Wald test and F distribution

Assume that the sample $X_i, i=1,\dots, n$, is iid normally distributed with unknown mean and variance $\theta = (\mu, \sigma^2)$. The Wald test statistic of testing if $\mu = \mu_0$ is $$W=\frac{n(\hat{\mu} - \mu_0)^2}{\hat{\sigma}^2}$$ where $\hat{\sigma}^2 = \frac{\sum_i (X_i - \hat{\mu})^2}{n}$ is the maximum-likelihood estimator of $\sigma^2$.

1. I was wondering if it is correct that $\frac{(n-1)W}{n^2}$ has a F distribution with $1$ and $n-1$ d.f. under null?
2. From a note, it seems that $W$ is an F distribution with $1$ and $n-1$ d.f. under null. Isn't this incorrect? Let $p_2$ be the dimension of $\mu$, and $p$ be dimension of $X_i$.

Under the assumption of normality we have a stronger result. The distribution of $W$ is exactly chi-squared with $p_2$ degrees of freedom if $σ^2$ is known. In the more general case where $σ^2$ is estimated using a residual sum of squares based on $n-p$ d.f., the distribution of $W/p_2$ is an $F$ with $p_2$ and $n-p$ d.f.

Thanks and regards!

To emphasize that the distance between the point estimate $\hat{\mu}$ and the hypothesized value $\mu$ is being scaled by its standard error, I find it easier to write $W$ as

$$W = \frac{(\hat{\mu} - \mu_0)^2}{\hat{\sigma}^2/n}.$$

Recall the relationship between the $\hat{\sigma}^2$ and the unbiased sample variance $s^2$:

$$\hat{\sigma}^2 = \frac{n-1}{n} s^2.$$

Now, notice that we can write

$$\frac{n-1}{n} W = \frac{(\hat{\mu} - \mu_0)^2}{s^2/n}.$$

This is useful for a couple of reasons. First, recognize that the square root of the righthand side is the statistic for the one-sample Student's t-test, which has a Student's t sampling distribution with $n-1$ degrees of freedom under the null hypothesis assuming the sample is iid normally distributed. That is,

$$\sqrt{\frac{n-1}{n} W} = \frac{\hat{\mu} - \mu_0}{s/\sqrt{n}} \sim t_{n-1}.$$

Next, recall a relationship between the Student's t and (central) F distributions: if $Y \sim t_{\nu}$, then $Y^2 \sim F$ with degrees of freedom 1 and $\nu$. Therefore,

$$\frac{n-1}{n} W \sim F$$ with degrees of freedom 1 and $n-1$.

The note that you linked in point 2 does not explicitly apply here. First, as you stated, $\sigma^2$ is unknown. Also, you have described the classic one-sample Student's t-test, whereas the link is describing a more general case (i.e., testing regression coefficients). The dimension concepts you quoted are referring to multivariate problems. You can see a connection though by noting that the dimension here is $p = 1$.