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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Confidence interval of a random variable

Given $Θ$ is an unknown, and $X$~$U(Θ-0.5,Θ+0.5)$ Is $[X-2,X+2]$ an 80% interval? Apparently, yes it is: $$P(X-2≤Θ≤X+2)=P(Θ-2≤X≤Θ+2)=1$$ The transition from $P(X-2≤Θ≤X+2)$ to $P(Θ-2≤X≤Θ+2)$ is ...
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Difference between mean response at $x_0$, and predicted value at $x_0$

I've read a few accounts on the difference between these two, but none have been very clear. So what exactly is the difference between 'mean response at $x_0$' and 'predicted value at $x_0$'? Also, ...
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Confidence interval for a probability

I'm currently studying the following problem in my textbook: Consider the confidence interval for an unknown probability $p$. Suppose that we want confidence level $0.95$ and that we want the length ...
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Central Limit Theorem with Confident Interval

Below is a question with two parts- a) and b) and further below the answers. I understand part a). However, part b) has me confused. I understand basic confidence intervals but this seems to be going ...
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How do I derive a formula for the prediction intervals for the sum of responses of two independent future observations? [closed]

So far I've tried using the formula for confidence intervals for a full rank model and trying to use that to get a prediction interval formula. What I don't understand is how to get a prediction ...
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Limits vs Confidence Interval, the difference?

Below is a question on "limits" and confidence interval. The answers are below. Why does question b) not require dividing by the square root of 10, but the confidence interval in question c) ...
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Central Limit Theorem "find sample size" question, with modulus inequality

I'm doing the below Central Limit Theorem question. I omitted sub-question a) and b) as they were unrelated. I have included the answer below the question. I half-understand the answer but I don't ...
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Why don't n-1 for df in this scenario

Simple statistics question, but can't figured out. What is the critical value t* for constructing a 99%, percent confidence interval for a mean with 11 degrees of freedom? My answer for this is 3....
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How to derive the formulae for Poisson confidence intervals?

I have been struggling to derive the formulae for CI of Poisson distribution parameters. Suppose I have a sample $X=(X_1,\dots,X_n)\in\text{Poisson}(\lambda)$. Parameter $\lambda=\theta$ is what I ...
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Determining the sample size to satisfy two tests

A survey of a university's students is to be carried out to estimate both the proportion, $P$, who own a bicycle and the average weekly spend on junk food, $\bar X$. It is desired to estimate $P$ with ...
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Calculate the confidence interval of the parameter representing a logical level

Basically, I have a big dataset including 150+k rows. It consists of completed flights by "N" air company. Each row in the Excel file represents each flight on a special date. The row ...
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statistics - find confidence interval from a given estimator

So I've been trying to solve this question: we have $x_i$~$N(\mu,4)$. an estimator $ T_1 = \frac{2x_1 +x_2}{3}$ we need to find confidence interval with $\alpha $ confidence level that is based on $T_!...
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Confidence Interval estimation for Cauchy distribution

Considering a Cauchy distribution with scale parameter known(=1) and unknown location parameter, how can I estimate the confidence interval of the location parameter. Estimating the parameters using ...
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Combining confidence intervals of multiple upper bounds

I am trying to bound a confidence interval of some expression given bounds and confidence intervals for the terms of that expression. Suppose I have the following bound: $$F \le \sum_{i=0}^{t-1} F_i$$ ...
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Calculating statistics for encryption times of multiple algorithms

I'm doing an assignment, that envolves timing multiple encryption/decryption algorithms, and I'm having trouble plotting them. My data is something in the lines of: Bytes Mean (ms) Confidence ...
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Confidence interval of transformed random variable

Let $X$ be a Normal random variable of mean $\mu$ and variance $\sigma^2$. Also let $g\colon \mathbb{R}\to\mathbb{R}_{>0}$ be a positive strictly increasing bijective function. I would like to find ...
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Interpretation of Confidence Interval in terms of Probability

Let $\hat{P}$ be the random variable of the sample proportion and $p$ be the population parameter. Let's form a $95\%$ interval estimate by approximating the distribution of $\hat{P}$ to normal We ...
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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 $\...
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Confidence interval for Poisson distribution

