List of top Regression Analysis Questions

Let a linear model \( Y = \beta_0 + \beta_1 X + \epsilon \) be fitted to the following data, where \( \epsilon \) is normally distributed with mean 0 and unknown variance \( \sigma^2>0 \): \[ \begin{array}{|c|c|c|c|c|c|} \hline x_i & 0 & 1 & 2 & 3 & 4 \\ \hline y_i & 3 & 4 & 5 & 6 & 7 \\ \hline \end{array} \] Let \( \hat{Y}_0 \) denote the ordinary least-square estimator of \( Y \) at \( X = 6 \), and the variance of \( \hat{Y}_0 \) be \( c\sigma^2 \). Then the value of the real constant \( c \) (rounded off to one decimal place) is equal to ________

Consider the multiple regression model \[ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \epsilon, \] where \( \epsilon \) is normally distributed with mean 0 and variance \( \sigma^2>0 \), and \( \beta_0, \beta_1, \beta_2, \beta_3 \) are unknown parameters. Suppose 52 observations of \( (Y, X_1, X_2, X_3) \) yield sum of squares due to regression as 18.6 and total sum of squares as 79.23. Then, for testing the null hypothesis \( H_0: \beta_1 = \beta_2 = \beta_3 = 0 \) against the alternative hypothesis \( H_1: \beta_i \neq 0 \) for some \( i = 1, 2, 3 \), the value of the test statistic (rounded off to three decimal places), based on one way analysis of variance, is ________

Let \( (X, Y, Z) \) be a random vector with the joint probability density function \[ f_{X,Y,Z}(x, y, z) = \begin{cases} \frac{1}{3}(2x + 3y + z), & 0<x<1, 0<y<1, 0<z<1, \\ 0, & \text{elsewhere}. \end{cases} \] Then which one of the following points is on the regression surface of \( X \) on \( (Y, Z) \)?

Consider the multiple linear regression model

\[ Y_i = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \dots + \beta_{22} x_{22,i} + \epsilon_i, \quad i = 1, 2, \dots, 123, \]

where, for \( j = 0, 1, 2, \dots, 22 \), \( \beta_j \)'s are unknown parameters and \( \epsilon_i \)'s are independent and identically distributed \( N(0, \sigma^2) \), \( \sigma>0 \), random variables. If the sum of squares due to regression is 338.92, the total sum of squares is 522.30 and \( R^2_{\text{adj}} \) denotes the value of adjusted \( R^2 \), then

\[ 100 R^2_{\text{adj}} = \underline{\hspace{2cm}} \]

(round off to 2 decimal places).

Consider the simple linear regression model \[ Y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \quad i = 1, 2, \dots, n \quad (n \geq 3), \] where \( \beta_0 \) and \( \beta_1 \) are unknown parameters and \( \epsilon_i \)'s are independent and identically distributed random variables with mean zero and finite variance \( \sigma^2>0 \). Suppose that \( \hat{\beta}_0 \) and \( \hat{\beta}_1 \) are the ordinary least squares estimators of \( \beta_0 \) and \( \beta_1 \), respectively. Define \( \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \), \( S_1 = \sum_{i=1}^n (x_i - \bar{x})^2 \), where \( y_i \) is the observed value of \( Y_i, i = 1, 2, \dots, n \). Then for a real constant \( c \), the variance of \( \hat{\beta}_0 + c \) is