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.
769
questions
0
votes
0
answers
32
views
Interpretation of a confidence interval
The textbook, Introduction to Probability by Anderson, Seppalainen, Valko, reads, on page 151,
Another task is to find the confidence interval around $\hat{p}$ that captures the true $p$, with a ...
- 1,314
0
votes
0
answers
16
views
Probability inequality for a statistic with lower bound constraint, where McDiarmid's inequality is used in proof
I'd like to prove a probability inequality involving a statistic and a random variable as in the following statement. My questions are two-fold:
Is the following proof using McDiarmid's inequality ...
- 319
-1
votes
1
answer
26
views
Confidence Interval and Hypothesis Test Disagree?
I created a problem where I am having my students complete a hypothesis test and confidence interval for proportions to highlight the connection between the two. However, in doing the problem I am ...
0
votes
0
answers
15
views
Which is the variance of a variable which is the linear sum of normally distributed random variables?
I have a random variable $x$ which is normally distributed with expceted value $\bar{x}$ and variance $\sigma$:
$$x\sim N(\bar{x},\sigma)$$
As you know, i can consider $\bar{x}$ a random variable that,...
0
votes
1
answer
17
views
A random sample of $50$ machines obtained that its average life is $\bar{x}=70$ months with a variance of $s^2=49$. Confidence interval for variance.
I need help with the part b) of this exercise.
A random sample of $50$ machines obtained that its average life is $\bar{x}=70$ months with a variance of $s^2=49$. Assume that they are normally ...
- 1,081
0
votes
2
answers
32
views
MLE, a book: Statistical modelling and inference using likelihood
I have a problem with a likelihood interval;
Here on the page $74$ in the example $4.1$ they write:
IN CASE the link doesn't download Please copy the link and paste it to your web browser.
The ...
- 3,911
0
votes
0
answers
24
views
How many observations are required to achieve a particular confidence interval for the mean?
Suppose that the standard deviation of an observation in a given population is
known to be $\sigma = 15$.
How many observations in a sample is needed to estimate $\overline{x}$ (the mean of the sample)...
- 23
0
votes
0
answers
25
views
Confidence Interval - Meaning and Interpretations
Could someone please help me understand what exactly "Confidence" in confidence interval actual means?
Does it mean, that, (on average or exactly?), we can say, that (for a 90% confidence ...
- 21
0
votes
0
answers
23
views
Inverting UMP tests to get Confidence Interval
Suppose X is an exponential random variable with mean $\mu$, i.e., $X ∼ \mu Exp(1)$.
Invert appropriate UMP tests of $H_0: \mu = \mu_0$ to find a 90% confidence interval for $\mu$ of the form $[∗,\...
- 85
0
votes
0
answers
10
views
Using a proportion test to see if the mean of two data sets are equal or not?
We have a normally distributed data set of size $n = 188$, with mean $\mu = 2913.29$ and standard deviation $\sigma = 697.5$.
The data set is split into two sets A and B of size $n_A$ = 95 and $n_B$ = ...
- 77
1
vote
1
answer
37
views
How to get a confidence interval for $n$ of binomial distribution with known $p$
Suppose $X \sim \mathrm{Bin}(n,p)$ is a binomial random variable with success probability $p$ and $n$ trials, where $p$ is known and $n$ is unknown.
Now only one observation of $X$ is given and I want ...
- 21
0
votes
0
answers
13
views
How to obtain confidence Intervals for minimum order statiscs
After computing the minimum order statics for iid normally distributed data as follow :
P(X1< x ) = 1-P(X1 > x )
= 1-P(X1 > x, X2 > x, ... ,Xn > x )
= 1-P(X1 > x) P( X2 > x) ... ...
- 25
0
votes
0
answers
19
views
The equality of variance test
I am facing a strange problem,
Imagine that you want to test whether the standard deviations of two populations are equal. Let us call the standard deviations of the first population $\sigma_1$ and ...
- 140
0
votes
0
answers
33
views
2 different form of confidence interval for variance estimator
For iid data {$X_n$} with mean $\mu$, variance $\sigma^2$, finite 4th order moment $\mu_4 = E(X_1-\mu)^4$. Let the estimator of $\sigma^2$ be $\sigma_n^2 =n^{-1}\sum_{t=1}^nX_t^2-(\bar X_n)^2,$
where $...
- 282
0
votes
0
answers
21
views
CI for Bernoulli parameter
$X_1, ..., X_n \in B(\theta)$. Find CI for $\theta$ with given $\alpha$
We know that $E[B]=\theta, D(X)=\theta(1-\theta)$
$\frac{\bar{X}-E[X]}{\delta/\sqrt{n}} \in N(0,1)=\frac{\bar{X}-\theta}{\theta(...
