List of top Standard Distributions Questions on Confidence Interval

Let \( \{x_1, x_2, \dots, x_n\} \) be a realization of a random sample of size \( n (\geq 2) \) from a \( N(\mu, \sigma^2) \) distribution, where \( -\infty<\mu<\infty \) and \( \sigma>0 \). Which of the following statements is/are true?
Let \( X_1, X_2, \dots, X_n \) be a random sample of size \( n (\geq 2) \) from a \( N(0, \sigma^2) \) distribution. For a given \( \sigma>0 \), let \( f_\sigma \) denote the joint probability density function of \( (X_1, X_2, \dots, X_n) \) and \( S = \{ f_\sigma : \sigma>0 \} \). Let \( T_1 = \sum_{i=1}^{n} X_i^2 \) and \( T_2 = \left( \frac{1}{n} \sum_{i=1}^n X_i \right)^2 \). For any positive integer \( v \) and any \( \alpha \in (0, 1) \), let \( \chi^2_{v, \alpha} \) denote the \( (1 - \alpha) \)-th quantile of the central chi-square distribution with \( v \) degrees of freedom. Consider testing \( H_0: \sigma = 1 \) against \( H_1: \sigma>1 \) at level \( \alpha \). Then which one of the following statements is true?