Tools Hypothesis Tests

Transcription

1 Tool Hypothei Tet The Tool meu provide acce to a Hypothei Tet procedure that calculate cofidece iterval ad perform hypothei tet for mea, variace, rate ad proportio. It i cotrolled by the dialog box how below: The iput field are: Tet: elect the radio butto for the tet you wih to perform. Tet are available for the mea of a ormal ditributio, the tadard deviatio of a ormal ditributio, a biomial proportio, ad a Poio rate. Alogide the radio butto, eter the value of the relevat ample tatitic. Sample : check thi box if you wih to compare two ample. Size: idicate the umber of obervatio i each ample. Null hyp: eter the value of the populatio parameter to be teted. Alt. hyp: elect a two-ided tet (NE for ot equal ) or oe-ided tet (LT for le tha or GT for greater tha ). 6 by StatPoit, Ic. Tool Hypothei Tet -

2 Alpha: the alpha rik of the tet (probability of rejectig a true ull hypothei). Thi value i ued to geerate cofidece iterval. Sigma kow: Whe tetig ormal mea, check thi box to perform a z-tet rather tha a t-tet. Equal igma: Whe comparig two ormal mea, check thi box to aume that the tadard deviatio of both populatio are equal (thi i the uual aumptio). Pre the OK butto to perform the tet, or the X butto to ed the procedure. If you wih to retur to the dialog box to make chage oce tatitic have bee calculated, elect Meu Recalc. 6 by StatPoit, Ic. Tool Hypothei Tet -

3 Tet of a Sigle Mea To tet the value of a igle populatio mea, eter:. Sample mea: the ample mea x.. Sample variace: If Sigma kow i ot checked, eter the ample variace. Otherwie, eter the kow value of the populatio variace σ. 3. Size: the ample ize. 4. Null hyp: the hypotheized value of the populatio mea μ. The example below tet the hypothei μ = 5 baed o x = 48.3, = 8.3, ad = 3. The output iclude:. A plot of the ormal ditributio with the mea pecified by the ull hypothei ad the idicated variace. 6 by StatPoit, Ic. Tool Hypothei Tet - 3

4 . A (-α)% two-ided cofidece iterval or oe-ided cofidece boud for μ. The limit are how graphically ad alo diplayed alogide Lower ad Upper i the tabular output. The two-ided cofidece iterval i calculated by σ x ± zα / () if σ i kow ad by ± t α /, () x if σ i ot kow. 3. Value of the tet tatitic for the ull hypothei. The tet tatitic i x μ o z = (3) σ / if σ i kow ad by x μo t = (4) / if σ i ot kow. 4. A P-value for the hypothei tet. If P < α, the the ull hypothei i rejected i favor of the alterative hypothei. P i calculated uig a tadard ormal ditributio if σ i kow ad from Studet t-ditributio with degree of freedom if σ i ot kow. For the ample data, the 95% cofidece iterval for the mea ru from 46.7 to The hypothei that the mea equal 5 i rejected at the 5% igificace level ice P <.5. 6 by StatPoit, Ic. Tool Hypothei Tet - 4

5 Tet to Compare Two Mea To compare the value of two populatio mea, eter:. Sample ad ample mea: the ample mea x ad x.. Sample ad ample variace: If Sigma kow i ot checked, eter the ample variace ad. Otherwie, eter the kow value of the populatio variace σ adσ. 3. Size: the ample ize ad. 4. Null hyp: the hypotheized value of the differece betwee the populatio mea Δ = μ. μ The ample below tet the hypothei Δ = baed o x = 48.3, x = 5.7, = 3, ad = 8, aumig equal igma. = 8.3, = 6., The output iclude: 6 by StatPoit, Ic. Tool Hypothei Tet - 5

6 . A plot of a ormal ditributio with the mea pecified by the ull hypothei ad the kow or etimated tadard error for the differece betwee the mea.. A (-α)% two-ided cofidece iterval or oe-ided cofidece boud for Δ. If σ ad σ are aumed to be kow, the the two-ided iterval i ( x x ) σ σ ± z (5) α / + If σ ad σ are etimated from the data ad aumed to be equal, the the two-ided iterval i where ad x ± tα /, ν p + (6) ( x ) ( ) + ( ) p = (7) + ν = + (8) If σ ad σ are etimated from the data ad ot aumed to be equal, the the two-ided iterval i: where x ± tα /, m + (9) ( x ) m = c + ( c) () ad / c = () / + / 3. Value of the tet tatitic for the ull hypothei. If σ ad σ are aumed to be kow, the tet tatitic i 6 by StatPoit, Ic. Tool Hypothei Tet - 6

