Degrees of Freedom in t-Distribution for Simple Regression without a Constant

[MaxManus]

Homework Statement

Simple regression without a constant
Y_i = Bx_i + eps_i for i = 1,2,...n
eps_i are independent and N(0, sigma^2) distributed, B and sigma^2 are unknown.

All my sums are from i = 1 to n
The question is: Explain why:
$$\frac{\hat{B} - B}{S} \sqrt{\sum{x_i^2}}$$
is t-distirbuted with n-1 degrees of freedom.

$$\hat{B}$$ is the least square estimator for B, and S^2 is the least square estiamtor for sigma^2


I'm not sure how to start solving the problem. My first idea was that this looket like a standard t-distribution for $$\hat{B}$$, but $$\sqrt{n} \neq \sqrt{\sum{x_i^2}}$$

Homework Statement

Homework Equations

The Attempt at a Solution

 

[MaxManus]

Can you say;
If
$$\frac{\hat{B} - B}{S}\sqrt{n}$$ is t-distributed then:
$$\frac{\hat{B} - B}{S} \sqrt{\sum{x_i^2}}$$
is t-distributed since n and the x are just numbers?
And can you go further and say that if the first have (n-1) degrees of freedom then the second equation also has to?