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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$95\%$ confidence interval with unknown standard deviation

The Question We have the results for the heart beats per minute of a sample of $15$ people: $$54, 63, 58, 72, 49, 92, 70, 73, 69, 104, 48, 66, 80, 64, 77$$ Give a $95\%$ confidence interval for the ...
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How to find the degrees of freedom of the confidence interval for difference between means

You have a sample of 9 men and a sample of 8 women. Are the degrees of freedom for the t value in your confidence interval on the difference between means 16? How is this obtained?
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Confidence Set for Regression Coeffcients Construct [closed]

enter image description here Ask for help about the problem in Confidence Set for Regressioni Coefficients Construction
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Constructing a pivot

$[X \mid \theta] \sim Uniform([1, \theta])$. Recall that for $U \sim Uniform([a, b])$, it holds that $c_1 U + c_2 \sim Uniform([c_1 a + b, c_1 b + c_2])$. Suppose given $\theta$, that $X_1, \ldots, ...
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How to solve this Confidence Level Question?

A company has conducted a test on $400$ people. Out of $400$ people, $224$ people like chocolate. Find the percentage of people who like chocolate in the $99\%$ confidence interval. The problem is ...
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Z confidence interval

$W_1, \ldots, W_5$ be iid $Normal(3.1, 5.2^2)$. What is the probability that $\left(\frac{\bar W - 3.1}{\hat \sigma / \sqrt{5}} \geq 2.1\right)$? I am not sure how to start. Is this equivalent to the ...
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Why does $2$ appear in $95\%$ confidence intervals?

I was told that, in linear regression, the $95\%$ CI for $\beta$ is $$\beta \in \left(\hat\beta-2 \sigma,\hat\beta+2\sigma\right), \text{ where } \sigma = \text{standard error}(\hat\beta).$$ My ...
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Confidence Interval for difference in means. Am I doing something wrong?

I have to do the following question but I've tried it multiple times but my confidence interval doesn't match with the answer key. Am I doing something wrong? An economist believes that a typical ...
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Large-sample confidence interval for a population proportion problem

I have this problem and i don't understand a part of its solution. A genetic model suggests that 80% of the plants grown from a cross between two given strains of seeds will be dwarf variety. After ...
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How to estimate error in Hoeffding's inequality for a sample stream?

I want to compute the average of a very long sequence $X=\{x_0,x_1,....\}$, where $x_i \in [0,a]$, with $a \in \mathbb{N}_1$. For simplicity let's assume $x_i \in [0,1]$. Since values are constantly ...
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Find the level of confidence given that standard deviation is 12 and there is sample size of 16:

A confidence interval for the mean price shows a $9.30 margin of error. What was the level of confidence? A. 99.9% B. 99.8% C. 99.7% D. 99.6% E. 99.5% I know that z(a/2) (std dev)/(sqrt n)=9.3 z(a/2)=...
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How does confidence interval affect the effective sample size?

So, here is the formula for variance of the estimated mean. Where $\epsilon$ stand for chosen units within which the estimated mean should lie in. Then there is the formula for sample size n: Where ...
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Of every 100 research paper using hyp. testing with $\alpha = 0.95$, 5 of them prove the wrong result?

I want to test my understanding of hypothesis testing. Is my hypothesis true? Of every 100 research paper using hypothesis testing with $\alpha = 0.95$, 5 of them find the wrong result All I know is,...
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Estimate sample total if standard error is given and sample is stratified ( confidence level - 95%).

Here I found a formula, to estimate the population total with some given confidence interval: where $\hat{t}$ is estimate for population total and $\hat{V}$ is estimate for the variance. How could I ...
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Confidence Interval, why is this unusual?

A Normal population has a known mean of $\mu = 50$ and a known variance of $\sigma = \sqrt 2$. A random sample of size $n = 16$ is selected from this population and find that the sample mean is $x = ...
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How to compute required sample size $n$, so that $SE$ would be 4 points from the computed mean with 95% confidence level?

I need to calculate sample size $n$, so that $SE$ of the mean would be 4 points. There are $55$ units in total in the $I$ group. $N_2 = 80$ in the $II$ group and $N_3=65$ in the $III$ group. The ...
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Wilcoxon ranked sign test and the pseudomedian

I need help understanding the Wilcoxon signed rank test and the pseudomedian. More concisely about the confidence interval of the pseudomedian. Say I have the observations $-2,3,6,10,20$. I will find ...
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How to find the confidence interval of a parameter estimator from an inverse Gaussian distribution?

