List of top Statistics Questions on Linear regression analysis asked in GATE ST

Consider the simple linear regression model \[ y_i = \alpha + \beta x_i + \epsilon_i, \quad i = 1, 2, \dots, 24, \] where $ \alpha \in \mathbb{R} $ and $ \beta \in \mathbb{R} $ are unknown parameters, the errors $ \epsilon_i $'s are i.i.d. random variables having $ N(0, \sigma^2) $ distribution, where $ \sigma>0 $ is unknown. Suppose the following summary statistics are obtained from a data set of 24 observations $ (x_1, y_1), \dots, (x_{24}, y_{24}) $: \[ S_{xx} = \sum_{i=1}^{24} (x_i - \bar{x})^2 = 22.82, \quad S_{yy} = \sum_{i=1}^{24} (y_i - \bar{y})^2 = 43.62, \quad S_{xy} = \sum_{i=1}^{24} (x_i - \bar{x})(y_i - \bar{y}) = 15.48, \] where $ \bar{x} = \frac{1}{24} \sum_{i=1}^{24} x_i $ and $ \bar{y} = \frac{1}{24} \sum_{i=1}^{24} y_i $. Then, for testing $ H_0: \beta = 0 $ against $ H_1: \beta \neq 0 $, the value of the $ F $-test statistic based on the least squares estimator of $ \beta $, whose distribution is $ F_{1,22} $, equals (rounded off to two decimal places):

For a given data $ (x_i, y_i) $, $ i = 1, 2, \dots, n $, with $ \sum_{i=1}^{n} x_i^2>0 $, let $ \hat{\beta} $ satisfy \[ \sum_{i=1}^{n} (y_i - \hat{\beta} x_i)^2 = \inf_{\beta \in \mathbb{R}} \sum_{i=1}^{n} (y_i - \beta x_i)^2. \] {Further, let } $ v_j = y_j - x_j $ and $ u_j = 2x_j $, for $ j = 1, 2, \dots, n $, and let $ \hat{\gamma} $ satisfy} \[ \sum_{i=1}^{n} (v_i - \hat{\gamma} u_i)^2 = \inf_{\gamma \in \mathbb{R}} \sum_{i=1}^{n} (v_i - \gamma u_i)^2. \] {If } $ \hat{\beta} = 10 $, then the value of $ \hat{\gamma} $ is:

Consider the following regression model \[ y_k = \alpha_0 + \alpha_1 \log k + \epsilon_k, \quad k = 1, 2, \dots, n, \] where $ \epsilon_k $ are independent and identically distributed random variables each having probability density function $ f(x) = \frac{1}{2} e^{-|x|}, \ x \in \mathbb{R}. $ Then which one of the following statements is true?

Consider the following regression model \[ y_t = \alpha_0 + \alpha_1 t + \alpha_2 t^2 + \epsilon_t, t = 1, 2, ...., 100, \] where $\alpha_0, \alpha_1, \alpha_2$ are unknown parameters and $\epsilon_t$'s are independent and identically distributed random variables each having $N(\mu, 1)$ distribution with $\mu \in \mathbb{R$ unknown. Then which one of the following statements is/are true?}