$X_{1}, X_{2}, ..., X_{n}$ is a random sample from $Poisson(\lambda)$ population. I need to show that when sample size n is large, the approximate two-sided (1-$\alpha$)% C.I. is $$ \left[ \bar{x} + ...
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Normal distributions and log transformations

Lets assume I have a origin distribution, lets call it Omega, which is heavily skewed and does not seem to be normally distributed. I now apply a log(n+1) function to it and get a normal distribution ...
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Confidence Interval for Markov Chain Probability

I have a simple transition model I am trying to use to predict the probability of two states. $$ \begin{bmatrix} p_{1,t+1}\\ p_{2,t+1} \\ \end{bmatrix}= \begin{bmatrix} p_{11} & p_{12} \\ ...
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Is choosing confidence interval bounds after observing data mathematically valid?

Let $D$ be a continuous distribution on the interval $[0,1]$ that is not known to us. We have no prior knowledge about $D$. For a given error tolerance $\delta$ we want to find bounds $a, b$ s.t. $b-a$...
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Confidence bound for a family of lines

I have a family of lines $y_i= a_ix+b_i$ where the pairs $(a_i,b_i)$ are generated by a random process. I can fix $x$ and generate a family of points $y_i$ for which I can estimate the mean and the ...
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Confidence Interval with the mean being the average of probabilities

Let's say that, in a video-game, every time you go to the blacksmith and there is a random (yet announced) chance he succeeds fixing your weapon. In this example it goes like this: 70% chance ...
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Finding a confidence interval for a binomial proportion without knowing the mean or variance?

I'm just learning statistics and I've been given an interesting problem to solve that I'm unsure how to approach. I've dealt with various tests (t-test, chisq test, confidence intervals etc.) but I'm ...
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Evaluation of the Poisson distribution parameter and its uncertainties using only one observation

I heard that, for example, the number of meteorites that hit the Earth follows the Poisson distribution. My question is how to estimate the Poisson parameter $\lambda$ and confidence interval for this ...
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Confidence interval for Poisson distribution using CLT.

In my lecture notes it is used that if $x_1,...,x_n$ is an i.i.d. sample from a Poisson distribution with parameter $\lambda$ then a confidence interval for $\lambda$ is given as $$\left[ \bar{x}+\...
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Assume $(X_i,Y_i)$ are i.i.d sequence of random vectors. Prove $\sqrt{n}((\bar X+ \bar Y)^2-(\mu_x+\mu_y)^2)\to_d N(0,k^2)$, Find confidence interval.

Assume $(X_i,Y_i)$ is an i.i.d sequence of random vectors with $E(X_1)=\mu_X>0, E(Y_1=\mu_Y>0,Var(X_1)=\sigma_{X}^2,Var(Y_1)=\sigma_{Y}^2, Cov(X_1,Y_1)=\sigma_{X,Y}^2$ with all covariances ...
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Using s to estimate $\sigma$ when finding the sample size in confidence interval questions

I am trying to learn sample confidence interval for $\mu$ , in this topic , there is a subtopic which is finding the sample size. I know that if $\sigma$ is given (standard deviation of population) , ...
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Is 95% confidance interval always included in minimum-maximum interval? [closed]

Is 95% confidance interval always included in minimum-maximum interval ? I have a set of altitudes (Z). I calculate the mean Z and the 95% confidance interval. Will the bounds of this confidance ...
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How do I calculate a 95% confidence interval for n=3?

Standard deviation (s): 23.888 $x_1=43.84$ $x_2=86.03$ $x_3=45.5$ $x_{avg}=58.45$ I need to create a 95% confidence interval. Should I use a t distribution? If so, my calculation for a 95% confidence ...
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Intuition and Proof of Confidence Interval for Sample Variation

Let $X_1, ..., X_n$ be a random sample from a normal distribution with an unknown $\mu$ and unknown $\sigma$. The sample mean of this sample is $\bar{X}$. The following graph shows the density curve ...
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Finding the interval of convergence using the ratio test

I found that the Taylor series for $h(t)=\ln(t)$ centered at $t_0=1$ is $$\sum^\infty_{n=0}(-1)^n\frac{1}{n+1}(t-1)^{n+1}$$ Now I want to find the interval of confidence for this series, so I use the ...
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Why when finding with a significance level $\alpha$ a confidence interval for the mean, the $\alpha$ appears as $\alpha/2$?

