Quiz: Properties of Random Variables

Question

We had a quiz given where we are supposed to pick out the incorrect answer. The options given are

• If $X_1 , \dots, X_n $ is a random sample from a normal population with population mean 1 and variance 2, then $ 0.5 \left((X_1 - 1)^2 + \cdots + (X_n - 1)^2 \right)$ follows a chi-square distribution with degrees of freedom $n$.

• If $T$ follows a t distribution with degrees of freedom $n$, then $T^2$ must follow an $F$ distribution

• If $X_1 , \dots, X_n $ is a random sample from a normal population with population mean 0 and variance 1, then $\frac{\sqrt{n} \hat{X}}{S} $ must follow a t distribution, where $\hat{X}$ denotes the sample mean and $S$ denotes the sample standard deviation.

• If $Y_1$ and $Y_2$ follow chi-square distributions with degrees of freedom $n_1$ and $n_2$ respectively, then $n_2 Y_1 / (n_1 Y_2)$ must follow an $F$ distribution

My chosen answer was the second option but apparently the correct answer is the last one. Can someone help me out with why???

Answer 1

In the last assertion note that there is no assumption that $Y_1$ and $Y_2$ are independent. Hence we cannot conclude the ratio will be $F$ distributed