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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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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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 ...
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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 ...
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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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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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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 = \...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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probability of intersection of confidence intervals

Suppose $\hat{\boldsymbol{\theta}} = \left(\hat{\theta}_1,\hat{\theta}_2\right)^{\top} \xrightarrow d N_2(\boldsymbol{\theta}, V)$ for some unknown $V.$ We have consistent estimator of $V:$ $\hat{V},$ ...
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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 ...
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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 $...
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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 ...
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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 ...
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Find Confidence Interval for a given sample mean without sample standard deviation

The question is: Assume that a single-digit random number is generated for 100 trials. Let X denote the number generated per trial. Suppose that x(sample mean) assumed the value 3.85 for these 100 ...
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How to calculate this confidence interval?

I numerically integrate a differential equation of the form $y’ = f(y, a, b)$ where the parameters $a$ and $b$ are real numbers. Also, I have many real numbers $v$ from $v_1$ to $v_m$ ($m$ is around ...
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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 ...
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Wilson interval estimation for multiple classes (confusion matrix)

Sorry to resurrect this, but after 9 years the links mentioned here (How can the Wilson Confidence Interval be adapted for more than 2 outcomes?) went dead ;) I am looking for similar help in adapting ...
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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 ...
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Help with sample size for finding average distance betwee geographical points

I have a program that I want to test through simulations (since I don't know exactly how it works). The program converts a precise location (lat, lon) to an approximate location (it adds noise and ...
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What is the difference between a coefficient in an OLS regression model and the mean difference in a Tukey's HSD test?

I'm wondering if someone could explain in high-school-level language what the difference is between these concepts? I'm looking at the results of an OLS regression for a single categorical variable (...
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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 ...
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Find between which values the 20% central will be at.

The measurements of the times of a group of children competing in a race of The 100-m dash follows a normal distribution with a mean of 13.4 s and a standard deviation of 1.2 s. Among which values ...
Acedium 20's user avatar
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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 ...
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introductory statistics - confidence interval

Consider the results of the group stage matches from two FIFA World Cups, namely the $2002$ tournament in Japan/South Korea and the 2018 tournament in Russia (N = 96). Let $X_1$ be the number of goals ...
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Intuition on bias corrected estimator in statistics

Say I have my original objective function $\|Y - X \beta\|_2^2$, and for some reason (other motivation), I want to add a penalty term and obtain a new objective function. The target estimator $\beta_0$...
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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 ...
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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 \ \...
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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{...
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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 = ...
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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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Confindece interval and hypothesis test for means

I need help with this exercise. The US Census Bureau conducts the Current Population Survey each year in its states and territories. The following table comes from the Census page and describes the ...
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How do I translate the coefficients of an OLS confidence interval into a range of actual values?

I'm using statsmodels to perform an OLS simple regression on the Palmer's penguins dataset. The regression uses birds' bill length as the sole independent variable and their body mass as the target. ...
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Estimating sample size based on partial results?

Say I have a fair 100-sided die that I rolled an unknown number of times. I know that I rolled a 7 exactly 219 times. Intuitively, I know there were probably around 21900 rolls (since ~1% of them ...
wlogan412's user avatar
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Constructing exact confidence interval for normal distribution

Suppose $X_1, ..., X_n, Y_1, ..., Y_m$ are independent random variables and $X_1, ..., X_n$ i.i.d.∼$\mathcal{N}(\mu_1, \sigma_1^2)$ and $Y_1, ..., Y_m$i.i.d.∼$\mathcal{N}(\mu_2, \sigma_2^2)$, with $\...
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How to compute a confidence interval if the sample is left-truncated?

If I have a sample of a lognormal random variable given by some values $x_1$, $x_2$, $x_3$, $x_4$, $x_5$,$x_6$, $x_7$, $x_8$, $x_9$, $x_{10}$, and this sample was truncated in some fixed value $x_0$; ...
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Statistical Question

I have a problem which goes like this: Assume you have two samples of the same population.Both samples contain textual instances.I know the size of one sample.I want to estimate the size of other ...
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Confidence bound for F-measure more accurate than normal approximation

I want to compute confidence interval for classifier's F-measure (true positive, true negative), using sample dataset. There is a related old post which suggest to use normal approximation to compute ...
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