# What is Confidence interval: Definition and 101 Discussions

In statistics, a confidence interval (CI) is a type of estimate computed from the statistics of the observed data. This gives a range of values for an unknown parameter (for example, a population mean). The interval has an associated confidence level that gives the probability with which an estimated interval will contain the true value of the parameter. The confidence level is chosen by the investigator. For a given estimation in a given sample, using a higher confidence level generates a wider (i.e., less precise) confidence interval. In general terms, a confidence interval for an unknown parameter is based on sampling the distribution of a corresponding estimator.This means that the confidence level represents the theoretical long-run frequency (i.e., the proportion) of confidence intervals that contain the true value of the unknown population parameter. In other words, 90% of confidence intervals computed at the 90% confidence level contain the parameter, 95% of confidence intervals computed at the 95% confidence level contain the parameter, 99% of confidence intervals computed at the 99% confidence level contain the parameter, etc.The confidence level is designated before examining the data. Most commonly, a 95% confidence level is used. However, other confidence levels, such as 90% or 99%, are sometimes used.
Factors affecting the width of the confidence interval include the size of the sample, the confidence level, and the variability in the sample. A larger sample will tend to produce a better estimate of the population parameter, when all other factors are equal. A higher confidence level will tend to produce a broader confidence interval.
Many confidence intervals are of the form

(
t

c

σ

T

,
t
+
c

σ

T

)

{\displaystyle (t-c\sigma _{T},t+c\sigma _{T})}
, where

t

{\displaystyle t}
is the realization of the dataset, c is a constant and

σ

T

{\displaystyle \sigma _{T}}
is the standard deviation of the dataset.Another way to express the form of confidence interval is a set of two parameters: (point estimate – error bound, point estimate + error bound), or symbolically expressed as (–EBM, +EBM), where (point estimate) serves as an estimate for m (the population mean) and EBM is the error bound for a population mean.The margin of error (EBM) depends on the confidence level.A rigorous general definition:
Suppose a dataset

x

1

,

,

x

n

{\displaystyle x_{1},\ldots ,x_{n}}
is given, modeled as realization of random variables

X

1

,

,

X

n

{\displaystyle X_{1},\ldots ,X_{n}}
. Let

θ

{\displaystyle \theta }
be the parameter of interest, and

γ

{\displaystyle \gamma }
a number between 0 and 1. If there exist sample statistics

L

n

=
g
(

X

1

,

,

X

n

)

{\displaystyle L_{n}=g(X_{1},\ldots ,X_{n})}
and

U

n

=
h
(

X

1

,

,

X

n

)

{\displaystyle U_{n}=h(X_{1},\ldots ,X_{n})}
such that:

P
(

L

n

<
θ
<

U

n

)
=
γ

{\displaystyle P(L_{n}<\theta <U_{n})=\gamma }
for every value of

θ

{\displaystyle \theta }

then

(

l

n

,

u

n

)

{\displaystyle (l_{n},u_{n})}
, where

l

n

=
g
(

x

1

,

,

x

n

)

{\displaystyle l_{n}=g(x_{1},\ldots ,x_{n})}
and

u

n

=
h
(

x

1

,

,

x

n

)

{\displaystyle u_{n}=h(x_{1},\ldots ,x_{n})}
, is called a

γ
×
100
%

{\displaystyle \gamma \times 100\%}
confidence interval for

θ

{\displaystyle \theta }
. The number

γ

{\displaystyle \gamma }
is called the confidence level.

View More On Wikipedia.org
1. ### Find a 95% confidence interval for population mean

I am refreshing on this... I think there is a mistake on the circled part in red...right? not correct symbol for sample mean...This is the part that i need clarity on. The other steps to solution are pretty easy to follow...as long as one knows the t-formula and also the knowledge to...
2. ### Correct statement related to confidence interval

I think (e) is wrong because the proportion should be out of the whole population, not only in a certain class and option (a) to (d) looks like identical to me so I answered (a), (b), (c) and (d) but my answer is wrong. Where is my mistake? Thanks
3. ### MHB Show that C gives a confidence interval for θ

