TESTS OF SIGNIFICANCE

Transcription

1 TESTS OF SIGNIFICANCE Seema Jaggi I.A.S.R.I., Library Aveue, New Delhi I applied ivetigatio, oe i ofte itereted i comparig ome characteritic (uch a the mea, the variace or a meaure of aociatio betwee two character) of a group with a pecified value, or i comparig two or more group with regard to the characteritic. For itace, oe may wih to compare two varietie of wheat with regard to the mea yield per hectare or to kow if the geetic fractio of the total variatio i a trai i more tha a give value or to compare differet lie of a crop i repect of variatio betwee plat withi lie. I makig uch compario oe caot rely o the mere umerical magitude of the idex of compario uch a the mea, variace or meaure of aociatio. Thi i becaue each group i repreeted oly by a ample of obervatio ad if aother ample were draw the umerical value would chage. Thi variatio betwee ample from the ame populatio ca at bet be reduced i a well-deiged cotrolled experimet but ca ever be elimiated. Oe i forced to draw iferece i the preece of the amplig fluctuatio which affect the oberved differece betwee group, cloudig the real differece. Statitical ciece provide a objective procedure for ditiguihig whether the oberved differece coote ay real differece amog group. Such a procedure i called a tet of igificace. The tet of igificace i a method of makig due allowace for the amplig fluctuatio affectig the reult of experimet or obervatio. The fact that the reult of biological experimet are affected by a coiderable amout of ucotrolled variatio make uch tet eceary. Thee tet eable u to decide o the bai of the ample reult, if i) the deviatio betwee the oberved ample tatitic ad the hypothetical parameter value, or ii) the deviatio betwee two ample tatitic, i igificat or might be attributed to chace or the fluctuatio of amplig. For applyig the tet of igificace, we firt et up a hypothei - a defiite tatemet about the populatio parameter. I all uch ituatio we et up a exact hypothei uch a, the treatmet or variate i quetio do ot differ i repect of the mea value, or the variability, or the aociatio betwee the pecified character, a the cae may be, ad follow a objective procedure of aalyi of data which lead to a cocluio of either of two kid: i) reject the hypothei, or ii) ot reject the hypothei. Tet of Sigificace for Large Sample For large (ample ize), almot all the ditributio ca be approximated very cloely by a ormal probability curve, we therefore ue the ormal tet of igificace for large ample. If t i ay tatitic (fuctio of ample value), the for large ample Z t - E(t) V(t) N (.)

2 Tet of Sigificace Thu if the dicrepacy betwee the oberved ad the expected (hypothetical) value of a tatitic i greater tha Z α time the tadard error (S.E.), hypothei i rejected at α level of igificace. Similarly if t E(t) Z α S.E(t), the deviatio i ot regarded igificat at 5% level of igificace. I other word the deviatio t - E(t), could have arie due to fluctuatio of amplig ad the data do ot provide ay evidece agait the ull hypothei which may, therefore be accepted at α level of igificace. If Z.96, the the hypothei H i accepted at 5% level of igificace. Thu the tep to be ued i the ormal tet are a follow: i) Compute the tet tatitic Z uder H. ii) If Z > 3, H i alway rejected iii) If Z < 3, we tet it igificace at certai level of igificace The table below give ome critical value of Z: Level of Sigificace Critical value (Z α ) of Z Two-tailed tet Sigle tailed tet % % % Tet for Sigle Mea A very importat aumptio uderlyig the tet of igificace for variable i that the ample mea i aymptotically ormally ditributed eve if the paret populatio from which the ample i draw i ot ormal. If x i ( i,,) i a radom ample of ize from a ormal populatio with mea μ ad variace, the the ample mea i ditributed ormally with mea μ ad variace /, i.e., x ~ N ( μ, / ) H : Populatio mea μ a give value μ ; H : μ μ Tet Statitic: x μ Z ~ N (,) If i ukow, the it i etimated by ample variace i.e., (for large ). Example..: A ample of 9 member ha a mea of 3.4 cm ad tadard deviatio (.d.).6 cm. I the ample draw from a large populatio of mea 3.5 cm? Solutio: II-74

