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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Determining precise value using imprecise measurements.

Suppose an accurate scale reports weight rounded to the nearest unit. If it reports the weight of an object as "6", all I know is that it could actually weigh anywhere between 5.5 and 6.5. But I need ...
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Do you know an interesting problem regarding confidence intervals?

I'm currently studying statistics and I was wondering if any of you have a good and interesting problem regarding confidence intervals (statistics) and its solution that you'd like to share with me. ...
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Determine the probability that any person is against government decisions

In one study, out of $80$ respondents, $23$ were against going to a concert. Determine if a person is against decisions to go to the concert if the confidence interval is 95%. Help me please. Thanks ...
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What sample size is needed to ensure a majority?

The results of a sample of voters showed that $55\%$ voted for a given candidate. It was determined that at a confidence level of $0.95$ that candidate would be the winner (i.e. would receive the ...
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Finding interval $I=[M_n,M_n+c]$such that $ P[I\ni \theta]\to .95$ as $n \to\infty $

With $X_i$ i.i.d. uniform random variables in $[0,\theta]$, for some $\theta>0$ and $M_n =\max(X_i)$ I am trying to find an interval $I$ of the form $I=[M_n,M_n+c]$, that does not depend on $\...
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the confidence intervals of an exponential distribution

Consider the random sample $X_1 \cdots X_n$ from a distribution with pdf $$f(x; \theta) = \dfrac {x}{2\theta^2}$$ if $0<x<2 \theta$. The most likelihood estimator of theta is given by: $\theta_{...
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Confidence Interval based on limiting distribution of the Maximum Likelihood Estimator Sigma and Sample Variance

Another statistics question I need some help with. Now, I am not too familiar with limiting distributions (had my course on it over a year ago). I have tried some things myself and ran some ...
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Show that the Value-at-Risk (VaR) at confidence level $c = 95\%$ is $\text{VaR} = 1$. [closed]

Suppose an asset has return $K = 1$ with probability $\frac{1}{2}$ and $K = −1$ with probabilty $\frac{1}{2}$. Show that the Value-at-Risk (VaR) at confidence level $c = 95\%$ is $\text{VaR} = 1$.
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conservative two-sided equal-tailed confidence interval

Consider a random sample $X_1,...,X_n$ from a Bernoulli distribution with unknown parameter $p$ that describes the probability that $X_i$ is equal to $1$. The maximum likelihood ($ML$) estimator for $...
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problem about confidence interval

first I have a random variable X, which has N samples(realizations): x1, x2, ...xN; and then another random variable Y, which is the average of some other k variables(assume they are from the same ...
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Why can we use empirical standard deviation when computing mean confidence intervals

I'm reviewing some basic statistics, and I'm asking myself questions on things I used to take for granted when I first saw them years ago. I'm going to state things as I understand them, so there ...
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Confidence interval and standard error for specific statistic. [closed]

What is the way to compute confidence interval for a specific statistic. Is it always just $\bar{x} +/- se$ or something else? Also if I want to calculate the $se$ for a specific statistic ( not using ...
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How to estimate the confidence interval for a proportion of two estimated values? [migrated]

I have estimated two values and confidence intervals (CIs), how can I estimate the CI of its ratio? For example, a new method has been developed claiming to reduce SO2 emission in power plants by 20%....
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Confidence Interval. Bernoulli Distribution

I am reviewing the construction of confidence intervals for a random sample with Bernoulli distribution. The book uses the statistics of the central limit theorem that distributes $N(0,1)$ to estimate ...
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The duration of a 75-watt lamp has a stdev of 25 hours. A sample of 35 has average 1014 hours. Find a confidence interval of 98%

It it known that the duration, in hours, of a 75-watt lamp has a standard deviation of 25 hours. A random sample of 35 lamps has an average time of life of 1014 hours. a) Build a confidence ...
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How to Find a Sample Size Given a Confidence Interval and Width?

I only found cookbook and don`t know why this work Step 1: Find z a/2 by dividing the confidence interval by two, and looking that area up in the z-table: Step 2: Multiply Step 1 by the standard ...
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Approximate Confidence Interval, based on limiting distribution

This is my first question here, so excuse the poor formatting. I am working on a statistics assignment and they asked me the following question: $Xi i.i.d. ∼ N(µ, σ^2)$ Give a 100(1 − α)% two-sided ...
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How to compute confidence interval for variance with unknown mean from a normal $(a,\sigma ^2)$ sample?