This is homework! I have $n$ samples $x_1\dots x_n$ which are i.i.d taken from an Inverse Gaussian distribution $X\sim IG(\mu, \lambda)$. $f_X(x) = \sqrt{\frac{\lambda}{2\pi x^3}}\exp\bigr({\frac{-\...
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Confidence interval for likelihood-ratio test

I've been trying to solve the following problem: Let $X_1,\ldots,X_n$ be a sample of a random variable $X\sim\text{Exp}(\lambda)$, i.e., the exponential distribution with parameter $\lambda>0$. ...
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Z test and confidence interval

I am having doubts about this problem: You have a set of $n = 2$ radar guns and the speed limit is your favorite speed limit between $30$ and $130$. You use a Z-test at a significance level of $5\%$ ...
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Which Stats test does this scenario require? (Not Hypothesis Test)

100 people take a treatment. 100 people take placebo. 200 total people. 20 get sick. 18 took placebo. 2 took treatment. This does not look like Hypothesis Test with a known population proportion.
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Confidence interval on multivariate mean

I have a dataset of $N_{T_k}$ observations tied to four variables ($0\leq Y_i\leq 100, i \in \{1,2,3,4\})$ during a given time period ($T_k$). These variables are correlated. For most observations, ...
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Lower bound of confidence interval

Given 10 nail lengths with $$\bar{x} = 59.98 \quad,\quad\sigma = 0.26$$ Q: What is the shortest average nail length on basis of the sample probe? You can freely choose the level of significance. I ...
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Find an interval $I$ such that the probability that the parameter belongs to $I$ goes to a particular value.

Let $X_i$ be i.i.d. uniform random variables in $[0,\theta]$, for some $\theta>0$ and let $M_n =\max(X_i)$. We have to find an interval $I$ of the form $I=[M_n,M_n+c]$, that does not depend on $\...
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Poisson or binomial confidence level for large samples and low observation rate, and low true value of p

Consider a more extreme case for a follow on to this question here about the Prussian soldier-face kicking example for Poisson random variables Relating the binomial probability distribution to the ...
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Is there evidence that means the sitting height is above 750?

Could anyone tell me how to solve the following problem? When designing a cinema, engineers consider the sitting eye height of the audience. a random sample of the eye sitting height of 20 men has ...
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Approximate confidence interval for binomial distribution, not quite right

My exercise goes as follows: Let Y~B($N,p)$ be a random variable. For big N, determine an approximate confidence interval with confidence level $1-\alpha$ for the parameter $p$. I got the following ...
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hypothesis test using a confidence interval

i need to test the hypothesis that a proportion is greater than 0.93 using a confidence interval. i cannot figure out how to do this. the basic information that i think i need to use is below. the ...
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How to find critical point of t distribution when 1-α can't be determine directly from the table.

I was following a tutorial on Bonferroni's method. As a challenge I was ask to Compute the Bonferroni simultaneous confidence interval For a 95 % overall confidence coefficient using the Bonferroni ...
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The confidence interval is $[2.663; 2.937]$?

The concentration of calcium in the blood for a given population follows a normal law of mean μ = 2.8 mmol / L and standard deviation σ = 0.7 mmol / L. If we take a sample of 100 people, what is the ...
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Confidence interval for a Sum approximated with the central limit theorem

According to the central limit theorem: $S\approx N(n*\mu, n*\sigma^2)$ Normally, a confidence interval is calculated according to: $(μ \pm 1.96*\sqrt{\frac{\sigma^2}{n}})$ Do you divide $\sigma^2$ by ...
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Do I have to use T-Student to calculate this confidence interval?

Given a distribution $X~Pois(λ)$ with unknown parameter $λ=E(X)$ with $14$ random observations given from this distribution. How do I calculate a confidence Interval at $95\%$ for $λ?$ Do I have to ...
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Finding the probability using a time interval

The game 'Go' s an abstract strategy board game for two players in which the aim is to surround more territory than the opponent. The time for player $1$ to place a stone follows Exponential ...
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How to find the confidence interval for a prediction using a multiple linear regression model?