0
votes
1
answer
48
views
Deriving CI for the mean: Wolfram Alpha giving different answer for equation and inequality [closed]
As a simple example, say I'm trying to derive the upper bound of a confidence interval for the mean. That is, I'm trying to solve for $\mu$ in
$$ \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}} < Z_{\...
1
vote
1
answer
53
views
two samples t-test and ANOVA
Suppose we want to know whether the effectiveness of two methods, methods A and methods B, is comparable. Suppose that the (scientific) criterion for determining whether the methods are identical is ...
- 140
0
votes
0
answers
34
views
Finding CLT for Gamma Distribution
It seems like a lot of examples for estimating the confidence intervals of a Gamma distribution, the parameter estimation involve one variable being known. I was wondering how to find a confidence ...
- 37
1
vote
0
answers
21
views
Confidence bounds on a mean of ratios
I am collecting $i=1\ldots N$ measurements of $x_i,y_i$ and computing the average ratio:
$r=\frac{1}{N}\sum_{i=1}^N \frac{x_i}{y_i}$. The mean ratio is supposed to be close to 0.5, if that matters and ...
- 43
0
votes
2
answers
65
views
Confidence interval for Binomial Distribution
I have recently heard that when constructing confidence intervals for a binomial distribution, with small probability of a success, and a large sample size it is best to use a Poisson distribution. I ...
- 115
1
vote
0
answers
27
views
Trial until first success. What is p?
Let's say I repeated some trial 90 times and got a success on my last attempt.
I guessing that such a confidence interval is different than if I had gotten 1 success at any point out of my 90 trials.
...
- 31
2
votes
1
answer
69
views
Construct a confidence interval for $\theta$
Let $X_{1}, \cdots, X_{n}$ be a random c.i.i.d sample such as, given $\theta$, $X_{1} \sim \mathcal{N}(0,\theta)$. Construct a confidence interval for $\theta$ using asymptotic results.
This question ...
- 31
0
votes
0
answers
25
views
Determining a suitable sample size for comparing two populations
I have a device to which a component is at EoL, and it has some sensitive sensors that necessitate tests to ensure equivalent accuracy with the new component. One proposed approach is to test a number ...
- 1
0
votes
0
answers
15
views
How to determine a confidence interval for the total number of trials given a number of successes
Given a binomial distribution with a known probability, is it possible to determine a confidence interval for the total number of trials that have occurred given a number of successes that have been ...
- 1
4
votes
1
answer
59
views
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 $...
- 5,343
0
votes
0
answers
20
views
How to calculate the total number of TVs in this problem?
Suppose there’s a small town of 2000 citizens in 1250 homes. Out of those we take sample from 60 homes were 100 people reside. Let $x_i$ be the number of people per house $i$, and $y_i$ the amount of ...
- 362
-1
votes
1
answer
28
views
Building Confidence Interval for Population Proportion [closed]
I'm having trouble understanding how the 95% confidence interval equation for population proportions gets simplified.
Here is the initial equation:
$$
Pr(\overline{X} - 2\hat{SE}(\overline{X}) ≤ p ≤ \...
- 3
2
votes
0
answers
33
views
Asymptotic Confidence Interval for ML-Estimate of Gamma-distribution
Assume I have a sample of $n \in \mathbb{N}$ data points $x_1,\ldots, x_n$, which are asummed to come from iid drawings of a Gamma-distribution. I may assume the shape paramter $k\in\mathbb{R}_+$ and ...
- 21
0
votes
0
answers
21
views
Relation between sample size and confidence interval in the Normal Distribution
I have been solving the next exercise:
"Consider a random variable X with Normal distribution of expected value μ and variance σ², with unknown values. Determine the random confidence interval ...
- 1
1
vote
1
answer
24
views
Confidence interval of 2 exponential random variables with different paramerts
So i have 2 independent samples, $X_1,X_2...,X_n$, where $X_i$~exp($\lambda$),
and $Y_1,Y_2...,Y_n$, where $Y_i$~exp($2\lambda$).I want to find a confidence interval for $\lambda$ with confidence ...
- 11
0
votes
1
answer
17
views
Confidence interval for a uniform distribution
I am reviewing the following example and am confused on a portion of it
Let X1,...,Xn be a random sample from the continuous uniform distribution on the interval from $0$ to $\theta$
Using the CDF ...
- 23
0
votes
0
answers
65
views
Confidence that a distribution is uniform
I run a distributed software making rounds.
Each round has a 35% probability of success.
I've been monitoring it for quite some time, and I have the intuition that the successes are happening in "...
- 9
9
votes
2
answers
245
views
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 ...
- 2,400
4
votes
2
answers
438
views
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 ...
- 2,400
0
votes
0
answers
21
views
MLE as a pivot for a sample of normally distributed random variables
Let $X_1, X_2, ..., X_n$ be independently distributed variables where $X_k \sim \mathcal{N}(k\mu, 1)$ for $k = 1, 2, ..., n$ and $\mu\in\mathbb{R}$ unknown.
