200912013PAC2PC13

1. La ventaja principal de la regresión múltiple respecto a la regresión simple es que nos permite usar más de la información disponible para estimar la variable dependiente.

V

F

The main advantage of multiple regression over simple regression is that it allows us to utilize more available information to estimate the dependent variable. In simple regression, we only consider one independent variable, whereas in multiple regression, we can include multiple independent variables. This enables us to capture the combined effects of these variables on the dependent variable, resulting in a more comprehensive and accurate estimation.

Explanation

The main advantage of multiple regression over simple regression is that it allows us to utilize more available information to estimate the dependent variable. In simple regression, we only consider one independent variable, whereas in multiple regression, we can include multiple independent variables. This enables us to capture the combined effects of these variables on the dependent variable, resulting in a more comprehensive and accurate estimation.

2. Las variables ficticias constituyen una técnica que puede utilizarse para incorporar datos cualitativos en las regresiones múltiples.

V

F

Las variables ficticias, también conocidas como variables dummy, son utilizadas en regresiones múltiples para representar variables cualitativas o categóricas. Estas variables toman el valor de 0 o 1 para cada categoría, permitiendo así incluir información cualitativa en el modelo de regresión. Por lo tanto, la afirmación de que las variables ficticias pueden utilizarse para incorporar datos cualitativos en las regresiones múltiples es verdadera.

Explanation

Las variables ficticias, también conocidas como variables dummy, son utilizadas en regresiones múltiples para representar variables cualitativas o categóricas. Estas variables toman el valor de 0 o 1 para cada categoría, permitiendo así incluir información cualitativa en el modelo de regresión. Por lo tanto, la afirmación de que las variables ficticias pueden utilizarse para incorporar datos cualitativos en las regresiones múltiples es verdadera.

3. Aunque en teoría es posible hacer cálculos de regresión múltiple a mano, muy pocas veces lo hacemos.

V

F

In the given statement, it is mentioned that although it is theoretically possible to perform multiple regression calculations by hand, it is rarely done. This implies that the statement is true and the correct answer is "V" for true.

Explanation

In the given statement, it is mentioned that although it is theoretically possible to perform multiple regression calculations by hand, it is rarely done. This implies that the statement is true and the correct answer is "V" for true.

4. Si se conoce la suma de cuadrados total y la suma de cuadrados de la regresión para una regresión múltiple, siempre se puede calcular la suma de cuadrados de error.

V

F

If the total sum of squares and the sum of squares of regression are known for a multiple regression, then it is indeed possible to calculate the sum of squares of error. The sum of squares of error represents the variability in the data that is not explained by the regression model.

Explanation

If the total sum of squares and the sum of squares of regression are known for a multiple regression, then it is indeed possible to calculate the sum of squares of error. The sum of squares of error represents the variability in the data that is not explained by the regression model.

5. Las regresiones simples de Y sobre X₁ y de Y sobre X₂ muestran que X₁, y X₂ son ambas variables explicativas significativas de Y. Pero una regresión múltiple de Y sobre X₁, y X₂ nos dice que ni X₁ ni X₂ son variables explicativas significativas para Y. Claramente, éste es un caso de multicolinealidad.

V

F

The explanation for the given correct answer is that the simple regressions of Y on X1 and Y on X2 show that both X1 and X2 are significant explanatory variables for Y. However, the multiple regression of Y on X1 and X2 indicates that neither X1 nor X2 are significant explanatory variables for Y. This suggests that there is multicollinearity present, which means that X1 and X2 are highly correlated with each other and are not independently contributing to the explanation of Y.

Explanation

The explanation for the given correct answer is that the simple regressions of Y on X1 and Y on X2 show that both X1 and X2 are significant explanatory variables for Y. However, the multiple regression of Y on X1 and X2 indicates that neither X1 nor X2 are significant explanatory variables for Y. This suggests that there is multicollinearity present, which means that X1 and X2 are highly correlated with each other and are not independently contributing to the explanation of Y.

6. Suponga que deseamos probar si los valores de Y en una regresión múltiple realmente dependen de los valores de X₁. La hipótesis nula para nuestra prueba será B₁ = 0.