I have a dataset of 100 observations and 5 numerical variables ($x_1, x_2, ..., x_{5}$) which I used to generate a linear model which to predict some value $y$. I plugged in some values into the model ...
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Can we derive a formula to calculate pointwise confidence intervals for empirical distribution functions via subsampling?

I am trying to derive a formula to calculate pointwise confidence intervals for empirical distribution functions using Hartigan's subsampling approach. The main idea is similar to bootstrapping, but ...
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Statistical significance in users research experiment

If a company sends out 100 emails to a user. These emails are opened at different hours of the day. During some of the hours, the user is more likely to open the mail. I want to understand how many ...
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How to determine the 90% confidence interval for the population proportion of students who found the math test easy?

Out of $120$ students, $85%$ found the math test easy. How to determine the $90%$ confidence interval for the population proportion of students who found the math test easy? I am not sure about my ...
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How to determine confidence interval using central limit theorem?

How to determine the confidence interval for the unknown theta parameter of the Uniform($[0, \theta]$)-distribution using the central limit theorem, considering that the significance level is given ...
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Monte Carlo simulation: confidence interval of the ratio of two beta distributions

I am trying to estimate the accuracy of a set of Monte Carlo simulation where my result is \begin{equation} C=1-\frac{P_X(1)}{(P_Y(1))^2} \end{equation} and $X$ and $Y$ are the results of two separate ...
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How many connections are needed for an equal distribution of connections per thread

I have a load balancer with $X$ threads. I have $Y$ servers that will each open a pool of connections to the load balancer. Each connection is assigned to a single thread at random. Currently, the ...
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Confidence interval for exponential distribution with MLE

Assume that $Y_1, Y_2,\dots, Y_n$ are iid samples from the exponential distribution with the density function $f_{\theta}(y) = \frac{1}{\theta}e^{\frac{-y}{\theta}}$. Assume that $\hat{\theta}$ is the ...
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Confidence interval of population standard deviation using measurements with uncertainties

Let's say I have a sample of N measurements which I use to calculate the standard deviation of said sample: $s=\sqrt{\frac{\sum{(x-\bar{x})^2}}{N-1}}$ I can use this value to place a 68% confidence ...
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Confidence interval of success of population given binomial distributed sample

I am working on an assignment in Statistics about Corona-tests. I am given that $n=56,276$ is the number of observations (people tested) of the pouplation, $N=5,825,337$ and $x=991$ is the number off ...
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Find the required sample size for the investigator.

An investigator interested in estimating a population mean wants to be 95% certain that the length of the confidence interval does not exceed 4. Find the required sample size for his study if the ...
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Using confidence interval versus using Hypothesis testing for inference

When is it more appropriate to use a confidence interval and more appropriate to use hypothesis testing when one wants to make inferences from a sample to a population? I think that is what confuses ...
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Interpreting Multiple Confidence Intervals with the Same Sample Size and Level of Significance

Let $\alpha=0.05$ be the level of significance. Let $N$ be the sample size of a valid random sample from the underlying population. Let $\mu$ be the population parameter to be approximated, $\bar{x}$ ...
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Confidence interval for $n$ of $X$~Bin$(n,0.3)$

I want to calculate a 90% confidence interval for $n$ (from $X$~Bin$(n,0.3)$) where the number of successes is known ($x=12$). I have found the smallest $n$ so that $P(X \le 12)\le 0.05$ is 61 and the ...
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Construct a 98% confidence interval for the population proportion of defective toys.

A toy company would like to estimate the proportions of toys produced by their younger workers. In a random sample of 300 toys, they found that 75 were defective. Construct a 98% confidence interval ...
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Propagation of confidence intervals

I'm seeking for a reference. I'll start with an example. Suppose that $p$ was estimated from data to be $\hat{p}$ with a $\alpha$-CI of $I_p$. The estimate was performed by an unknown method. So, all ...
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Constructing confidence interval with limit result involving population parameter

$X\sim B(n,p), \hat{p}_{n} = \frac{X}{n}$ I am trying to invert the following statement to obtain a confidence interval. $P(-1.96 \leq \frac{\sqrt n(\hat{p}_n-p)}{\sqrt{p(1-p)}} \leq 1.96)=0.95$ Not ...

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