Hey! 😊 For $n \in \mathbb{N}$ we consider the discrete statistical product model $(X, (\mathbb{P}_{\theta})_{\theta\in \Theta})$ with $X=\mathbb{N}^n$, $\Theta=\mathbb{N}$ and $p_{\theta}(x_i)=\frac{1}{\theta}1_{\{1\leq x_i\leq \theta\}}$ forall $x_i\in \mathbb{N}$, $\theta\in \Theta$. Let...
4. ### MHB Determine the confidence interval

Hey! 😊 (a) In winter, the roads around Lake Macatawa are salted. To study the impact of this on Lake Macatawa, students took $32$ water samples from the western basin of the lake and Sodium content (in parts per million, ppm) determined. As a result, the students have receive the following...
5. ### MHB Confidence interval of difference

Hey! :giggle: From two populations of small and medium enterprises, we collect two random and independent samples of size $n_1 = 64$ and $n_2 = 36$ enterprises respectively. If we calculate for the companies of the first sample average sales of $9$ thousand euros and a variation of $4$, while...
6. ### How Do I Calculate Confidence Intervals for Two Means?

Here is the question I'm struggling with (Q1) : I just... I just don't understand what my first step is. Whats my barx1 and barx2? (bar x = mean, x1 = subscript 1) My thoughts on approaching this question : barX1 - barX2 `~ N(u1-u2, sd1^2/n1 + sd2^2/n2) Find Z value when p = 0.975, z = + or...
7. ### I Confidence interval on Standard deviation

I read that confidence interval on standard deviation can be found using chi-square distribution. If I have a sample size N=500, and sample standard deviation= 3 with mean=0, and I need a 95% confidence, I wasn't sure what to set for degrees of freedom in chi-square formula. Is the degrees of...
8. ### I Confidence Interval vs. One sided Hypothesis Test

I learned that the confidence interval and a two sided hypothesis test (test for difference) agree if both have the same significance level. Is that also true for one sided hypothesis tests? For instance, must a right-tailed Hypothesis test with alpha = 0.05 agree with the 95% confidence level...
9. ### MATLAB To what extent is the fit to experimental data good?

I have experimental spectrum in which y-axis is intensity values, and x-axis is frequency values. Int - array of experimental intensities (y-axis). w - array of frequencies (x-axis). I know the view of theoretical function that must describe the obtained spectrum. I explicitly set the function...
10. ### I Convergence of 2 sample means with 95% confidence

I tried to derive an equation for one sample mean to converge to another sample mean within a 95% confidence interval, but I know I am wrong. Can someone tell me what I did wrong, and what is the correct formula? Suppose: ##\hat{x_1},\hat{\sigma_1},N## are a sample mean, standard deviation...
11. ### B Chebyshev inequality, confidence intervals, etc

Hello. I am bewildered by so many different notions of probability distribution percentages, i.e. the proportion of values that lie within certain standard deviations from the mean. (1) There is a Chebyshev's inequality: - for any distribution with finite variance, the proportion of the...
12. ### MHB Confidence Interval, p-value and Critical Region

Hello, I have been struggling with this question for a few days now would appreciate being walked through it! :) Data are collected on the wingspans of adult robins. For N=20 birds, the sample mean and variance are given by \overline{x}=9.5cm and s^{2}=2.6^{2}cm^{2} a) If we assume that the...
13. ### I T-distribution, standard error - how to interpret

Hello. Math is not my strongest part, yet, so I will be very grateful for your help and for giving the answer in sort of "plain" English, meaning that it will be great if you don't go too deep into the rabbit hole using difficult terminology, as these distribution issues are truly difficult for...
14. ### MHB Construct the 99% confidence interval estimate

A simple random sample of 400 high school football players was taken in order to estimate the mean weight of all high school football players. The sample has a mean of 198 lbs. and a standard deviation 26 lbs. Construct the 99% confidence interval estimate of the mean weight of all high school...
15. ### MHB What is the formula for determining the length of a confidence interval?

