Undergrad Distribution of a sample random variable

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The statistics T, defined as T = (X̄ - 7) / √(s²/15), follows a t-distribution with 14 degrees of freedom. To determine the distribution of T², it is noted that T² can be expressed as a ratio involving a chi-squared distribution. Specifically, T² = (Z² * (n-1)) / χ²(n-1), where Z is a standard normal variable and χ²(n-1) represents a chi-squared distribution with n-1 degrees of freedom. Consequently, T² follows an F-distribution with parameters (1, 14). Understanding these relationships is crucial for statistical analysis involving sample means and variances.
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what distribution follows
$X_1, X_2, ..., X_{15}$ are independently to each other and follow $N (7, 3^2)$ what distribution the following statistics follow$T = \frac{(\bar{X}− 7)}{\sqrt{s^2/15}}$i know this follow t distribution $t_(n-1) =t_{14}$but how do i find what distribution $T^2$ follows, can i just multiply it?$T = (\frac{(\bar{X}− 7)}{\sqrt{s^2/15}})^2=\frac{Z^2*(n-1)}{\chi_{(n-1)}}$
 
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Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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