3 Tet of Sigificace H : The ample ha bee draw from the populatio with mea μ 3.5 cm H : μ 3.5 (two tailed tet) Here x 3.4 cm, 9, μ 3.5 cm,.6 cm Uder H, Z Sice Z <.96, we coclude that the data doe ot provide ay evidece agait the ull hypothei H which may therefore be accepted at 5% level of igificace.. Tet for Differece of Mea Let x ( x ) be the mea of a ample of ize ( ) from a populatio with mea μ (μ ) ad variace ( ). Therefore x ), ~ N (μ, x ~ N(μ, ) ) The differece ( x - x) i alo a ormal variate Tet Statitic: x x E (x x Z ) ~ N (,) S.E (x x ) Z x x (μ μ ) + Uder the ull hypothei H : μ μ Z Therefore (x x ) + ~ N (,) (x x Z ), If + If i ot kow, the it etimate i ued + ˆ + II-75

4 Tet of Sigificace.3 Tet for Sigle Proportio Suppoe i a ample of ize, x be the umber of pero poeig the give attribute. x The oberved proportio of uccee p x E(p) E ( ) E (x) P (populatio proportio) PQ ad V(p), Q - P The ormal tet for the proportio of uccee become p - E (p) Z S.E (p) p - P PQ/ ~ N (,) Example.3.: I a ample of people, 54 are rice eater ad the ret are wheat eater. Ca we aume that both rice ad wheat are equally popular at % level of igificace. Solutio: It i give that, x No. of rice eater 54, 54 p ample proportio of rice eater.54, P Populatio proportio of rice eater.5. H : Both rice ad wheat are equally popular; H : P.5 p - P Z.53 PQ/.5 x.5/ Sice computed Z <.58 at % level of igificace, therefore H o i ot rejected ad we coclude that rice ad wheat are equally popular..4 Tet for Differece of Proportio Suppoe we wat to compare two populatio with repect to the prevalece of a certai attribute A. Let x (x ) be the umber of pero poeig the give attribute A i radom ample of ize ( ) from t ( d ) populatio. The ample proportio will be x x p, p Let P ad P be the populatio proportio. P V(p ) Q P, V(p ) Q, Tet Statitic: p Z p ( P P ) ~ N (,) P Q P Q + Uder H : P P P i.e. o igificat differece betwee populatio proportio II-76

5 Tet of Sigificace Z p p PQ( + ). Tet of Sigificace for Small Sample I thi ectio, the tatitical tet baed o t, ad F are give.. Tet Baed o t-ditributio.. Tet for a Aumed Populatio Mea Suppoe a radom ample x,..,x of ize ( ) ha bee draw from a ormal populatio whoe variace i ukow. O the bai of thi radom ample the aim i to tet H : μ μ H : μ μ (two-tailed) μ > μ (right-tailed) μ < μ (left-tailed) Tet tatitic: x μ t ~ t / where x x i, (xi x) The table givig the value of t required for igificace at variou level of probability ad for differet degree of freedom are called the t table which are give i Statitical Table by Fiher ad Yate. The computed value i compared with the tabulated value at 5 or percet level of igificace ad at (-) degree of freedom ad accordigly the ull hypothei i accepted or rejected... Tet for the Differece of Two Populatio Mea Let x (x ) be the ample mea of a ample of ize ( ) from a populatio with mea μ (μ ) ad variace of the two populatio be ame, which i ukow. H : μ - μ δ Sice x ~ N (μ, / ) ; x ~ N (μ, / ). Therefore, x - x ~ N (μ - μ, + ). Tet tatitic: Uder H x x δ t ~ t + + Sice i ukow, therefore, it i etimated from the ample ( ) ( ) + + II-77

