly treated except that they receve a low proten det, wegh 8, 0, 4, 0 and 3 ounces. Test at 5 per cent level whether there s sgnfcant evdence that addtonal proten has ncreased the weght of the chckens. Use assumed mean (or A ) 0 for the sample of 7 and assumed mean (or A ) 8 for the sample of 5 chckens n your calculatons. Soluton: Takng the null hypothess that addtonal proten has not ncreased the weght of the chckens we can wrte: H 0 : µ µ H a : µ > µ (as we want to conclude that addtonal proten has ncreased the weght of chckens) Snce n the gven queston varances of the populatons are not known and the sze of samples s small, we shall use t-test for dfference n means, assumng the populatons to be normal and thus work out the test statstc t as under: t X X bn gσs + bn gσs + n + n n n wth d.f. (n + n ) From the sample data we work out X, X, σ s and σ s sample one and low proten det sample as sample two) as shown below: (takng hgh proten det sample as