V

F

The correct answer is "V" because the statement is stating that the null hypothesis for the test is B1 = 0, which means that the coefficient for X1 in the multiple regression model is equal to zero. This implies that the values of Y do not depend on the values of X1, suggesting that X1 is not a significant predictor in the regression model.

Explanation

The correct answer is "V" because the statement is stating that the null hypothesis for the test is B1 = 0, which means that the coefficient for X1 in the multiple regression model is equal to zero. This implies that the values of Y do not depend on the values of X1, suggesting that X1 is not a significant predictor in the regression model.

7. Suponga que una regresión múltiple ha producido la siguiente ecuación: Ŷ = 5.6 + 2.8X₁ – 3.9X₂ + 5.6X₃. Si X₁, X₂ y X₃ tienen valor de cero, entonces se esperaría que Y tuviera el valor de 5.6.

V

F

If X1, X2, and X3 have a value of zero, then according to the given regression equation, the predicted value of Y (Ŷ) would be 5.6. This is because the coefficient of X1, X2, and X3 is zero, which means they do not contribute to the predicted value of Y. Therefore, the only constant term in the equation, 5.6, would be the expected value of Y when all the independent variables have a value of zero.

Explanation

If X1, X2, and X3 have a value of zero, then according to the given regression equation, the predicted value of Y (Ŷ) would be 5.6. This is because the coefficient of X1, X2, and X3 is zero, which means they do not contribute to the predicted value of Y. Therefore, the only constant term in the equation, 5.6, would be the expected value of Y when all the independent variables have a value of zero.

8. Una medida de nuestra de incertidumbre acerca del valor exacto de un coeficiente de regresión múltiple es el ______________del coeficiente

Varianza

Error estandar

The error estándar is a measure of uncertainty or variability in the exact value of a multiple regression coefficient. It represents the average amount by which the estimated coefficient may differ from the true population coefficient. A smaller error estándar indicates greater precision and less uncertainty in the coefficient estimate, while a larger error estándar suggests more variability and less confidence in the estimate.

Explanation

The error estándar is a measure of uncertainty or variability in the exact value of a multiple regression coefficient. It represents the average amount by which the estimated coefficient may differ from the true population coefficient. A smaller error estándar indicates greater precision and less uncertainty in the coefficient estimate, while a larger error estándar suggests more variability and less confidence in the estimate.

9. Una relación lineal entre variables explicativas con toda seguridad producirá multicolinealidad en el modelo de regresión.

V

F

A linear relationship between explanatory variables will likely result in multicollinearity in the regression model. This is because when there is a strong linear relationship between two or more explanatory variables, it becomes difficult for the model to distinguish the individual effects of each variable on the response variable. This leads to high correlation between the variables, which is known as multicollinearity. Multicollinearity can cause issues in the regression analysis, such as unstable and unreliable coefficient estimates, difficulty in interpreting the effects of individual variables, and increased standard errors.

Explanation

A linear relationship between explanatory variables will likely result in multicollinearity in the regression model. This is because when there is a strong linear relationship between two or more explanatory variables, it becomes difficult for the model to distinguish the individual effects of each variable on the response variable. This leads to high correlation between the variables, which is known as multicollinearity. Multicollinearity can cause issues in the regression analysis, such as unstable and unreliable coefficient estimates, difficulty in interpreting the effects of individual variables, and increased standard errors.

10. Para determinar si una regresión es significativa como un todo, se calcula un valor observado de F y se compara con un valor obtenido de una tabla.

V

F

The statement is true. In order to determine if a regression is significant as a whole, an observed F value is calculated and compared with a value obtained from a table. This is a common method used in statistical analysis to assess the significance of a regression model.

Explanation

The statement is true. In order to determine if a regression is significant as a whole, an observed F value is calculated and compared with a value obtained from a table. This is a common method used in statistical analysis to assess the significance of a regression model.

11. Si una regresión incluye a todos los factores explicativos relevantes, los residuos serán aleatorios.

V

F

If a regression includes all relevant explanatory factors, it means that the model is capturing all the variables that can explain the dependent variable. In this case, the residuals, which are the differences between the actual and predicted values, will be random. This is because the model has accounted for all the factors that could explain the variation in the dependent variable, leaving no systematic patterns in the residuals. Therefore, the statement "V" (true) is correct.

