Timeline for Central Limit Theorem and Confidence Interval
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 17, 2023 at 19:36 | comment | added | James C | @Dave I see. Thanks! | |
| Jan 17, 2023 at 19:28 | comment | added | Dave | @JamesC That sounds like a good topic for a new question to post separately. // You're getting closer when you have the $\sqrt n$ only on the left, but it is not quite right. The fully correct version would give some kind of arrow to denote convergence, perhaps even with a $d$ up top to denote convergence in distribution: $\sqrt n \bar X_n \overset{d}{\rightarrow} N(\mathbb E\left[X\right], \sigma)$. The standard way to write the theorem would have the convergence be to $N(0,1)$, but that's just a matter of some algebra. | |
| Jan 17, 2023 at 19:24 | comment | added | James C | @Dave Another question. Hope you don't mind. In practice, is there a $n$-value that we use to apply CLT? We know that $\sqrt n \bar X_n$ converges to a normal distribution, but we do not know how fast the convergence is. | |
| Jan 17, 2023 at 19:20 | comment | added | James C | @Dave so if I fix my statement as $\sqrt{n} \bar X_n \sim \mathcal N(\mathbb E[X], \sigma)$, then the statement becomes correct. Is this what you mean? | |
| Jan 17, 2023 at 19:11 | comment | added | Dave | @JamesC Did you read what Aksakal wrote in the link by Jbowman? "Since your right hand side is changing, it's difficult to prove statements about this convergence relation, and I mean impossible by difficult." If so, what remains unclear? | |
| Jan 17, 2023 at 19:09 | comment | added | James C | @jbowman Would you elaborate why my definition of CLT is incorrect? en.wikipedia.org/wiki/Central_limit_theorem#Classical_CLT | |
| Jan 16, 2023 at 23:48 | comment | added | Glen_b | @James Your notation is off (among other things, you conflate standard deviations with estimates). With CIs write a pivotal quantity (or in this case an asymptotically pivotal quantity) and use not just CLT but also Slutsky's theorem to get a suitable distribution for the pivotal quantity, and hence the CI for the parameter. Different CIs dont need to have the same length, you just need 1-alpha of them to overlap the parameter. | |
| Jan 16, 2023 at 22:15 | comment | added | jbowman | 1. The confidence intervals having the same length a) will never occur if the random variates are from a continuous distribution and b) is irrelevant. 2. Your definition of the central limit theorem is incorrect: see stats.stackexchange.com/questions/174734/… for some clarification. | |
| Jan 16, 2023 at 21:57 | history | asked | James C | CC BY-SA 4.0 |