I calculated that the MLE for $\mu$ is ...
- 137
3
votes
1
answer
123
views
Confidence Intervals of Markov Chains?
I understand that a Discrete Time Markov Chain (https://en.wikipedia.org/wiki/Markov_chain) is closely related to a Multinomial Distribution (https://en.wikipedia.org/wiki/Multinomial_distribution).
...
- 2,400
0
votes
0
answers
11
views
How to estimate confidence interval for each found parameters by solving minimization problem?
I solve minimization problem (non-linear regression) with N initial guesses (as usually, N = 3 or 4). And I want to calculate confidence itnerval for each N parameter. How can I do this, if I hav only ...
- 1
0
votes
1
answer
25
views
How to determine the confidence interval in one tailed test?
Suppose a question is given to find the confidence interval at 95% confidence level.
If it is a one tailed test, how to calculate the confidence interval?
The critical value will be 1.645 or 1.96 in ...
0
votes
1
answer
39
views
How to find confidence intervals when not a normal population
What I know:
r.v. $X_1, \dots, X_n \sim i.i.d \ Po(\lambda)$.
From the central limit theorem, it follows that
$$
\sqrt{n}(\bar{X} - \lambda) \xrightarrow{L} \mathcal{N}(0, \lambda).
$$
The following ...
- 313
0
votes
1
answer
28
views
A calculating problem concerning the C.I. of a random variable with known pdf.
Define $f(x)={N \choose{N \alpha}}x^{N \alpha}(1-x)^{N-N\alpha}$, where $f$ is defined on $(0,1)$. We further define $p(x)=\frac{f(x)}{\int_{0}^{1}f(u)du}$. Not hard to see $p(x)$ is actually a pdf on ...
- 121
0
votes
0
answers
21
views
estimating confidence interval and covariance with singular design matrix
I would like to obtain the confidence interval for a linear system $Gm=d$, except that the design matrix $G$ is singular. To solve for the model, I used the Moore-Penrose inverse. To estimate the ...
- 3
0
votes
1
answer
34
views
How many Monte Carlo simulations must I run to get a $95\%$ confidence interval for some error E
Suppose I want to use Monte Carlo to compute some probability $p$. A single MC simulation will run for $R$ iterations and calculate $p$ as the fraction of 'successes' (each iteration gives failure and ...
- 1
0
votes
0
answers
32
views
Problem on probability distributions, hypothesis testing, confidence intervals
i am doing an assignment with the following questions and I would like to ask for help since i am a bit confused...
A geodesic drone is using LIDAR, a laser system for measuring distances, to measure ...
- 1
0
votes
1
answer
41
views
Minimum sample size for achieving the desired margin of error
When trying to find a confidence interval for an unknown population mean, we can achieve a desired margin of error by ensuring that our sample size is large enough. The textbook I'm using gives the ...
user637978
0
votes
1
answer
26
views
Confidence intervals and probability [closed]
If I am given an interval, say .51-1.49, of 100 random variables, distributed N(1,9). How would I calculate the probability that the mean of the 100 variables is in said interval. I know how I would ...
- 27
0
votes
0
answers
28
views
Width of Clopper-Pearson interval reaches maximum when $x=N/2$?
Suppose we perform $N$ trials and find $x$ successes, then the $1-\alpha$ Clopper-Pearson confidence interval is $[B(\frac{\alpha}{2};x,N-x+1),B(1-\frac{\alpha}{2};x+1,N-x)]\ (0<\alpha<1,0<x&...
- 1
0
votes
0
answers
28
views
Using Sample Variance to Estimate Population Variance and Sample Size
I am a freshman to statistics, and recently faced a problem with which I need help clarifying.
Suppose there are 130 boxes of similar sizes, each containing a random number of chocolate cookies. 15 ...
- 1
0
votes
1
answer
26
views
Estimating a highest probability density (HPD) interval
QUESTION: (I translated it into English so you might find somewhat awkward..)
A company wants to get information about the lifespan of certain machines. Let's say that the lifespan of the machines ...
- 109
1
vote
0
answers
399
views
Nonasymptotic confidence level of Bernoulli R.V.
Let $𝑋_1,…,𝑋_𝑛$ be i.i.d. Bernoulli random variables with some unknown parameter $𝑝∈(0,1)$. Then which of the following is/are valid confidence interval(s) for $𝑝$ with nonasymptotic confidence ...
- 207
0
votes
0
answers
19
views
How to remove $\theta$ from the exponent? (confidence interval)
I'm solving the following mathematical statistics problem
Find a 95% confidence interval for $\theta$ where
$X_1,...,X_n\sim Beta(\theta,1)$
I started by finding a function in which $\theta$ isn't a ...
- 1