Explanation

If a regression includes all relevant explanatory factors, it means that the model is capturing all the variables that can explain the dependent variable. In this case, the residuals, which are the differences between the actual and predicted values, will be random. This is because the model has accounted for all the factors that could explain the variation in the dependent variable, leaving no systematic patterns in the residuals. Therefore, the statement "V" (true) is correct.

12. Suponga que en la ecuación de regresión múltiple Ŷ = 24.4 + 5.6X₁ + 6.8X₂, Ŷ es el peso (en libras) y X₂ es la edad (en años). Por cada año adicional en la edad, entonces, se puede esperar que el peso aumente en 24.4 libras.

V

F

The statement is false. According to the given equation, for each additional year in age (X2), the weight (Ŷ) is expected to increase by 6.8 pounds, not 24.4 pounds.

Explanation

The statement is false. According to the given equation, for each additional year in age (X2), the weight (Ŷ) is expected to increase by 6.8 pounds, not 24.4 pounds.

13. Podemos formar un modelo de regresión de segundo grado si multiplicamos por 2 los valores observados de una variable independiente.

V

F

Multiplicar por 2 los valores observados de una variable independiente no nos permite formar un modelo de regresión de segundo grado. La formación de un modelo de regresión de segundo grado requiere la inclusión de términos cuadráticos en la variable independiente, no simplemente multiplicar los valores observados por un factor constante. Por lo tanto, la afirmación es falsa.

Explanation

Multiplicar por 2 los valores observados de una variable independiente no nos permite formar un modelo de regresión de segundo grado. La formación de un modelo de regresión de segundo grado requiere la inclusión de términos cuadráticos en la variable independiente, no simplemente multiplicar los valores observados por un factor constante. Por lo tanto, la afirmación es falsa.

14. Suponga que intenta establecer un intervalo de confianza para un valor de Y a partir de una ecuación de regresión múltiple. Si existen 20 elementos en la muestra y se utilizan cuatro variables independientes en la regresión, deberá usar 16 grados de libertad cuando obtenga un valor de la tabla t.

V

F

When establishing a confidence interval for a value of Y from a multiple regression equation, the degrees of freedom to be used when obtaining a value from the t-table are not determined by the number of elements in the sample or the number of independent variables used in the regression. The degrees of freedom for a multiple regression are calculated based on the sample size and the number of independent variables, which in this case would be 20 - 4 = 16 degrees of freedom. Therefore, the correct answer is F.

Explanation

When establishing a confidence interval for a value of Y from a multiple regression equation, the degrees of freedom to be used when obtaining a value from the t-table are not determined by the number of elements in the sample or the number of independent variables used in the regression. The degrees of freedom for a multiple regression are calculated based on the sample size and the number of independent variables, which in this case would be 20 - 4 = 16 degrees of freedom. Therefore, the correct answer is F.

15. El error estándar del coeficiente b₂ en una regresión múltiple se denota con s₂.

V

F

The statement is false. The standard error of the coefficient b2 in multiple regression is typically denoted as se(b2) or s(b2), not s2. The standard error represents the variability or uncertainty in the estimated coefficient, and it is used to calculate confidence intervals and perform hypothesis tests.

Explanation

The statement is false. The standard error of the coefficient b2 in multiple regression is typically denoted as se(b2) or s(b2), not s2. The standard error represents the variability or uncertainty in the estimated coefficient, and it is used to calculate confidence intervals and perform hypothesis tests.

16. Agregar variables adicionales a una regresión múltiple siempre reducirá el error estándar de la estimación.

V

F

Adding additional variables to a multiple regression will not always reduce the standard error of estimation. The standard error of estimation measures the accuracy of the regression model in predicting the dependent variable. Adding more variables can sometimes lead to overfitting, where the model becomes too complex and starts fitting the noise in the data rather than the underlying pattern. This can increase the standard error of estimation and reduce the model's predictive power. Therefore, the statement is false.

Explanation

Adding additional variables to a multiple regression will not always reduce the standard error of estimation. The standard error of estimation measures the accuracy of the regression model in predicting the dependent variable. Adding more variables can sometimes lead to overfitting, where the model becomes too complex and starts fitting the noise in the data rather than the underlying pattern. This can increase the standard error of estimation and reduce the model's predictive power. Therefore, the statement is false.

