1. It is said that SUV-type vehicles are overturned on the road. Particularly it

Question

1. It is said that SUV-type vehicles are overturned on the road. Particularly it is said that the Toyota brand vehicles turn more than those of luxury brand Lexus manufactured by the same company. A sample of 100 vehicles of the same model for each Toyota brand and Lexus, which were involved in accidents, was selected. The results obtained are presented in the table below.

Model/ Brand

Overturned vehicles

Sample size

Land Cruiser/ Toyota

25

100

LX 570/ Lexus

35

100

a. Perform a hypothesis test to corroborate whether the ratio of overturned vehicles difference between the two models is greater for the Toyota model when compared to the Lexus model. Use a significance level of 0.05

b. Calculate p-value

2.In a biotechnology process properly validated (it was proven that the process occurs, in the same way, each time it is run) three samples of the product were taken during a manufacturing stage to verify the active ingredient percentage. The investigator suspects that the first sample influences in the result of the second sample and that the second influences in the result of the third sample. For this reason, the investigator decided to compare a few lots to verify if the samples are independent. Table below presents the results of this experiment. Perform a variance analysis to check the independence between samples. Use a significance level of 0.05

Lot

1st Sample

2nd Sample

3rd Sample

A

93.502

92.319

92.451

C

91.177

92.230

92.097

D

87.304

87.496

88.001

D2

81.275

80.564

80.386

G

79.865

79.259

79.209

G2

81.722

80.931

81.351

Model/ Brand

Overturned vehicles

Sample size

Land Cruiser/ Toyota

25

100

LX 570/ Lexus

35

100

QUESTION 1 Problema 1. Se comenta que los vehículos tipo SUV se vuelcan en la carretera. Particularmente se comenta que los vehículos marca Toyota se vuelcan m que aquellos de la marca lujosa Lexus fabricados por la misma compañía. Se seleccionó una muestra de 100 vehículos del mismo modelo para cada marc Toyota y Lexus, que estuvieron involucrados en accidentes. Los resultados obtenidos se presentan en la tabla 1. Tabla 1: Datos Problema 1 Modelo/Marca Land Cruiser/Toyota LX 570/Lexus Vehículos volcados 25 35 Tamaño de Muestra 100 100 a. Realice una prueba de hipótesis para corroborar si la razón de la diferencia de vehículos volcados entre los dos modelos es mayor para el modelo Toyota al comparar con el modelo Lexus. Utilice un nivel de significancia de 0.05. (20 puntos) b. Calcule el p-value (7 puntos)

Explanation / Answer

1. Null Hypothesis : H0 : Ratio of overturned vehicles difference for Toyota Model is same as the Ratio of overturned vehicles for Lexus Model.

Alternative Hypothesis : Ha : Ration of overturned vehicles for Toyota Model is more than the Ratio of overturned vehicles for Lexus Model.

Here p1 = 0.25 and p2 = 0.35

Pooled estimate p = ( 25 + 35)/ 200 = 0.30

Standard error of the proportions se0 = sqrt [p(1-p) * (1/N1 +1/N2)] = sqrt [0.3 * 0.7 * 2/100] = 0.0648

so Z = (p1 -p2 )/se0 = (0.35 - 0.25)/ 0.0648 = 1.54

p - value = Pr ( Z > 1.54) = 0.0618

here we can say that p - value is greater than the significance level alpha = 0.05 so we can reject the null hypothesis and can conclude that there is no significant difference between Ratio of overturned vehicles difference for Toyota Model is same as the Ratio of overturned vehicles for Lexus Model.

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