As in the video above, let R_1,\ldots ,R_ n\stackrel{iid}{\sim } extsf{Ber}(p) for some unknown parameter p. We estimate p using the estimator \hat{p}=\displaystyle \overline{R}_ n=\frac{1}{n}\sum _{i=1}^{n} R_ i.

Question

As in the video above, let R_1,\ldots ,R_ n\stackrel{iid}{\sim } 	extsf{Ber}(p) for some unknown parameter p. We estimate p using the estimator \hat{p}=\displaystyle \overline{R}_ n=\frac{1}{n}\sum _{i=1}^{n} R_ i.

For a fixed number \alpha, after applying the CLT (and doing some algebra), we obtained

\lim _{n	o \infty } \mathbf{P}\left(\left[\overline{R}_ n-\frac{q_{\alpha /2}\sqrt{p(1-p)}}{\sqrt n},\overline{R}_ n+\frac{q_{\alpha /2}\sqrt{p(1-p)}}{\sqrt n}ight]
i pight) = 1-\alpha .
 
Which of the quantities in the equation above is random and which is deterministic?
(Choose one for each column.)

\overline{R}_ n:	n:	q_{\alpha /2}:	p:

random

deterministic
unanswered

random

deterministic
unanswered

random

deterministic
unanswered

random

deterministic
unanswered
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GPT 3.5 · Answer

In the equation above: \overline{R}_ n: deterministic n: deterministic q_{\alpha /2}: deterministic p: random

As in the video above, let R_1,\ldots ,R_ n\stackrel{iid}{\sim } \textsf{Ber}(p) for some unknown parameter p. We estimate p using the estimator \hat{p}=\displaystyle \overline{R}_ n=\frac{1}{n}\sum _{i=1}^{n} R_ i.