Hey! :o A research institute wants to establish a confidence interval for the quota of working people in a city. Let $\hat{p}_n$ be the estimated quota, based on a sample of size $n$. It is assumed that $n> 30$. How can one determine the length of the confidence interval? Generally this is...
16. ### MHB Is my calculation of the confidence interval for $\mu$ correct? (Wondering)

Hey! :o The variable $X$ is normally distributed with unknown expected value $\mu$ and unknown variance $\sigma^2$. I want to determine the confidence interval for $\mu$ for $n=22; \ \overline{X}_n=7.2; \ S'=4; \ 1-\alpha=0.90$. Is $S'$ the standard deviation? (Wondering) In some notes...
17. ### Confidence Interval based on two sample tests

Homework Statement The following two samples are corrosion values for 20 untreated pipes (Xi) and 20 surface treated pipes (Yi) ∑Xi = 950, ∑Xi2 = 46344 and ∑Yi = 1092, ∑Yi2 = 62136 We want to examine if there is basis to claim that the surface treatment reduce corrosion. a) 1: Find a 95%...
18. ### MHB Confidence interval to stay awake while listening to the Dean

Confidence Interval or the lack of it... It is 1581 Anno Domini. At the Undergraduate School of UMUC, besides Assistant Academic Director of Mathematics and Statistics, I am also the Undergraduate School-appointed CPA, Coffee Pot Attendant. It is a very important office sponsored by the...
19. ### Confusion between z and t values

Homework Statement Hi, Alright so I have some confusion on when to use specific tests and the z vs t test. Given this example (not my homework) could someone please clarify. Alright say you have a random sample of size 200. You find the sample mean to be 10 and the sample standard deviation...
20. T

### What is the correct method for determining confidence intervals for the mean?

Homework Statement Both images are 2 consecutive pages of my notes . In this theory , i was told to use z-distribution when the sample size is large ( more than 30) and the standard deviation of the population , σ is known) . However , in the 2nd image , i was told to replace σ with s ( sample...
21. ### I How To Find 68% Confidence Interval After Running MCMC

This is not a homework problem so I'm asking it here. I just ran an MCMC and found the best fit parameters for a model of massive gravity. I now need to find the said 68% Confidence Interval interval for those parameters. I have never done anything like this before so I'm clueless where to begin...
22. ### I Confidence interval or Standard deviation?

I have a question regarding when to go for standard deviation σ or confidence interval ci. I am not sure of how to interpret ci. It seems to give me an option to choose the level of confidence, but I don't see why the the ci is divided by the root of the number of samples. Any pointers would be...
23. ### MHB Are the statements about the confidence interval correct?

Hey! :o We have a 90%-confidence interval. I want to check if the following statements are correct. 1. If double the sample, the possibility that the value that we are looking for is out of the confidence interval is smaller. 2. The bigger the standard error, the smaller the confidence...
24. ### MHB  95% Confidence Interval for Staying Awake at UMUC

This is the word problem I am faced with... I have provided my answer. Please give input if I am right or wrong. It is 1581 Anno Domini. At the Undergraduate School of UMUC, besides Assistant Academic Director of Mathematics and Statistics, I am also the Undergraduate School-appointed CPA...
25. ### Proof that an interval is a confidence interval for Geom(q)

Hello Physicsforum Homework Statement I have a problem proving this: Given C(x)=[0, 3/x] for all x\in\chi, with \chi=\Omega being the sample space and P_q=Geom(q) being the geometric distribution. I have to show that C(x) is a confidence Interval for q but I don't know how to get started...
26. ### Confidence Intervals for not integers numbers ratio

Hi, I’m having a problem with a particular case of binomial proportion. I want calculate a confidence Intervals for a binomial proportion for an efficiency. This kind of intervals are usually defined for ratios between integers numbers but in my case I had to subtract from both numerators and...
27. ### Confidence interval for the weighted mean

I run an algorithm that outputs a normally distributed random number with unknown mean and variance. I need to know the mean of those numbers. To do that, i repeatedly run that algorithm to obtain many numbers, then I calculate the mean and the confidence interval for the mean. Over the last...
28. ### T-Distribution and confidence interval)

I have a question I've been working on and I've noticed that my solution differs from the textbook solution. I would have considered it a mistake, but the next 4 questions are similar with the same solution "mistake"? Aside from not converting the % to 0.885, I am wondering about part b. It...
29. ### MHB Confidence Interval for Mean $\mu$: Let $X_1, X_2, \dots, X_n$

