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Confidence Intervals and Hypothesis Testing Formulas, Study Guides, Projects, Research of Statics

Asia Europe University Statics

Formulas for calculating confidence intervals and hypothesis testing in statistics, including z-scores, t-scores, chi-square, F-distribution, and various probability distributions. It also includes formulas for Mann-Whitney U test, Wilcoxon rank-sum test, sign test, and Kruskal-Wallis test.

Typology: Study Guides, Projects, Research

2013/2014

Uploaded on 08/19/2021

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INTERVALOS DE CONFIANZA

Estimación

Condiciones

Intervalo de Confianza Bilateral

𝜇

𝜎2 (𝐶𝑜𝑛𝑜𝑐𝑖𝑑𝑎)

𝑥−𝑧(∝

2)𝜎

√𝑛≤𝜇≤𝑥+𝑧(∝

2)𝜎

√𝑛

𝜇1−𝜇2

𝜎12,𝜎22 (𝐶𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠)

(𝑥𝟏−𝑥𝟐)−𝑧(∝

2)√𝜎12

𝑛1+𝜎22

𝑛2≤𝜇1−𝜇2≤(𝑥𝟏−𝑥𝟐)+𝑧(∝

2)√𝜎12

𝑛1+𝜎22

𝑛2

𝜇

𝜎2 (𝑑𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎)

𝑃𝑜𝑏𝑙𝑎𝑐𝑖ó𝑛 𝑁𝑜𝑟𝑚𝑎𝑙

𝑥−𝑡(∝

2,𝑛−1) 𝑠

√𝑛≤𝜇≤𝑥+𝑡(∝

2,𝑛−1) 𝑠

√𝑛

𝜇1−𝜇2

𝜎12= 𝜎22

𝐷𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠

(𝑥𝟏−𝑥𝟐)−𝑡(∝

2,𝑣)𝑠𝑝√1

𝑛1+1

𝑛2≤𝜇1−𝜇2≤(𝑥𝟏−𝑥𝟐)+𝑡(∝

2,𝑣)𝑠𝑝√1

𝑛1+1

𝑛2

𝑠𝑝=√(𝑛1−1)𝑠12+(𝑛2−1)𝑠22

𝑛1+𝑛2−2 ; 𝑣 =𝑛1+𝑛2−2

𝜇1−𝜇2

𝜎12≠ 𝜎22

𝐷𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠

𝑃𝑜𝑏𝑙𝑎𝑐𝑖𝑜𝑛𝑒𝑠

𝑛𝑜𝑟𝑚𝑎𝑙𝑒𝑠

(𝑥1−𝑥2)−𝑡(∝

2,𝑣)√𝑠12

𝑛1+𝑠22

𝑛2≤𝜇1−𝜇2≤(𝑥𝟏−𝑥𝟐)+𝑡(∝

2,𝑣)√𝑠12

𝑛1+𝑠22

𝑛2

𝑣= (𝑠1

2

𝑛1+𝑠2

2

𝑛2)2

(𝑠1

2

𝑛1)2

𝑛1+1+(𝑠2

2

𝑛2)2

𝑛2+1−2

𝜎2

𝑃𝑜𝑏𝑙𝑎𝑐𝑖ó𝑛 𝑁𝑜𝑟𝑚𝑎𝑙

(𝑛−1)𝑠2

𝜒(𝛼

2,𝑛−1)

2≤𝜎2≤(𝑛−1)𝑠2

𝜒(1−𝛼

2,𝑛−1)

2

𝜎12

𝜎22

𝑃𝑜𝑏𝑙𝑎𝑐𝑖𝑜𝑛𝑒𝑠

𝑛𝑜𝑟𝑚𝑎𝑙𝑒𝑠

𝑠12

𝑠22𝑓(1−∝

2,𝑛2−1,𝑛1−1)≤𝜎12

𝜎22≤𝑠12

𝑠22𝑓(∝

2,𝑛2−1,𝑛1−1)