7 z ( x x ) Δ σ + = () σ If σ ad σ are etimated from the data ad aumed to be equal, the tet tatitic i t ( x x ) = ~ t v (3) p + Δ If σ ad σ are etimated from the data ad ot aumed to be equal, the tet tatitic i t ( x x ) = ~ t m (4) Δ + 4. A P-value for the hypothei tet. If P < α, the the ull hypothei i rejected i favor of the alterative hypothei. P i calculated uig a tadard ormal ditributio if the variace are kow ad from Studet t ditributio if the variace are ot kow. For the ample data, the 95% cofidece iterval for the differece betwee the mea ru from to.75. The hypothei that the differece betwee the mea equal i ot rejected at the 5% igificace level ice P.5. 6 by StatPoit, Ic. Tool Hypothei Tet - 7

8 Tet of a Sigle Variace To tet the value of a igle populatio variace, eter:. Sample variace: the ample variace.. Size: the ample ize. 3. Null hyp: the hypotheized value of the populatio variace σ. The ample below tet the hypothei σ = 5 baed o = 8.3, ad = 3. The output iclude:. A plot of a caled chi-quare ditributio with a vertical lie draw at the value of the variace pecified by the ull hypothei.. A (-α)% two-ided cofidece iterval or oe-ided cofidece boud for σ. The two-ided iterval i 6 by StatPoit, Ic. Tool Hypothei Tet - 8

9 ( ) ( ), χ α χ /, α /, (5) 3. Value of the tet tatitic for the ull hypothei. The tet tatitic i ( ) χ = (6) σ 4. A P-value for the hypothei tet. If P < α, the the ull hypothei i rejected i favor of the alterative hypothei. P i calculated uig a chi-quare ditributio with degree of freedom. For the ample data, the 95% cofidece iterval for the variace ru from.6 to 33.. The hypothei that the variace equal 5 i ot rejected at the 5% igificace level ice P.5. 6 by StatPoit, Ic. Tool Hypothei Tet - 9

10 Tet to Compare Two Variace To compare the value of two populatio variace, eter:. Sample ad ample variace: the ample variace. Size: the ample ize ad. ad. 3. Null hyp: the hypotheized value of the ratio of the populatio variace ρ =. σ /σ The ample below tet the hypothei that ρ = baed o = 8. = 8.3, = 6., = 3, ad The output iclude:. A plot of a caled F ditributio with a vertical lie draw at the value of the variace ratio pecified by the ull hypothei.. A (-α)% two-ided cofidece iterval or oe-ided cofidece boud for ρ. The two-ided iterval i 6 by StatPoit, Ic. Tool Hypothei Tet -

11 F, α /,, F α /,, (7) 3. Value of the tet tatitic for the ull hypothei. The tet tatitic i / F = (8) ρ 4. A P-value for the hypothei tet. If P < α, the the ull hypothei i rejected i favor of the alterative hypothei. P i calculated uig a F ditributio with ad degree of freedom. For the ample data, the 95% cofidece iterval for the ratio of the variace ru from.33 to.48. The hypothei that the ratio of the variace equal i ot rejected at the 5% igificace level ice P.5. 6 by StatPoit, Ic. Tool Hypothei Tet -

12 Tet of a Sigle Proportio To tet the value of a igle proportio, eter:. Sample proportio: the ample proportio p.. Size: the ample ize. 3. Null hyp: the hypotheized value of the populatio proportio θ. The ample below tet the hypothei θ =.5 baed o p =.46 ad = 3. The output iclude:. A plot of a biomial ditributio with a vertical lie draw at the value of the mea pecified by the ull hypothei.. A (-α)% two-ided cofidece iterval or oe-ided cofidece boud for θ. The two-ided iterval i 6 by StatPoit, Ic. Tool Hypothei Tet -

13 v v F α /, v, v + vf α /, v, v v3fα, v + v F 4 3 v v /, 3, 4 α v v /, 3, 4 (9) where v p = () v = ( p ) () + v = ( p ) () 3 + v4 = ( p) (3) 3. For large ample, the value of the tet tatitic for the ull hypothei. The tet tatitic i p θ z = (4) θ ( θ ) 4. A P-value for the hypothei tet. If P < α, the the ull hypothei i rejected i favor of the alterative hypothei. For large ample, P i calculated uig a tadard ormal ditributio. For mall ample, P i calculated directly from the cumulative biomial ditributio. For a two-ided tet { F ( p, θ ),( F ( p, ))} P = mi θ (5) B B For the ample data, the 95% cofidece iterval for the populatio proportio ru from.43 to.58. The hypothei that the populatio proportio equal.5 i ot rejected at the 5% igificace level ice P.5. 6 by StatPoit, Ic. Tool Hypothei Tet - 3