17. Ciertos patrones en los signos de los residuos de un modelo de regresión de segundo grado indican que sería mejor utilizar un modelo lineal.

V

F

The statement suggests that certain patterns in the residuals of a second-degree regression model indicate that it would be better to use a linear model. However, the correct answer is false. This means that the statement is incorrect and there is no evidence or indication that using a linear model would be better based on the patterns in the residuals of a second-degree regression model.

Explanation

The statement suggests that certain patterns in the residuals of a second-degree regression model indicate that it would be better to use a linear model. However, the correct answer is false. This means that the statement is incorrect and there is no evidence or indication that using a linear model would be better based on the patterns in the residuals of a second-degree regression model.

18. Aunque es posible hacer inferencias acerca de la regresión como un todo, no es posible hacer inferencias acerca de los coeficientes de regresión estimados.

V

F

The statement "Aunque es posible hacer inferencias acerca de la regresión como un todo, no es posible hacer inferencias acerca de los coeficientes de regresión estimados" translates to "Although it is possible to make inferences about the regression as a whole, it is not possible to make inferences about the estimated regression coefficients." The correct answer is false because it is possible to make inferences about the estimated regression coefficients.

Explanation

The statement "Aunque es posible hacer inferencias acerca de la regresión como un todo, no es posible hacer inferencias acerca de los coeficientes de regresión estimados" translates to "Although it is possible to make inferences about the regression as a whole, it is not possible to make inferences about the estimated regression coefficients." The correct answer is false because it is possible to make inferences about the estimated regression coefficients.

19. El error estándar de los datos de la población se denota por s_e.

V

F

The statement "El error estándar de los datos de la población se denota por se" is false. The correct notation for the standard error of the population data is "σ" (sigma), not "se". The symbol "se" is commonly used to represent the standard error of the sample data. Therefore, the correct answer is F (False).

Explanation

The statement "El error estándar de los datos de la población se denota por se" is false. The correct notation for the standard error of the population data is "σ" (sigma), not "se". The symbol "se" is commonly used to represent the standard error of the sample data. Therefore, the correct answer is F (False).

20. Si existe un alto nivel de correlación entre las variables explicativas, por lo general es posible separar las contribuciones de estas variables en una regresión.

V

F

If there is a high level of correlation between the explanatory variables, it is generally not possible to separate the contributions of these variables in a regression. This is because high correlation between variables can lead to multicollinearity, which makes it difficult to determine the individual effects of each variable on the dependent variable.

Explanation

If there is a high level of correlation between the explanatory variables, it is generally not possible to separate the contributions of these variables in a regression. This is because high correlation between variables can lead to multicollinearity, which makes it difficult to determine the individual effects of each variable on the dependent variable.

21. Cuando se utiliza una variable ficticia con valores 0 y 1, es muy importante asegurarse de que los ceros y unos se usen de acuerdo con la práctica estándar. Invertir la codificación destruirá completamente los resultados de la regresión múltiple.

V

F

In multiple regression analysis, a dummy variable is often used to represent categorical variables with two levels. These dummy variables take values of 0 and 1 to indicate the presence or absence of a certain category. It is important to follow the standard practice of using 0 and 1 appropriately in order to interpret the results correctly. However, inverting the coding of the dummy variable, such as using 1 for the absence and 0 for the presence, will completely distort the results of the multiple regression analysis.

Explanation

In multiple regression analysis, a dummy variable is often used to represent categorical variables with two levels. These dummy variables take values of 0 and 1 to indicate the presence or absence of a certain category. It is important to follow the standard practice of using 0 and 1 appropriately in order to interpret the results correctly. However, inverting the coding of the dummy variable, such as using 1 for the absence and 0 for the presence, will completely distort the results of the multiple regression analysis.

22. Como r² = 1 – (Y – Ŷ)²/(Y – Ŷ)², r² es equivalente a:

1 – SCR/SCT

1 – SCE/SCT

1 – SCE/SCR

1 – SCT/SCR

1 – SCT/SCE

The correct answer is 1 - SCE/SCT. This is because r2, also known as the coefficient of determination, is calculated by subtracting the sum of the squared errors of the regression model (SCE) from the total sum of squares (SCT) and then dividing it by SCT. Therefore, 1 - SCE/SCT is the correct formula to calculate r2.

