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

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

 

[Architeuthis Dux]

I can provide a response to this question. First, we need to understand the concept of degrees of freedom in statistics. Degrees of freedom refer to the number of independent pieces of information available to estimate a parameter or variable. In this case, we have n+1 random variables, but we are only estimating the parameters \mu and \sigma^2. Therefore, the degrees of freedom for this problem would be n-1.

To find the constant c, we can use the definition of a t-distribution, which is given by:

t=\frac{Z}{\sqrt{\frac{\chi^{2}}{n}}}

where Z is a standard normal variable, \chi^2 is a chi-square variable with n-1 degrees of freedom, and n is the sample size. In our case, n is n+1, but since we are estimating two parameters, the degrees of freedom for \chi^2 would be n-1. Therefore, we can rewrite the equation as:

t=\frac{Z}{\sqrt{\frac{\chi^{2}}{n-1}}}

Substituting in the given values for \bar{x} and s^2, we get:

t=\frac{\frac{\bar{x}-\mu}{\sigma/\sqrt{n}}}{\sqrt{\frac{\frac{1}{n}\sum_{i=1}^{n}{\left(x_{i}-\bar{x}\right)^{2}}}{n-1}}}

Simplifying this equation, we get:

t=\frac{\bar{x}-\mu}{s\sqrt{\frac{n}{\left(n-1\right)\sigma^{2}}}}

Comparing this to the given equation, we can see that c=\sqrt{\frac{n}{\left(n-1\right)\sigma^{2}}}

Therefore, to ensure that t follows a t-student distribution, we need to multiply the given equation by \sqrt{\frac{n}{\left(n-1\right)\sigma^{2}}} . This would result in:

t=\sqrt{\frac{n}{\left(n-1\right)\sigma^{2}}}\frac{\bar{x}-x_{n+1}}{s}

The degrees of freedom for this t-distribution would be n-1, as we have n+1 random variables but are estimating two parameters.

In conclusion, the constant c for the given equation is \