𝑓(∝

2,𝑛2−1,𝑛1−1)=1

𝑓(1−∝

2,𝑛1−1,𝑛2−1)

𝑝

𝑃𝑜𝑏𝑙𝑎𝑐𝑖𝑜𝑛𝑒𝑠

𝑛𝑜𝑟𝑚𝑎𝑙𝑒𝑠

𝑝−𝑧(∝

2)√𝑝(1−𝑝)

𝑛≤𝑝≤𝑝+𝑧(∝

2)√𝑝(1−𝑝)

𝑛

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INTERVALOS DE CONFIANZA

Estimación Condiciones Intervalo de Confianza Bilateral

𝜇 𝜎^2 (𝐶𝑜𝑛𝑜𝑐𝑖𝑑𝑎) 𝑥̅^ −^ 𝑧(∝ 2 )

√𝑛^

2 )

𝜇 1 − 𝜇 2 𝜎 12 , 𝜎 22 (𝐶𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠) (𝑥̅𝟏^ −^ 𝑥̅𝟐)^ −^ 𝑧(∝ 2 )√

2 )

√𝜎^1

2 𝑛 1

𝑃𝑜𝑏𝑙𝑎𝑐𝑖ó𝑛 𝑁𝑜𝑟𝑚𝑎𝑙

2 ,𝑛−^1 )

2 ,𝑣)

𝑠𝑝√^

𝑛 1 +^

𝑛 2 ≤^ 𝜇^1 −^ 𝜇^2 ≤^

2 ,𝑣)

𝑠𝑝√^

𝑛 1 +^

𝑛 1 + 𝑛 2 − 2 ;^ 𝑣^ =^ 𝑛^1 +^ 𝑛^2 −^2

2 ,𝑣)

√𝑠^1

2 𝑛 1

2 ,𝑣)

√𝑠^1

2 𝑛 1

(𝑠^1

2 𝑛 1 +^

2

(𝑠^1

2

(𝑠^2

2

𝜎^2 𝑃𝑜𝑏𝑙𝑎𝑐𝑖ó𝑛 𝑁𝑜𝑟𝑚𝑎𝑙

(𝑛 − 1 )𝑠^2

2 ,𝑛−^1 )

2 ≤^ 𝜎

2 ≤ (𝑛^ −^1 )𝑠

2

2 ,𝑛−^1 )

2

2 ,𝑛^2 −^1 ,𝑛^1 −^1 )^

2 ,𝑛^2 −^1 ,𝑛^1 −^1 )

2 ,𝑛^2 −^1 ,𝑛^1 −^1 )^

2 ,𝑛^1 −^1 ,𝑛^2 −^1 )

2 )

√𝑝̂^ (^1 −^ 𝑝̂^ )

𝑛 ≤^ 𝑝^ ≤^ 𝑝̂^ +^ 𝑧(∝ 2 )

√𝑝̂^ (^1 −^ 𝑝̂^ )

𝑝 1 − 𝑝 2 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑐𝑖ó𝑛 𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙

2 )

PRUEBAS DE HIPÓTESIS

𝑯𝟎 Condiciones Estadístico calculado

𝜇 = 𝜇 0 𝜎^2 (𝐶𝑜𝑛𝑜𝑐𝑖𝑑𝑎)

𝜇 = 𝜇 0 𝜎^2 (𝐷𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎)