6 Tet of Sigificace x x δ t + ~ t + If δ, thi tet reduce to tet the equality of two populatio mea. Example...: A group of 5 plot treated with itroge at kg/ha. yielded 4, 39, 48, 6 ad 4 kg wherea ecod group of 7 plot treated with itroge at 4 kg/ha. yielded 38, 4, 56, 64, 68, 69 ad 6 kg. Ca it be cocluded that itroge at level 4 kg/ha. icreae the yield igificatly? Solutio: H : μ μ, H : μ < μ x 46, x 57, t -.7.6( + ) 5 7 ~ t Sice t <.8 (value of t at 5% ad d.f.), the yield from two doe of itroge do ot differ igificatly...3 Paired t-tet for Differece of Mea Whe ad the two ample are ot idepedet but the ample obervatio are paired together, the thi tet i applied. Let (x i, y i ), i,.., be a radom ample from a bivariate ormal populatio with parameter (μ, μ,,, ρ). Let d i x i - y i H : μ - μ μ Tet tatitic: d μ t ~ t- / where d d i, (d i d)...4 Tet for Sigificace of Oberved Correlatio Coefficiet Give a radom ample (x i, y i ), i,, from a bivariate ormal populatio. We wat to tet the ull hypothei that the populatio correlatio coefficiet i zero i.e. H : ρ ; H : ρ r Tet Statitic: t ~ t, r where r i the ample correlatio coefficiet. H i rejected at level α if t > t - (α/). Thi tet ca alo be ued for tetig the igificace of rak correlatio coefficiet.. Tet of Sigificace Baed o Chi-Square Ditributio.. Tet for the Variace of a Normal Populatio Let x, x,,x ( ) be a radom ample from N(μ, ). H :. II-78

7 Tet of Sigificace Tet tatitic: xi μ ~, whe μ i kow xi x ~, whe μ i ot kow Table are available for at differet level of igificace ad with differet degree of freedom... Tet for Goode of Fit A tet of wide applicability to umerou problem of igificace i frequecy data i the tet of goode of fit. It i primarily ued for tetig the dicrepacy betwee the expected ad the oberved frequecy, for itace, i comparig a oberved frequecy ditributio with a theoretical oe like the ormal. H : the fitted ditributio i a good fit to the give data; H : ot a good fit. Tet tatitic: If O i ad E i, i,, are repectively the oberved ad expected frequecy of i th cla, the the tatitic ( ) Oi E i ~ E -r - i where r i the umber of parameter etimated from the ample, i the umber of clae after poolig. H i rejected at level α if calculated > tabulated (α) Example..: I a F populatio of chillie, 83 plat with purple ad 69 with opurple chillie were oberved. I thi ratio coitet with a igle factor ratio of 3:? Solutio: O the hypothei of a ratio of 3:, the frequecie expected i the purple ad opurple clae are 85 ad 75 repectively. Frequecy Oberved (O i ) Expected (E i ) O i - E i Purpoe No-purple (O E ) i i.7 E i Here i baed o oe degree of freedom. It i ee from the table that the value of.7 for with d.f correpod to a level of probability which lie betwee.5 ad.7. It i cocluded that the reult i o-igificat...3 Tet of Idepedece Aother commo ue of the tet i i tetig idepedece of claificatio i what are kow a cotigecy table. Whe a group of idividual ca be claified i two way the reult of the claificatio i two way the reult of the claificatio ca be et out a follow: Cotigecy table -r- II-79

8 Tet of Sigificace Cla A A A 3 B 3 B 3 B Such a table givig the imultaeou claificatio of a body of data i two differet way i called cotigecy table. If there are r row ad c colum the table i aid to be a r x c table. H : the attribute are idepedet H : they are ot idepedet Tet tatitic: c r j H i rejected at level α if (O ij - Eij ) Eij > ~ (r-)(c-) (r -)(c-).3 Tet of Sigificace Baed o F-Ditributio.3. Tet for the Compario of Two Populatio Variace Let x i, i,, ad x j, j,, be the two radom ample of ize ad draw from two idepedet ormal populatio N ( μ, ) ad N ( μ, ) repectively. ad are the ample variace of the two ample. (xi x ) ad j (x x ) j x xi, x x j j H : the ratio of two populatio variace i pecified Tet tatitic: Aumig > F ~ F, Uder H : F ~ F,. Table are available givig the value of F required for igificace at differet level of probability ad for differet degree of freedom. The computed value of F i compared with the tabulated value ad the iferece i draw accordigly. II-8