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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.

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A little help with confidence interval estimation (for poker database analysis)

I'm a poker player, and I do a lot of database study to understand the frequencies of different profiles of players, to figure out how often they're bluffing and folding in different situations. I'll ...
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Inequality between length of confidence intervals

For given statistical model $(\mathcal{X},\mathcal{B},\mathcal{P})$, where $\mathcal{P}=\{B(1,\theta)^{\otimes n}:\theta\in (0,1)=\Theta \}$, $B(1,\theta)$ is a Bernoulli-trial with success ...
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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 $θ$ ...
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Confidence interval for difference between means

I have the following information about data: $$\sum_{i=1}^{12}x_i=2114; \;\sum_{i=1}^{12}y_i=2144;\;\sum_{i=1}^{12}x_i^2=373160;\;\sum_{i=1}^{12}y_i^2=383666$$ And I want to calculate the confidence ...
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Finding α-Quantiles of χ2 Distribution for Variance Estimation

I posted this same question yesterday but it got closed because i hadn't met the guidlines for questions, my apologies guys, so i'm going to re-write it better this time. Exercise 17. Let $X_1, \ldots,...
mathmath's user avatar
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Confidence Interval for Regression Values

From this article, I understand the idea behind the Confidence Interval for the Individual Response. However I don't understand the part on ...
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Calculating confidence interval while fitting multiple datasets simultaneously.

I am dealing with six experimental datasets of two species interaction and I want to fit these six datasets to a coupled ode system simultaneously to get a set of estimated parameters and ...
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Likelihood ratio test of Poisson distribution

I'm given this problem: Let $X_1,...X_{100}$ be a random sample from a Poisson distribution with mean $\lambda$. Consider testing the hypothesis $H_0$: $\lambda=1$ vs $H_1$: $\lambda<1$. Consider ...
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Confidence Interval for Expected Value of Binomial Distribution

been trying to wrap my head around this for a while. What is the confidence interval for the expected value for a binomial distribution? Let's say for a sample, I throw a coin 7 times and only 1 is ...
joelleoqiyi's user avatar
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Missing N in confidence interval for binomial distribution

I am studying on confidence interval for binomial distribution recently. I know general formula for confidence interval is $$ z1⋅\frac{σ}{\sqrt{n}} $$ if I substitute the formula for variance for ...
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Confidence Interval of Continuous Statistic Model

Let $(\mathcal{X}\subseteq \mathbb{R}^d, \mathcal{F}=\mathcal{B}(\mathbb{R}^d)\vert_{\mathcal{X}},\{P_{\vartheta}|\vartheta \in \Theta\})$ be a continuous statistic model, $\rho:\mathcal{X}\times \...
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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 $X_i\sim N(\theta,\theta^2)$ ...
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Poisson distribution coverage probability for confidence interval.

I have been trying to solve this problem 9.24 of Statitical inference by Casella and Berger. I was able to understand in the theory that confidence interval for Poisson distribution is based on chi-...
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RMS/quadratic mean and confidence interval

After searching on the web without success, I'm asking for help here. I wasn't taught statistics, and I get lost in a lot of formulas that I often find hard to understand. I'm also not used to posting ...
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Confidence interval 100 coin tosses, 52 heads?

I am really confused about the confidence interval for n=100 tosses and k=52 heads. If I estimate the probability $\hat{p}$: $\hat{p} = \frac{52}{100} = 0.52$ $Var(\hat{p}) = pq = 0.52 * 0.48 = 0.2496$...
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Definition of confidence interval - intuition

In the context of the confidence interval for parameter $\theta$ with confidence level $1-\alpha$ I was always dealing with such formulation, $P(\theta \in \mathbb{T}_n)=1-\alpha$, with the "=&...
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Is this a general form for the confidence interval of a uniform distribution?

Let $X_1, \ldots, X_n$ a random sample with $X_i \sim \mathcal{U}[0, \theta]$ (where $\mathcal{U}$ = uniform dist). Let $Y = \max(X_1, \ldots, X_n)$, the MLE of $\theta$. It can be proven that $U =...
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Confidence interval on the quantile of expected values

Given $s_i = E_{\delta \sim p}[ f(x_i + \delta)]$ define $q=\mathrm{Quant}({s_1, \dots, s_n}, \alpha)$ as the $\alpha$ quantile. One way to obtain a confidence interval for q is to obtain individual ...
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1 answer
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Confidence interval for mean based on a single trajectory of a first-order autoregressive process

I am currently studying Statistics for Spatial Data, revised edition, by Cressie. Chapter 1.3 STATISTICS FOR SPATIAL DATA: WHY? says the following: 1.3 STATISTICS FOR SPATIAL DATA: WHY? Some simple ...
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When using pivots to create confidence intervals, it seems that pivots are always depending on θ?

