How to Determine the Constant c for a t-Student Distribution in Statistics?

[kcirick]

Question:
Consider n+1 mutually independent random variables $x+i$ from a normal distribution $N(\mu ,\sigma ^{2})$. Define:

$$\bar{x} = \frac{1}{n} \sum_{i=1}^{n}{x_{i}}$$ and $$s^{2}=\frac{1}{n}\sum_{i=1}^{n}{\left(x_{i} - \bar{x}\right)^{2}}$$

Find the constant c so that the statistic

$$t= c\frac{\bar{x} - x_{n+1}}{s}$$

follows a t-student law. Find the degrees of freedom. Justify.

What I have so far:
Not much, but for a statistic to follow Student T distribution with n-1:

$$t=\frac{\bar{x}-\mu}{s / \sqrt{n}}$$

Because we have n+1 random variables, and s has n degrees of freedome, the resulting student t distribution will have n degree of freedoms (if s is sample standard deviation, it should have n-1 degrees of freedom). I also just expanded the above equation:

$$t=c \frac {\frac{1}{n}\sum_{i=1}^{n}{x_{i}}-x_{n+1}}{\left(\frac{1}{n}\sum_{i=1}^{n}\left(x_{i} - \bar{x}\right)^{2}\right)^{\frac{1}{2}}}$$

But what next? Can someone help? thanks!

 

[EnumaElish]

What are the two differences between the formula

$$t= \frac{\bar{x} - x_{n+1}}{s}$$

and

$$t=\frac{\bar{x}-\mu}{s / \sqrt{n}}$$
?