√𝜎^1

2 𝑛 1 +^

𝑠𝑝√ 𝑛^1

1

+ 𝑛^1

2

𝑛 1 + 𝑛 2 − 2 ;^ 𝑣^ =^ 𝑛^1 +^ 𝑛^2 −^2

√𝑠^1

2 𝑛 1 +^

(𝑠^1

2 𝑛 1 +^

2

𝜎^2 = 𝜎 02 𝜒𝑐𝑎𝑙

2 = (𝑛^ −^1 )𝑠

2 𝜎 02

𝑵: 𝑁ú𝑚𝑒𝑟𝑜 𝑡𝑜𝑡𝑎𝑙 𝑑𝑒 𝑑𝑎𝑡𝑜𝑠

3. Suma del cuadrado total “SCT”

𝑆𝐶𝑇 = ∑ ∑(𝑥𝑖𝑗 − 𝑀)^2

𝑛𝑖

𝑗=

𝑘

𝑖=

4. Suma de cuadrados del tratamiento “SCTR”

𝑆𝐶𝑇𝑅 = ∑ 𝑛𝑖 (𝑥̅𝑖 − 𝑀)^2

𝑘

𝑖=

5. Suma de los cuadrados de los errores “SCE”

𝑆𝐶𝐸 = ∑ ∑(𝑥𝑖𝑗 − 𝑥̅𝑖)^2

𝑛𝑖

𝑗=

𝑘

𝑖=

6. Cuadrado medio total “CMT”

7. Cuadrado medio del tratamiento “CMTR”

8. Cuadrado medio de los errores “CME”

9. Valor estadístico calculado

PRUEBA DE KRUSKAL - WALLIS

𝜒^2 Cal =

𝑅𝑖^2

𝑘

𝑖=

𝜒^2 𝑇𝑒𝑜 = 𝜒^2 (∝,𝑘−1)

PRUEBA DE AJUSTE DE KOLMOGOROV - SMIRNOV

PRUEBA DE RANGO CON SIGNO DE WILCONXON

Tipo de Prueba Para probar 𝑯𝟎 Contra 𝑯𝒂 𝒘𝒄𝒂𝒍 𝒘𝒕𝒆𝒐

Dos colas 𝜇 1 = 𝜇 0 𝜇 1 ≠ 𝜇 0 min(𝑤+, 𝑤−) 𝑤(∝,𝑛)

Cola superior 𝜇 1 ≥ 𝜇 0 𝜇 1 < 𝜇 0 𝑤+^ 𝑤(∝,𝑛)

Cola inferior 𝜇 1 ≤ 𝜇 0 𝜇 1 > 𝜇 0 𝑤−^ 𝑤(∝,𝑛)

PRUEBA DE LA SUMA DE RANGOS DE WILCONXON

Tipo de Prueba Para probar 𝑯𝟎 Contra 𝑯𝒂 𝒘𝒄𝒂𝒍 𝒘𝒕𝒆𝒐

Dos colas 𝜇 1 = 𝜇 0 𝜇 1 ≠ 𝜇 0 𝑤 1 𝑜 𝑤 2 𝑤(𝑛 1 ,𝑛 2 )

Cola superior 𝜇 1 ≥ 𝜇 0 𝜇 1 < 𝜇 0 𝑤 1 𝑤(𝑛 1 ,𝑛 2 )

Cola inferior 𝜇 1 ≤ 𝜇 0 𝜇 1 > 𝜇 0 𝑤 2 𝑤(𝑛 1 ,𝑛 2 )

PRUEBA DE SIGNO

Tipo de Prueba Para probar 𝑯𝟎 Contra 𝑯𝒂 𝑹𝑪𝒂𝒍 𝑹𝒕𝒆𝒐

Dos colas 𝑀𝑒 1 = 𝑀𝑒 0 𝑀𝑒 1 ≠ 𝑀𝑒 0 min(𝑟+, 𝑟−) 𝑅(𝛼,𝑛)

Cola superior 𝑀𝑒 1 ≥ 𝑀𝑒 0 𝑀𝑒 1 < 𝑀𝑒 0 𝑟+^ 𝑅(𝛼,𝑛)

Cola inferior 𝑀𝑒 1 ≤ 𝑀𝑒 0 𝑀𝑒 1 > 𝑀𝑒 0 𝑟−^ 𝑅(𝛼,𝑛)

PRUEBA U DE MANN - WHITNEY

La prueba U Mann – Whitney se utiliza para probar hipótesis acerca de la media de dos poblaciones:

𝑘) ;^ 𝑉𝑎𝑟(𝑥) = 𝜎

2 =∪ 2 [𝛤 (1 +^2

2 (1 +^1

𝑘)] ;^ 𝑉𝑥

𝛤^2 (1 +^1 𝑘)

Nota: 𝛤(𝑟) = ∫ 0 ∞ 𝑡𝑟−1𝑒−𝑡𝑑𝑡 ; 𝛤(𝑟 + 1) = 𝑟𝛤(𝑟)

4. Distribución Log Normal:

𝑓(𝑥) = (^) 𝑥 𝜎𝑙𝑛𝑥^1 √2𝜋 𝑒 −^12 (𝑙𝑛𝑥 − 𝜇𝜎𝑙𝑛𝑥𝑙𝑛𝑥 ) 2 𝑥 > 0 ; 𝜇𝑥 = 𝑒 𝜇𝑙𝑛𝑥+

1 2 𝜎^2 𝑙𝑛𝑥^ ; 𝜎^2 𝑥 = 𝜇^2 𝑥[𝑒𝜎^2 𝑙𝑛𝑥^ − 1] ; 𝑣𝑥 = √𝑒𝜎^2 𝑙𝑛𝑥 (^) − 1

ln(𝑥) − 𝜇𝑙𝑛𝑥 𝜎𝑙𝑛𝑥

Confidence Intervals and Hypothesis Testing Formulas, Study Guides, Projects, Research of Statics

Related documents

Partial preview of the text

Download Confidence Intervals and Hypothesis Testing Formulas and more Study Guides, Projects, Research Statics in PDF only on Docsity!

INTERVALOS DE CONFIANZA

Estimación Condiciones Intervalo de Confianza Bilateral

𝜇 𝜎^2 (𝐶𝑜𝑛𝑜𝑐𝑖𝑑𝑎) 𝑥̅^ −^ 𝑧(∝ 2 )

√𝑛^

𝜇 1 − 𝜇 2 𝜎 12 , 𝜎 22 (𝐶𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠) (𝑥̅𝟏^ −^ 𝑥̅𝟐)^ −^ 𝑧(∝ 2 )√

√𝜎^1

𝑠𝑝√^

𝑛 1 +^

𝑛 2 ≤^ 𝜇^1 −^ 𝜇^2 ≤^

𝑠𝑝√^

𝑛 1 +^

𝑛 1 + 𝑛 2 − 2 ;^ 𝑣^ =^ 𝑛^1 +^ 𝑛^2 −^2

√𝑠^1

√𝑠^1

(𝑠^1

(𝑠^1

(𝑠^2

(𝑛 − 1 )𝑠^2

2 ≤^ 𝜎

2 ≤ (𝑛^ −^1 )𝑠

√𝑝̂^ (^1 −^ 𝑝̂^ )

𝑛 ≤^ 𝑝^ ≤^ 𝑝̂^ +^ 𝑧(∝ 2 )

√𝑝̂^ (^1 −^ 𝑝̂^ )

PRUEBAS DE HIPÓTESIS

𝑯𝟎 Condiciones Estadístico calculado

𝜇 = 𝜇 0 𝜎^2 (𝐶𝑜𝑛𝑜𝑐𝑖𝑑𝑎)

𝜇 = 𝜇 0 𝜎^2 (𝐷𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎)