I am really confused on how to find a pivot, it seems that it's just always derived randomly and on top of that even though it's defined that a pivot should nto depend on θ the pivot function always ...
Ann Flus's user avatar
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How do I find the confidence interval for the MLE of a Bradley Terry Model?

I am trying to find out how to find the confidence interval of the Bradley Terry Model. I have the log likelihood equation, and I know I need to use the Fisher Information which is the negative ...
John Smith's user avatar
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Logistic regression: why do +-2*std.error of predicted values differ from 95% confidence intervals of odds ratios?

I have a logistic regression model, where a binary response variable is being explained by a categorical variable which has three classes. When we look at the 95% confidence intervals of the odds ...
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159 views

Finding Asymptotic Confidence Intervals for Parameter θ in Uniform and Exponential Distributions

Let $(X_1, \ldots, X_n)$ be an i.i.d. random sample. Determine asymptotic confidence Let $(X_1, \ldots, X_n)$ be an i.i.d. random sample. Determine asymptotic confidence intervals at level $\gamma \in ...
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2 votes
1 answer
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Two sample test - distribution of pooled variance estimator

I am attending a statistics course this semester and although it is offered by the math department the precise assumptions underlying the main theorems are not provided, let alone the proofs. That ...
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Finding a worst case upper bound for the Poisson mean

Assume that $X_1,\ldots X_m \sim Poi(\lambda_0)$ (iid). Now the goal is to get a worst-case upper bound for $\lambda_0$, which is also consistent when $X_1=...=X_m=0$, or more generally, when $S_m = \...
FreddyGrit's user avatar
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0 answers
51 views

Calculate $Var\left(\frac{X_1+\cdots+X_n}{n}\right)$ and estimate $\sigma_{\bar{X}}$

Suppose that in a sample of size $n = 100$ from an AR(1) process with mean $\mu$ $$X_t - \mu = \phi (X_{t-1}-\mu) + Z_t$$ where ${Z_t} \sim \operatorname{WhiteNoise}(0, \sigma^2), \sigma^2 = 2, \phi = ...
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Pivotal Quantity for Normal Distribution

Suposse a random sample of size $n$ from a Nomal distribution $X_{i}\sim N(\mu,\sigma^{2})$, for the following random variables: (1) $\frac{\overline{X}-\mu}{S/\sqrt{n}}\sim t(n-1)$ and (2) $\frac{(n-...
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Confidence interval for Ratio of slope in linear regression

The question I'm trying to solve is as follows; "Company-wide, one unit of TV advertising on the above coded scale would cost $3, 250, 000. The Vice President for Marketing wants to know how many ...
ethan's user avatar
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1 vote
0 answers
57 views

Use prediction interval to get possible values for x given y?

I have the linear model $y = \beta x + \alpha + \epsilon$ with $\epsilon$ i.i.d normally distributed with variance $\sigma^2$. I fit the linear regression using OLS and compute a prediction interval ...
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How do i make prediction intervals without normally distributed residuals?

I have made predictions for the amount of conversions of a particular website. After predicting the amount for every day. I looked for ways to get a prediction interval for every day. The residuals ...
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Prediction interval for AR(1) forecast

This link (and others, e.g. slides 43 and 46 of this) say that: Where all the coefficients in the model are point estimates, we could calculate the MSE to generate distributions for the distribution ...
Cyclopropane's user avatar
0 votes
1 answer
70 views

Please help me derive the formula for upper bound for one sided confidence interval $\bar{x} + z_{\alpha}(\frac{\sigma}{\sqrt{n}})$?

I want to derive for myself the known formula for the upper bound for one sided confidence interval $\bar{x} + z_{\alpha}(\frac{\sigma}{\sqrt{n}})$ for mean $\mu$ for a sample of size $n$ from a ...
Alex's user avatar
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Accounting for an uncertainty in the number of categories of the multinomial distribution

Assume that we have an unfair die, and our task is to determine the probabilities of rolling a 1, a 2, a 3 and so on by rolling the die. Unfortunately we have no way of knowing how many sides the die ...
uLoop's user avatar
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0 answers
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Determine if a $95\%$ confidence interval for $\theta$ is equal to a given set

So I need to decide if the following statement is true or false: Let $X_1,X_2,...,X_n \sim N(\theta,1)$. Then a $95\%$ confidence interval for $\theta$ is all of the values of $\theta$ that satisfies: ...
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0 answers
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Comparing Confidence Given by Concentration Inequalities and Central Limit Theorem

Given an i.i.d. sample $X_1,\dots, X_n$ from Ber($p$), by Chebyshev's inequality,we have $$\text{Pr}(|\overline{X}-p|\le \epsilon)\ge 1-\frac{p(1-p)}{n\epsilon^2}.$$ Therefore, if $\text{Pr}(|\...
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Is non-nominal confidence interval coverage normal in practice?