When mean is known, we note that $\frac{\bar{(X-a)^2}n}{\sigma ^2}$ has a Chi-squared distribution with $n$ degrees of freedom. However, what to do when $a$ is unknown? I can't just substitute sample ...
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Identifying Problems with a Statistical Analysis Plan

So, I've been provided with a supposedly flawed analysis plan that I have to assess and find issues with. We're considering a trial of a certain drug, used to treat high blood pressure. Patients will ...
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What can be inferred from low sample size binomial statistics?

Consider the binomial (or Bernoulli) process with probability $p$ of passing or not passing a certain test. Say you have a limited sample size $n$ (because the test is hard to run, or the population ...
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Statistical significance of a classifier's precision

Suppose I have a sample of $n$ data points (examples) that have to be classified into one of two classes (positive and negative). Let's say I have a method to generate a score for each example. The ...
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Statistical significance? [closed]

I have a company's satisfaction score for two different years. Past studies have shown that when a satisfaction score has increased over the past year and is also higher than the national average ($75....
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Minimum length of confidence interval for $\theta$ when $X_1,\ldots,X_n\sim\operatorname{Beta}( \theta,1)$

I am asked to find the minimum length confidence interval for a Beta distribution with parameters $a=\theta$ and $b=1$ and probability $1-\alpha$. I have found that $-2\theta \sum_i \log(X_i)$ has ...
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Determining whether to apply unequal or equal variance case for finding the confidence interval for difference between two means

The following data represents the total time taken, in days, to deliver books ordered through two online sellers. The sample delivery times are collected and reveal the following information: Find ...
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Confidence interval for categorical data

I have the following data: ND NI SC PI PA Green 27 3 9 4 7 Blue 24 14 14 6 8 I want to do the following: Write the multinomial model ...
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Calculating the margin of error for a one-sample confidence interval for proportions

I recently saw a question that gave a 95% confidence interval for one-sample t-interval for means as (5.6, 7) and was asked to find the point estimate and the margin of error. The solution said that ...
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To estimate mean of two populations.

Suppose that you are given with two set of sample $F = \{ x_1, x_2,\ldots,x_n\}$ and $M=\{y_1, y_2,\ldots, y_m\}$ ($n=100,$ $m=100$) $x_i$ (or $y_i$) represents the height of a female (or male) ...
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Exact $(1-\alpha)100\%$ confidence interval for $\theta$ in $\operatorname{Gamma}(4, \frac{1}{\theta})$

I am trying to solve the following problem for exercise purposes: Let $ X_1, ..., X_n $ be independent and identically distributed random variables with the probability density function: $$ ...
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25 views

Calculate confidence level from a given confidence interval

I'm fairly comfortable calculating the confidence interval. But now I'm seeing a problem where I'm giving a confidence interval $CI(27.6621, 30.3379)$ and I'm requested to calculate the confidence ...
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Construct an appropriate pivot using $Y(n)$ and $\theta$.

$Y_1,...,Y_n$ denote a random sample of size $n \geq 4$ from distribution with density function: $f(y)= 4 y^3/\theta^4$, 0 involving an unknown parameter $\theta$, ($\theta>0$). $Y(n)= \max \{ Y_1,...
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Will a higher bandwidth go thinner confidence intervals on the Nadaraya-Watson estimtor?

The confidence interval for Nadaraya-Watson estimator is: $CI_{1-\alpha}{m(x)} = \hat{m_h} \pm z_{1-\frac{a}{2}} \sqrt{\frac{||K||^2_2\sigma^2}{nh\hat{f}_h(x)}} $ I was just wondering if we have a ...
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How do I interpret coverage probability?

