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Questions tagged [confidence-interval]

In statistics, a confidence interval (CI) is a type of interval estimate (of a population parameter) that is computed from the observed data.

The confidence level is the frequency (i.e., the proportion) of possible confidence intervals that contain the true value of their corresponding parameter. In other words, if confidence intervals are constructed using a given confidence level in an infinite number of independent experiments, the proportion of those intervals that contain the true value of the parameter will match the confidence level.

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Probability vs Confidence

My notes on confidence give this question: An investigator is interested in the amount of time internet users spend watching TV a week. He assumes $\sigma = 3.5$ hours and samples $n=50$ users and takes the sample mean to estimate the population…
hongsy
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What is the (fully rigorous) definition of a confidence interval?

In a nutshell: what is the (fully rigorous) definition of a confidence interval? In page $92$ of Wasserman's All of Statistics, it is written that A $1 − α$ confidence interval for a parameter $θ$ is an interval $C_n = (a, b)$ where $a =…
Sam
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Calculating the "Spreads" for Different Outcomes in Dice Rolls?

Suppose I roll a 6-sided die 100 times and observe the following data - let's say that I don't know the probability of getting any specific number (but I am assured that each "trial" is independent from the previous "trial"). Below, here is some R…
stats_noob
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Sufficient statistics function for $N(\theta, c\theta^2)$ and symmetrical confidence interval using $\bar{X}$

Exercise: Let $X_1, \dots, X_n$ be a random sample from the Normal Distribution $N(\theta,c\theta^2)$ where $c > 0$ is a known constant and $\theta \in \mathbb R$ an unknown parameter. i) Find a sufficient statistics function for $\theta$. ii)…
Rebellos
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Why use 95% confidence interval?

May I ask why $95\%$ confidence is so commonly used? Does it have anything to do with $\frac{d}{d\alpha}e_n(\alpha)$, where $e_n(\alpha) = Z_{\alpha/2}\frac{S_n}{\sqrt n}$? (My professor asks me to evaluate this derivative at $\alpha = 0.05$, given…
PassingBy
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Why does confidence interval need to be symmetric about the point estimate and contiguous?

This post could be entitled as "Interpretation of confidence interval 2" according to the prior question. (Interpretation of confidence interval) I've read the QA but still not gotten to the point. I am not the same person who asked the previous…
mriryt
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Confidence interval for parameter of normal distribution $X_i\sim N(\theta,\theta^2)$ with equal mean and standard deviation

A sample $X_1,\dots,X_n$ is drawn from the normal distribution $N(\theta,\theta^2)$. I am asked to find a $90\%$ confidence interval for the population mean $\theta$. Let $X_i\sim N(\theta,\theta^2)$ with $$\mathbb{E}(X_i)=\theta \text{ and }…
Tutusaus
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Can we rely on Confidence Intervals?

Suppose the mean is in (7.6,8.4) with 95% confidence. I understand that this means 95% of the confidence intervals from different samples will contain the population mean. But, what is the significance of this particular interval on its own. Since I…
Archer
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Confidence interval interpretation difficulty

I have seen a lot of questions in this forum related to what my question is, but I didn't find any convincing answer. So I would to like to put this question: When we are dealing with 95% confidence interval we mean that if we repeat process of…
user1825567
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Is a $90\%$ confidence interval really $90\%$ confident?

Let's say you are estimating a population proportion, which you model as binomial. One source of error already is using the normal approximation to the binomial when getting your critical values. But what bothers me more is that the theoretically…
doctorpigeonhole
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Interpretation of confidence interval

Say I have a 95% confidence interval of the sample mean. Does that mean there is a 95% probability that the population mean lies within that interval?
Hamza
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paired t-test vs Welch's t-test.

Need to find a 95% confidence interval for $E(Z)$ where $Z=X-Y$ using both paired t-test and Welch's t-test. For one what is the main difference between them and for two how do you do it? Need help studying for a test so the answer can be generic.
daultongray8
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Can we find a better algorithm to solve the following sequential game?

Let $\sigma_0, \sigma_1, \sigma_2, \dots$ be a sequence in $\{-1,+1\}$ and $T \in \mathbb{N}$ a time horizon. Consider the following game. At each time step, we're asked if we want to give an answer $X_t \in \{-1,1\}$, or to abstain and skip to the…
Bob
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Can A Probability Ever Be Outside of $0$ and $1$?

Recently, I have been studying the Multinomial Probability Distribution Suppose you go to a casino and there is a game that involves rolling a six-sided die (i.e. one dice). However, you are not told what is the probability that this die lands on…
stats_noob
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Finite-sample confidence interval for the sample mean of N iid Beta-Binomial random variables

Let $m,n,l,N$ be 3 integers, and $C_1,\dots,C_N$ i.i.d. Beta-Binomial RV with the following distribution: $C_i\sim\frac{1}{m}\text{Binom}(m,M)$ where $M\sim\text{Beta}(n+1-l,l)$ The sample mean is $\bar{C}=\frac1N\sum_{i=1}^N C_i$ I would like to…
DeltaIV
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