14 Tet to Compare Two Proportio To compare the value of two proportio, eter:. Sample ad ample proportio: the ample proportio p ad p.. Size: the ample ize ad. 3. Null hyp: the hypotheized value of the differece betwee the populatio proportio Δ = θ. θ The ample below tet the hypothei Δ = baed o p =.46, p =.4, = 3, ad = 8. The output iclude:. A plot of a approximatig ormal ditributio for the differece betwee the proportio with a vertical lie draw at the value pecified by the ull hypothei.. A (-α)% two-ided cofidece iterval or oe-ided cofidece boud for Δ. The two-ided iterval i ( p p ) ± z / p( p) / + p ( p ) / α (6) 6 by StatPoit, Ic. Tool Hypothei Tet - 4

15 3. Value of the tet tatitic for the ull hypothei. If the hypotheized Δ =, the tet tatitic i z = Δˆ / ˆ( θ ˆ) θ / ˆ ˆ (7) + θ ( θ ) / where ( p + p ) ( + ) ˆ θ = (8) / If the hypotheized Δ, the tet tatitic i ( p p Δ )/ p( p) / + p ( p ) / z = (9) 4. A P-value for the hypothei tet. If P < α, the the ull hypothei i rejected i favor of the alterative hypothei. P i calculated uig a tadard ormal ditributio. For the ample data, the 95% cofidece iterval for the differece betwee the proportio ru from -.4 to.. The hypothei that the differece betwee the proportio equal i ot rejected at the 5% igificace level ice P.5. 6 by StatPoit, Ic. Tool Hypothei Tet - 5

16 Tet of a Sigle Rate To tet the value of a igle rate, eter:. Sample rate: the ample rate r.. Size: the ample ize. Note that doe ot have to be a iteger i thi cae, ice it repreet the ize of the amplig iterval or regio over which the rate wa meaured (ofte time or pace). 3. Null hyp: the hypotheized value of the populatio proportio λ. The ample below tet the hypothei λ = baed o r = 8.5 ad = 5. The output iclude:. A plot of a Poio ditributio with a vertical lie draw at the value of the mea pecified by the ull hypothei.. A (-α)% two-ided cofidece iterval or oe-ided cofidece boud for λ. The two-ided iterval i 6 by StatPoit, Ic. Tool Hypothei Tet - 6

17 ,r χα /,( r+ ) χ α /, (3) 3. For large ample, the value of the tet tatitic for the ull hypothei. The tet tatitic i r λ z = (3) λ 4. A P-value for the hypothei tet. If P < α, the the ull hypothei i rejected i favor of the alterative hypothei. For large ample, P i calculated uig a tadard ormal ditributio. For mall ample, P i calculated directly from the cumulative Poio ditributio. For a two-ided tet: { F ( r, λ ),( F ( r, ))} P = mi λ (3) P P For the ample data, the 95% cofidece iterval for the populatio rate ru from 6.4 to.47. The hypothei that the rate equal i ot rejected at the 5% igificace level ice P.5. 6 by StatPoit, Ic. Tool Hypothei Tet - 7

18 Tet to Compare Two Rate To compare the value of two rate, eter: 4. Sample ad ample rate: the ample rate r ad r. 5. Size: the ample ize ad. 6. Null hyp: the hypotheized value of the differece betwee the populatio rate Δ = λ. λ The ample below tet the hypothei Δ = baed o r = 8.5, r =5., = 5, ad = 5. The output iclude:. A plot of a approximatig ormal ditributio for the differece betwee the rate with a vertical lie draw at the value pecified by the ull hypothei.. A (-α)% two-ided cofidece iterval or oe-ided cofidece boud for Δ. The two-ided iterval i 6 by StatPoit, Ic. Tool Hypothei Tet - 8

19 ( r r ) ± z ˆ ˆ α / λ / + λ / (33) 3. Value of the tet tatitic for the ull hypothei. If the hypotheized Δ =, the tet tatitic i where ( r r )/ ˆ λ / ˆ + λ / z = (34) ( r + r ) ( + ) ˆ λ = (35) / If the hypotheized Δ, the tet tatitic i ( Δˆ Δ )/ r / r / z = + (36) 4. A P-value for the hypothei tet. If P < α, the the ull hypothei i rejected i favor of the alterative hypothei. P i calculated uig a tadard ormal ditributio. For the ample data, the 95% cofidece iterval for the differece betwee the two rate ru from -.5 to The hypothei that the differece betwee the rate equal i rejected at the 5% igificace level ice P <.5. 6 by StatPoit, Ic. Tool Hypothei Tet - 9