I would love to have feedback on my interpretation on coverage probability. I am studying how changing error distributions affects the coverage probability of predicting new observations. The ...
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Statistics question on errors and probability with Coronavirus

Recently a survey in California suggested that considerably more people have been exposed to COVID-19 than previously thought. However, the testing equipment used apparently has a 1.7% false positive ...
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90th percentile of a randomly included sample

I have a set, $S_1$, of $n$ numbers, $a_1 < a_2 < ... < a_n$. I have a process whereby each number has a $p=50\%$ chance of inclusion in the next set, $S_2$. My final value is then min($S_2$...
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Confidence interval problem for when chi-squared distribution is the pivotal that is used

I had a problem where a $1-\epsilon$ confidence interval for some parameter (namely, the variance of a normal distribution with known mean) needed to be built. The pivotal (independent of parameter) ...
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Find exact confidence interval for uniform distribution

In my homework I have $X_1,...,X_n$ which are all uniformly distributed on $(0,\theta)$ I have concluded that the $MLE=\hat{\theta}=max(X_1,...,X_n)$ because: $L_x(\theta)= \frac{1}{\theta^n}$ so ...
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Probability distribution of the infinity norm of a Gaussian vector

Let $N \geq 1$ be an integer. Let $X$ be a standard $\mathbb{R}^N$ Gaussian vector (all components are $\mathcal{N}(0, 1)$ and i. i. d.). Let $A \in \mathcal{M}_N(\mathbb{R})$ be a deterministic ...
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Confusion regarding Confidence Intervals

Suppose we have $2$ population parameters $p_1$ and $p_2$, such that the $90$% ( symmetric ) confidence intervals for $p_1$ and $p_2$ are given by $ (0.411, 0.498) $ and $(0.473, 0.567)$ respectively. ...
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Hoeffding Inequality for Confidence Interval (In Phase Retrieval)

I am reading a paper which is related to phase retrieval theory. In Eq. $3.13$ (Page $14$), the authors state that " Setting $T_n = \sqrt{2\beta \log n}$, then a simple application of Hoeffding’s ...
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Algebra with variables having confidence intervalls

In my work I have come across a situation where I need to derive values from values having confidence intervalls. I want to calculate the confidence intervals for my derived values. I am quite rusty ...
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Misconception of confidence intervals.

Let A and B be events such that $P(A)=P(B)=0.95$. I know that $$P(A\cup B) = P(A)+P(B)-P(A\cap B)$$ and hence $P(A \cap B) \geq 0.9$ How can I use the information above to disprove the misconception ...
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Sample size in Confidence Intervals

In repeating confidence interval experiments, are we allowed to take samples of different size every time? Because a confidence interval of 95% means that if the sampling process is repeated infinite ...
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Function of a confidence interval

If we have a confidence interval for a given parameter $\theta$ given as $[\theta_l, \theta_u]$ ($l$ is for lower and $u$ is for upper) at confidence level $\gamma$, and we have a monotone (Borel) ...
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How come we can use the $Z$ table (or $t$ table) when finding confidence intervals?

I am currently learning about confidence intervals and the following question keeps bugging me. How come that when we find a confidence interval I can just use the $Z$ or $t$ table? Let's say that I ...
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Monte carlo simulation confidence interval coverage

$Y_{i}=\beta x_{i}+\epsilon_{i}$ where $ \epsilon_{i} \sim N(0,\tau)\ x_i$ are covariates and the profile likelihood of $ \beta $ is $\ l_p(\beta) = \ l_p(\beta,\hat\tau(\beta)) =-\frac{n}{2}log\...
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Standard deviation, standard error and confidence interval difference explanation

I write some code to generate random weights (I define low as 50Kg and high as 100Kg) of males, then generate 100 samples containing 100 measurements (weights per sample) i.e. 100 samples and each ...
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Confidence interval for binomial success parameter approximated by normal distribution

Given is a group of 82 people, 7 of those are at risk for some disorder. The overall goal is to model the proportion p of people with the disorder - and it is clear that this can be modeled by a ...
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Are the means of the two data sets statistically different?

A 95% confidence interval for the difference of means for two data sets A and B was constructed. The confidence interval is (-2.1,.5). Are the means of the two data sets statistically different? I ...
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Find the confidence interval of square of the mean $\mu^2$

Suppose I have a population, I know the variance $\sigma^2$, but I don't know the mean $\mu$. How to find the confidence interval for the square of the mean $\mu^2$ ? It's well known that the CI of ...
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Asymptotic point wise confidence band using empirical dist func

The question is as follows: Let $X_1, X_2, . . . , X_n$ be an i.i.d. sample and $\hat F_n(x)$ the empirical distribution function. Assume that $\hat F_n(x)$ is normally distributed and each $x$ is ...

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