√𝜎^1

𝑠𝑝√ 𝑛^1

+ 𝑛^1

𝑛 1 + 𝑛 2 − 2 ;^ 𝑣^ =^ 𝑛^1 +^ 𝑛^2 −^2

√𝑠^1

(𝑠^1

𝜎^2 = 𝜎 02 𝜒𝑐𝑎𝑙

2 = (𝑛^ −^1 )𝑠

𝑵: 𝑁ú𝑚𝑒𝑟𝑜 𝑡𝑜𝑡𝑎𝑙 𝑑𝑒 𝑑𝑎𝑡𝑜𝑠

3. Suma del cuadrado total “SCT”

𝑆𝐶𝑇 = ∑ ∑(𝑥𝑖𝑗 − 𝑀)^2

4. Suma de cuadrados del tratamiento “SCTR”

𝑆𝐶𝑇𝑅 = ∑ 𝑛𝑖 (𝑥̅𝑖 − 𝑀)^2

5. Suma de los cuadrados de los errores “SCE”

𝑆𝐶𝐸 = ∑ ∑(𝑥𝑖𝑗 − 𝑥̅𝑖)^2

6. Cuadrado medio total “CMT”

7. Cuadrado medio del tratamiento “CMTR”

8. Cuadrado medio de los errores “CME”

9. Valor estadístico calculado

PRUEBA DE KRUSKAL - WALLIS

𝜒^2 Cal =

𝑅𝑖^2

𝜒^2 𝑇𝑒𝑜 = 𝜒^2 (∝,𝑘−1)

PRUEBA DE AJUSTE DE KOLMOGOROV - SMIRNOV

PRUEBA DE RANGO CON SIGNO DE WILCONXON

Tipo de Prueba Para probar 𝑯𝟎 Contra 𝑯𝒂 𝒘𝒄𝒂𝒍 𝒘𝒕𝒆𝒐

Dos colas 𝜇 1 = 𝜇 0 𝜇 1 ≠ 𝜇 0 min(𝑤+, 𝑤−) 𝑤(∝,𝑛)

Cola superior 𝜇 1 ≥ 𝜇 0 𝜇 1 < 𝜇 0 𝑤+^ 𝑤(∝,𝑛)

Cola inferior 𝜇 1 ≤ 𝜇 0 𝜇 1 > 𝜇 0 𝑤−^ 𝑤(∝,𝑛)

PRUEBA DE LA SUMA DE RANGOS DE WILCONXON

Tipo de Prueba Para probar 𝑯𝟎 Contra 𝑯𝒂 𝒘𝒄𝒂𝒍 𝒘𝒕𝒆𝒐

Dos colas 𝜇 1 = 𝜇 0 𝜇 1 ≠ 𝜇 0 𝑤 1 𝑜 𝑤 2 𝑤(𝑛 1 ,𝑛 2 )

Cola superior 𝜇 1 ≥ 𝜇 0 𝜇 1 < 𝜇 0 𝑤 1 𝑤(𝑛 1 ,𝑛 2 )

Cola inferior 𝜇 1 ≤ 𝜇 0 𝜇 1 > 𝜇 0 𝑤 2 𝑤(𝑛 1 ,𝑛 2 )

PRUEBA DE SIGNO

Tipo de Prueba Para probar 𝑯𝟎 Contra 𝑯𝒂 𝑹𝑪𝒂𝒍 𝑹𝒕𝒆𝒐

Dos colas 𝑀𝑒 1 = 𝑀𝑒 0 𝑀𝑒 1 ≠ 𝑀𝑒 0 min(𝑟+, 𝑟−) 𝑅(𝛼,𝑛)

Cola superior 𝑀𝑒 1 ≥ 𝑀𝑒 0 𝑀𝑒 1 < 𝑀𝑒 0 𝑟+^ 𝑅(𝛼,𝑛)

Cola inferior 𝑀𝑒 1 ≤ 𝑀𝑒 0 𝑀𝑒 1 > 𝑀𝑒 0 𝑟−^ 𝑅(𝛼,𝑛)

PRUEBA U DE MANN - WHITNEY

La prueba U Mann – Whitney se utiliza para probar hipótesis acerca de la media de dos poblaciones:

𝑘) ;^ 𝑉𝑎𝑟(𝑥) = 𝜎

2 =∪ 2 [𝛤 (1 +^2

2 (1 +^1

𝑘)] ;^ 𝑉𝑥

𝛤^2 (1 +^1 𝑘)

DISTRIBUCIÓN GAMMA

(^1 𝜆)

𝑥(𝛼−1)^ ∗ 𝑒−𝜆𝑥