I am finalizing an academic paper and am having a bit of trouble with confidence interval estimation. Using $Z$-approximation, I calculated the estimated confidence interval using $$\widehat{\mu} \pm ...
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0 answers
34 views

Can I use assume a Normal Distribution to calculate a Confidence Interval for multiple random variables with different means and std each?

Let's say, I've a table with a list of trips I need to make this week and I want to calculate a confidence interval for the average trip duration time for all the trips I'll make in the week. Var ...
Moises Rojo's user avatar
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1 answer
58 views

Locating variable based on confidence interval for exponential distribution

$X$ is a random variable and its PDF is: $$ f(x;θ) = \begin{cases} θe^{-θx}, & x > 0 \\\ 0, & x \le 0 \end{cases} $$ Now, I would like to find the confidence interval regarding parameter $...
Sonamu's user avatar
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1 vote
1 answer
190 views

Asymptotic confidence interval for gamma distribution

Let $X_1,...,X_n$ be a sample from a gamma distribution with parameters $(θ,λ)$. I need to find an asymptotic confidence interval for $θ$ confidence level $α$, When $λ$ is unknown. I don't understand ...
wxist's user avatar
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0 answers
171 views

Confidence interval using Chebyshev's inequality

$X_1,...,X_n$ is a sample, $X_1 = ξ + η$, where $ξ,η$ are independent random variables, $ξ ∼ R[0,θ], η ∼ Bin(1,θ)$. I want to establish a confidence interval for $θ$ confidence level $1−α$ using ...
wxist's user avatar
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1 vote
1 answer
63 views

What will be the distribution of the expected marks?

in a multiple choice exam you have an unlimited supply of Questions,in Which a correct answer fetches 4 points a wrong answer a penalty of 1 mark.if you randomly select one of the four options ...
Arjenton's user avatar
1 vote
1 answer
54 views

Finding the probability that the true proportion $p$ is higher than $0.30$

Here is the exact wording of the question I am struggling to answer. The manager of a store wants to know more about the proportion of customers who are visiting the store for the first time. She ...
Matthew's user avatar
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1 vote
0 answers
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Calculation of lower range confidence interval

Question We observe $x$, the maximum of $n$ values in a random sample from the uniform distribution between $0$ and $c$, where $c > 0$. Find an exact lower range $100(1 - \alpha)\%$ confidence ...
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What is the geometrical difference between $Z_{\alpha/2}$ and $E_{a}$?

What is the geometrical difference between $Z_{\alpha/2}$ and $E_{a}$? Let's say we have a generator of toys and the weight is distributed with a standard deviation of $4kg$ and a mean of $5kg$ for ...
Acedium 20's user avatar
1 vote
1 answer
46 views

Find the number of measurements so that our iuncertainty is less than 2%

The results of a certain experiment whose true value is about 10.0 cm, follow a normal distribution with standard deviation of 1.0 cm. How many measurements will we have to do so that our uncertainty ...
Acedium 20's user avatar
0 votes
1 answer
40 views

Confidence Interval Inequality Simplification

In the Wikipedia for Hoeffding's Inequality, it says that $\alpha \leq 2e^{-2\varepsilon^2n}$ implies that $n \geq \frac{\log(2/\alpha)}{2\varepsilon^2}$. When I work through the intermediate steps, I ...
James Larsen's user avatar
1 vote
0 answers
51 views

Calculations of standard errors

Problem Let $Y_1 \sim N(μ_1,1),Y_2 \sim N(μ_2,1), Y_3 \sim N(μ_3,1)$ and also assume that these three random variables are mutually independent. The observed sample values are $y_1 = 2, y_2 = 0 \ \...
Hmmmmm's user avatar
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1 vote
0 answers
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Construction of confidence intervals for a Gamma random variable

Problem Suppose that $Y_1,Y_2,Y_3$ denote a random (independent) sample of size $3$ from a distribution with parameter $\lambda$ defined by the following probability density function: $$\begin{...
Hmmmmm's user avatar
  • 333
1 vote
0 answers
118 views

Construction of confidence interval for a normally distributed parameter

Problem Let $Y_1 \sim N(μ_1,1),Y_2 \sim N(μ_2,1), \ \mathrm{and} \ Y_3 \sim N(μ_3,1)$ and also assume that these three random variables are mutually independent. The observed sample values are $y_1 = ...
Hmmmmm's user avatar
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0 answers
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How to calculate the confidence interval of non-normal large volume data sample?

There are tens of thousands of data, and the sample mean is 5.04, the sample standard deviation is 3.97, the skewness is 3.74, and the kurtosis is 23.721. How can we calculate confidence intervals for ...
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