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2. ejercicios de prueba de hipótesis

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Ejercicios de Prueba de Hipótesis

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2. ejercicios de prueba de hipótesis

  1. 1. Luis Alberto Garcia Aguilar 2°“B”
  2. 2. PRUEBA DE UNA SOLA MUESTRA CON RESPECTO A UNA SOLA MEDIA (VARIANZA CONOCIDA) Una empresa eléctrica fabrica focos que tienen una duraciónque se distribuye de forma aproximadamente normal con una media de 800 horas y una desviación estándar de 40 horas. Pruebe la hipótesis de que µ≠800 horas si una muestra aleatoria de 30 focos tiene una duración promedio de 788 horas. Utilice un nivel de significancia de 0.04. Datos H0: µ1=800 H1: µ2=788 σ=40 horas X=788 Significancia=0.04
  3. 3. Con la resolución del ejercicio se llega a la conclusión de que la duración media delos focos si corresponde a 800 horas por lo que la hipótesis nula es aceptada. Zona de aceptacion Zona de Zona de Rechazo Rechazoz=-1.75 z=1.7 z=-1.64 5
  4. 4.  Se lleva a cabo un experimento para comparar el desgaste por abrasivo de dos diferentes materiales laminados. Se prueban 12 piezas del material 1 mediante la exposición de cada pieza a una maquina para medir el desgaste. 10 piezas del material 2 se prueban de manera similar. En cada caso, se mide la profundidad del desgaste. Las muestras del material 1 dan un desgaste promedio de 85 unidades con una desviación estándar muestral de 4, mientras que las muestras del material 2 dan un promedio de 81, desviación estándar muestral de 5. ¿Podemos concluir con un nivel de significancia del 0.05 que el desgaste abrasivo del material 1 excede el del material 2 en 2 unidades?
  5. 5. Representemos con µ₁ y µ₂ las medias poblacionalesdel desgaste abrasivo para el material 1 y 2,respectivamente.Las regiones criticas unilaterales rechaza a H₀: µ₁ - µ₂ = d₀ cuando t > tαn₁+n₂-2
  6. 6.  Cálculos: x1 85 s1 4 n1 12 x2 81 s2 5 n 2 10 De aquí: (85 81) 2sp (11)(16 ) (9)( 25 ) = t = 12 10 - 2 4.478 (1 / 12 ) (1 / 10 ) P = P(T>1.04) ≈ 0.16 Decisión: No rechazar Somos incapaces de concluir que el desgaste abrasivo del material 1 excede el del material 2 en mas de dos unidades
  7. 7.  Una marca de nueces afirma que, como máximo, el 6% de las nueces están vacías. Se eligieron 300 nueces al azar y se detectaron 21 vacías. 1.Con un nivel de significación del 1%, ¿se puede aceptar la afirmación de la marca?
  8. 8.  1 Enunciamos las hipótesis nula y alternativa: H0 : p ≤ 0.06 H1 : p >0.06 2Zona de aceptación α = 0.01 zα = 2.33. Determinamos el intervalo de confianza:
  9. 9. Decisión : Aceptamos la hipótesis nula H0. Con un nivel de significación del 1%. Si se mantiene el porcentaje muestral de nueces que están vacías y 1-α = 0.95, ¿qué tamaño muestral se necesitaría para estimar la proporción de nueces con un error menor del 1% por ciento?
  10. 10.  Se lleva a cabo un experimento para comparar el desgaste por abrasivo de dos diferentes materiales laminados. Se prueban 12 piezas del material 1 mediante la exposición de cada pieza a una maquina para medir el desgaste. 10 piezas del material 2 se prueban de manera similar. En cada caso, se mide la profundidad del desgaste. Las muestras del material 1 dan un desgaste promedio de 85 unidades con una desviación estándar muestral de 4, mientras que las muestras del material 2 dan un promedio de 81, desviación estándar muestral de 5. ¿Podemos concluir con un nivel de significancia del 0.05 que el desgaste abrasivo del material 1 excede el del material 2 en 2 unidades?
  11. 11.  Representemos con µ₁ y µ₂ las medias poblacionales del desgaste abrasivo para el material 1 y 2, respectivamente. H₀: µ₁ - µ₂ = 2 H₁: µ₁ - µ₂ ≠ 2 α = 0.05 Región critica: con v= 20 grados de libertad t > 1.725 Las regiones criticas unilaterales rechaza a H₀: µ₁ - µ₂ = d₀ cuando t > tαn₁+n₂-2
  12. 12.  Cálculos: x1 85 s1 4 n1 12 x2 81 s2 5 n 2 10  De aquí: (11)(16 ) (9)( 25 ) (85 81) 2 t sp 12 10 - 2 = 4.478 (1 / 12 ) (1 / 10 ) P = P(T>1.04) ≈ 0.16 Decisión: No rechazar Somos incapaces de concluir que el desgaste abrasivo del material 1 excede el del material 2 en mas de dos unidades
  13. 13. Ho: La distribución normal con un α de 0,01es una buena descripción del proceso de fabricación del producto X.H1: La distribución normal con un α de 0,01 no es una buena descripción del proceso de fabricación del producto X.
  14. 14. LI LS Frec Frec. Xk D Fr Acom 35000 40000 6 6 37500 -3 40000 45000 15 21 42500 -2 45000 50000 58 79 47500 -1 50000 55000 139 218 52500 0 55000 60000 66 284 57500 1 60000 65000 11 295 62500 2 65000 70000 5 300 67500 3 300Xk D Frec*D Frec*D² Prob. Frec. Frec. Esp Esperada Esperada acom37500 -3 -18 54 0.0087 2.6142500 -2 -30 60 0.0677 20.31 22.9247500 -1 -58 58 0.2428 72.84 72.8452500 0 0 0 0.3687 110.61 110.6157500 1 66 66 0.2385 71.55 71.5562500 2 22 44 0.0654 19.62 21.9667500 3 15 45 0.0078 2.34 -3 327
  15. 15. Frecuencia Frecuenciaobservada esperada Fo-Fe (Fo-Fe)^2 (Fo-Fe)^2/Fe 21 22.92 -1.92 3.69 0.16 58 72.84 -14.84 220.23 3.02 139 110.61 28.39 805.99 7.29 66 71.55 -5.55 30.80 0.43 16 21.96 -5.96 35.52 1.62 X² 12.52
  16. 16. Un constructor afirma que se instalan bombas de calor en 70%de todas las casas que se construyen hoy en día en la ciudadde Richmond. ¿Estaría de acuerdo con esta afirmación si unainvestigación de casas nuevas en esta ciudad muestra que 8de 15 tienen instaladas bombas de calor? utilice un nivel designificancia de 0.10
  17. 17. 1. H0: p=0.72. H1: p=0.73. α= 0.104. Estadística de prueba: Variable binomial X con p= 0.7 y n= 155. Cálculos x=8 y np0=(15)(0.7)=10.5. por tanto, de la tabla A.1, el valor P calculado es6. P= 2P(X ≤ 8 cuando p=0.7)= 0.1622>0.107. Por lo tanto no se pude rechazar H0, por lo que concluimos que no ha razón suficiente para dudar de la afirmación del constructor.

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