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.
For a fixed number \alpha, after applying the CLT (and doing some algebra), we obtained
\lim _{n\to \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}\right]\ni p\right) = 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
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random
deterministic
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random
deterministic
unanswered
(The submit button is activated only after you have answered each question.)
In the equation above:
\overline{R}_ n: deterministic
n: deterministic
q_{\alpha /2}: deterministic
p: random
In the equation above:
- $\overline{R}_ n$ is a random quantity, as it is the sample mean of the random variables $R_1, \ldots, R_n$.
- $n$ is a deterministic quantity, as it represents the fixed number of observations in the sample.
- $q_{\alpha /2}$ is a deterministic quantity, as it represents the critical value corresponding to the given value of $\alpha$, which can be determined from a standard normal distribution table or statistical software.
- $p$ is a deterministic quantity, as it is the unknown parameter being estimated, but it is not directly involved in the sampling process or calculation of the confidence interval.
So the correct answer is:
random | deterministic