Explanation

The correct answer is 1 - SCE/SCT. This is because r2, also known as the coefficient of determination, is calculated by subtracting the sum of the squared errors of the regression model (SCE) from the total sum of squares (SCT) and then dividing it by SCT. Therefore, 1 - SCE/SCT is the correct formula to calculate r2.

23. Suponga que un fabricante de juguetes desea determinar si sus juguetes rojos se venden más que sus juguetes azules. Recolecta datos concernientes a los niveles de ventas, color, precio y edad promedio de las personas a las que van dirigidos. Introduce todos estos datos en un paquete estadístico; la ecuación de regresión múltiple resultante es Ŷ = 70,663 – 713X₁ – 59.6X₂ + 66.4X₃, donde Ŷ representa los niveles de ventas en unidades, X₁ es el color (0 para azul, 1 para rojo), X₂ es el precio al menudeo (en dólares) y X₃ la edad promedio (en años). ¿Cuáles de las siguientes afirmaciones son verdaderas si las variables correspondientes al precio y la edad se mantienen constantes?

Deben venderse 713 unidades más de juguetes rojos que de juguetes azules.

Deben venderse 713 unidades menos de juguetes rojos que de juguetes azules.

Los niños siempre preferirán un juguete azul a uno rojo.

B) y c), pero no a).

Ninguna de las anteriores

The given equation of multiple regression suggests that for every unit increase in the color variable (from blue to red), the predicted level of sales decreases by 713 units. Therefore, it can be concluded that there should be 713 units less of red toys sold compared to blue toys, assuming that the price and age variables are held constant.

Explanation

The given equation of multiple regression suggests that for every unit increase in the color variable (from blue to red), the predicted level of sales decreases by 713 units. Therefore, it can be concluded that there should be 713 units less of red toys sold compared to blue toys, assuming that the price and age variables are held constant.

24. 1. El análisis de residuos en un modelo de regresión lineal se hace para determinar el valor correcto de s_e.

V

F

The analysis of residuals in a linear regression model is not done to determine the correct value of "se" (standard error). Residual analysis is used to assess the goodness of fit of the model and to check for any patterns or outliers in the residuals. The standard error is a measure of the variability of the estimated regression coefficients, and it is typically calculated as part of the model estimation process.

Explanation

The analysis of residuals in a linear regression model is not done to determine the correct value of "se" (standard error). Residual analysis is used to assess the goodness of fit of the model and to check for any patterns or outliers in the residuals. The standard error is a measure of the variability of the estimated regression coefficients, and it is typically calculated as part of the model estimation process.

25. Las señales de la presencia posible de multicolinealidad en una regresión múltiple son:

Valores t significativos para los coeficientes.

Errores estándar bajos para los coeficientes.

Todos los anteriores.

Ninguno de los anteriores

The correct answer is "El brusco aumento de un valor t para el coeficiente de una variable explicativa cuando se elimina del modelo otra variable". This is because a sudden increase in the t-value for a coefficient when another variable is removed from the model suggests the presence of multicollinearity. Multicollinearity occurs when there is a high correlation between independent variables, making it difficult to determine the individual effects of each variable on the dependent variable.

Explanation

The correct answer is "El brusco aumento de un valor t para el coeficiente de una variable explicativa cuando se elimina del modelo otra variable". This is because a sudden increase in the t-value for a coefficient when another variable is removed from the model suggests the presence of multicollinearity. Multicollinearity occurs when there is a high correlation between independent variables, making it difficult to determine the individual effects of each variable on the dependent variable.

26. Suponga que ha realizado una regresión múltiple y encuentra que el valor de b, es 1.66. Sin embargo, los datos obtenidos a partir de la experiencia pasada indican que el valor de B₁ , debería ser 1.34. Usted desea probar, al nivel de significancia de 0.05, la hipótesis nula de que B, sigue siendo 1.34. Suponiendo que tiene acceso a todas las tablas que pueda necesitar, ¿qué otra información requiere para poder realizar la prueba?

