In the video titled " Estimating Var[S2]" of Week 20, at around 13:05 , Pratyush Sir says that when

X_i\sim N(\mu ,\sigma ^2)

then

\frac{\bar{X}-\mu}{\sigma}\sim N(0,1)

Here, is the corresponding screenshot:

But, I think as

\bar{X}\sim N(\mu,\frac{\sigma^2}{n})

the correct formaulation should be:

\frac{\bar{X}-\mu}{\sigma/\sqrt n}\sim N(0,1)

As a result, at around 15:04,

when Sir says that

\frac{n(\bar{X}-\mu)}{\sigma^2}

can be seen as

n\bar{Z}^2

it should only be

\bar{Z}^2

Please look into the matter once and clarify my doubt. Correct me if I am wrong !