I just can't get some things in the reasoning that gives us the confidence interval for the mean. I am going to state the argumentation and let me know where am I wrong. Given a sample $X_1,...X_n$ ...
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Interval confidence of a poisson distribution for different sample sizes

Assume I have an infinite sequence of natural numbers $n_1, \ldots$ where each number comes from a poisson distribution $P(k)$ with $k$ unknown. I want to estimate $k$: I take the first $N$ numbers $...
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Why is my method not applicable, one-sided confidence interval

I have "solved" a question in mathematical statistics, but unfortunately received an incorrect answer and I would appreciate any help in understanding why my method did not work in this case....
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Pivotal method to verify a confidence interval for $\theta$

Let $X$ be a random variable with p.d.f.: $$f(X|\theta) = \frac{e^{x-\theta}}{(1+e^{x-\theta})^2}$$ where $-\infty<x<\infty$ and $-\infty<\theta<\infty$ Use the pivotal method to verify ...
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Are there unusual one-sided confidence intervals?

Suppose $X$ is a continuous random variable with known distribution. I am able to observe one value of $Y := X+\theta$ where $\theta$ is an unknown parameter, and I want to build a one-sided ...
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Usage of $\hat p$ and $p_0$ in confidence intervals/hypothesis testing

Cheers, I have a question about the usage of $\hat p$ and $p_0$ when talking about the confidence intervals of proportion problems. My professor and my textbook both state that for Bernoulli trials (...
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Why is standard deviation calculated differently for finding Z scores and confidence intervals?

Suppose that as a personnel director, you want to test the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. ...
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Two-sided confidence interval for the sample for the mean length

A sample of $n \in \mathbb{N}$ screws is taken from a large batch of screws that are produced. The length of a bolt is approximated as normally distributed with variance $4$ and the lengths of the ...
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Confidence in Mean Time-til-absorption for Absorbing Markov Model

I am modeling the failure rate of a certain piece of equipment as a Markov Model that has both "up" and "down" states, where the "down" states are absorbing. I have ...
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True confidence interval for the parameter of a known distribution

I have $X$ which has values in ${0, 1, 2}$. And i'd like to know if i could compute a 95% confidence interval for the mean of n samples from this distribution. I know $P(X=0), P(X=1)$ and $P(X=2)$. I ...
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Deriving confidence interval for Bernoulli proportion

I want to derive from scratch $(1 - \alpha)$ CI for Bernouli proportion. My result differs from the well known result I'm not sure why: My derivation We are looking for such $l, r$ that $P(\theta \in [...
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Confidence interval for the variance of exponential distribution?

I know that if $X \sim \operatorname{Exp}(\theta)$, and $Y=\theta X \implies Y \sim \operatorname{Exp}(1)$. I want a confidence interval for $E(X)= \frac{1}{\theta}$ and $\operatorname{Var}(X)= \frac{...
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Credible interval for gamma prior

Let's cosnider $X_1, X_2,...,X_n \sim \textrm{Poisson}(\lambda)$ i.i.d, and $\textrm{Gamma}(\alpha, \beta)$ as prior. Then posterior distribuion is $\textrm{Gamma}(\sum_i{X_i} + \alpha, n + \beta)$. ...
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When do we use p-value or confidence intervals

Let's take a simple example, where we are given the daily average costs of a student 1.2 0.8 0.6 1.1 1.2 0.9 1.5 0.9 1.0. And we have to determine whether we can reject the hypothesis $H_0$: the mean ...
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Establish an interval estimate for the population mean

Suppose a sample of 50 is taken from the population with standard deviation 15 and that the sample mean is 100. Establish an interval estimate for the population mean that is a) 90% certain to include ...
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interval estimate formula of mean

What formula should I use to establish an interval estimate for the population mean, if I have unknown distribution, standard deviation of the population, sample size & mean?
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How can I use a taylor series variance approximation to construct a confidence interval?

I read that the variance of the ratio of counts from a multinomial distribution can be approximated as $$ var\left[\frac{X_1}{X_2}\right] \approx \frac{1}{N}\left(\frac{p_1}{p_2}\right)^2\left(\frac{1}...
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