Grados de libertad.

Sbi.

Se.

A) y b), pero no c).

A) y c), pero no b).

To perform the hypothesis test, you need the degrees of freedom (df) and the standard error (se). The degrees of freedom represent the number of independent pieces of information available for estimating the parameter. The standard error measures the variability of the estimated parameter. These two pieces of information are necessary to calculate the test statistic and determine the p-value, which will be compared to the significance level of 0.05 to make a decision about the null hypothesis. Therefore, you need both the degrees of freedom and the standard error, but you do not need the Sbi in this case.

Explanation

To perform the hypothesis test, you need the degrees of freedom (df) and the standard error (se). The degrees of freedom represent the number of independent pieces of information available for estimating the parameter. The standard error measures the variability of the estimated parameter. These two pieces of information are necessary to calculate the test statistic and determine the p-value, which will be compared to the significance level of 0.05 to make a decision about the null hypothesis. Therefore, you need both the degrees of freedom and the standard error, but you do not need the Sbi in this case.

27. Para la regresión múltiple Ŷ = a + b₁X₁ + b₂X₂, utilizada para estimar Y = A + B₁X₁ + B₂X₂, la forma de un intervalo de confianza posible para B, es

B1 – tsbl, B1 + tsbl.

B1 – tse, B1 + tse.

B1 – tsbl, b1 + tsbl

B1 – tse b1 + tse

B1 + tsbl, B1 + tsbl

The given answer suggests that the possible interval of confidence for the coefficient B1 in the multiple regression equation is b1 – tsbl to b1 + tsbl. This means that the estimated value of B1 can vary within this interval with a certain level of confidence. The terms tsbl and tse likely represent critical values from a t-distribution, which are used to calculate the margin of error for the estimate.

Explanation

The given answer suggests that the possible interval of confidence for the coefficient B1 in the multiple regression equation is b1 – tsbl to b1 + tsbl. This means that the estimated value of B1 can vary within this interval with a certain level of confidence. The terms tsbl and tse likely represent critical values from a t-distribution, which are used to calculate the margin of error for the estimate.

28. En la ecuación Y = A = B₁X₁ + B₂X₂, Y es independiente de X₁ si:

B2 = 0.

B2 = –1.

B1 = 1.

Todos los anteriores

Ninguno de los anteriores

In the equation Y = A + B1X1 + B2X2, Y is not independent of X1 if B2 = 0. This means that the value of X2 does not affect the value of Y. However, if B2 is not equal to 0, then the value of X2 does have an impact on the value of Y. Therefore, the correct answer is "Ninguno de los anteriores" (None of the above).

Explanation

In the equation Y = A + B1X1 + B2X2, Y is not independent of X1 if B2 = 0. This means that the value of X2 does not affect the value of Y. However, if B2 is not equal to 0, then the value of X2 does have an impact on the value of Y. Therefore, the correct answer is "Ninguno de los anteriores" (None of the above).

29. Hemos dicho que el error estándar de una estimación tiene n – k – 1 grados de libertad. ¿Qué significa k en esta expresión?

El número de elementos de la muestra.

El número de variables independientes en la regresión múltiple.

La media de los valores de la muestra de la variable dependiente.

Todos los anteriores.

Ninguno de los anteriores.

The correct answer is "El número de variables independientes en la regresión múltiple." This is because the degrees of freedom in the error term of a regression model is given by n - k - 1, where n is the number of observations and k is the number of independent variables. Therefore, k represents the number of independent variables in the multiple regression model.

Explanation

The correct answer is "El número de variables independientes en la regresión múltiple." This is because the degrees of freedom in the error term of a regression model is given by n - k - 1, where n is the number of observations and k is the number of independent variables. Therefore, k represents the number of independent variables in the multiple regression model.

30. Se puede utilizar una distribución normal para aproximar la distribución t para regresión múltiple siempre que el número de grados de libertad (n menos el número de coeficientes de regresión estimados) sea:

Menor que 40.

Mayor que 10.

Igual que 5.

Mayor que 50

Ninguno de los anteriores.

not-available-via-ai

Explanation

not-available-via-ai

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200912013pac2pc13