Hello! (Wave) Let $X_1, X_2, \dots, X_n$ a random sample with $E(X_i)=\mu$, $Var(X_i)=\sigma^2 \forall i$. For $0<a<0.5$: show that for any $k \in [0,1]$, the interval $$\left( \overline{X}-z_{k=a} \frac{\sigma}{\sqrt{n}}, \overline{X}+z_{(1-k)=a} \frac{\sigma}{\sqrt{n}}\right)$$ is a...
30. ### Conceptual Question Dealing with Confidence Intervals

Homework Statement 1.) When we collect more data for our confidence interval (A) Confidence Increases (B) Confidence Decreases (C) n decreases (D) The length of the interval decreases (E) The length of the interval increases 2.) We wish to determine a confidence interval for the population...
31. ### MHB Predicting election outcome based on confidence interval

Is it possible to predict an election outcome based on a confidence interval for the population proportion?
32. ### Confidence interval of two sample tests

Homework Statement Some car tires can develop what is known as "heel and toe" wear if not rotated after a certain mileage. To assess this issue, a consumer group investigated the tire wear on two brands of tire, A and B, say. Fifteen cars were fitted with new brand A tires and thirteen with...
33. ### MHB Help with making a two sided confidence interval

I am working on a problem for my Statistics class and I am looking for some clarification and some direction. The questions asks about a study that has been done on the diameter of 50 screws (both digital and vernier scale calipers). The differences in the measured diameters had the following...
34. ### The Meaning of a 95% T Confidence Interval for the Mean

Homework Statement A diet pill is given to 9 subjects over six weeks. The average difference in weight (follow up - baseline) is -2 pounds. What would the standard deviation have to be for the 95% T confidence interval to lie entirely below 0? ANSWER: Around 2.6 pounds or less Refer to the...
35. ### Confidence interval for a cohort

Homework Statement Latency Time(years): Number of patients 0: 2 1: 6 2: 9 3: 33 4: 49 5: 66 6: 52 7: 37 8: 18 9: 11 10: 4 (I don't know how to make a proper table with latex... tried \being{tabular}{l r} but this doesn't work) Homework Equations When the variance is...
36. ### Calculating the confidence interval of your data

Hello guys, I would like to calculate the confidence interval of the data in which the data is correct. In otherwords I would like to know how much confidence we have that the data is correct. Could anyone direct me where to start? Thank you.
37. ### Finding sample size given info on confidence interval

Homework Statement Suppose that an investigator believes that virtually all values in the population are between 38 and 70. The appropriate sample size for estimating the true population mean within 2 units with 95% confidence level is approximately how much? Homework EquationsVariance of the...
38. ### Statistics - Confidence interval

Homework Statement Suppose you have a bucket containing a lot of balls with different colors. You randomly pick 50 balls, 9 of which are red (X = 9, where X ~ N(μ, σ²)). The probability of picking a red ball is 15%. From this you want to construct a 95% confidence interval for the standard...
39. ### Questions related to sufficient statistics and confidence interval

Homework Statement Suppose X1, X2, ..., Xn constitute a random sample from a population following N(μ,θ) where μ is known. (a)Find a sufficient statistic for θ. (b)Use the maximum likelihood estimator of θ to construct a confidence interval for θ with confidence level 1-α. Homework...
40. ### 95% confidence interval on the mean

Homework Statement What does it mean? Is this the interval such that 95% of the means from a sample mean distribution (frequency distribution of means of sample size n) are a certain number of standard deviations from the population mean? Homework Equations The Attempt at a Solution
41. ### Find the 90% two-sided confidence interval for the mean score.

The distribution of grades is know to follow a normal distribution, but the mean and variance are unknown. A random sample of 12 students produced the following scores: 59, 84, 68, 93, 49, 77, 82, 75, 81, 58, 70, 80 Find the 90% two-sided confidence interval for the mean score.
42. ### Confidence Interval Calculation

See attached for problem Working: Mean: 1203.26 Standard Deviation: 7.047 for 99% confidence interval, level of significance = 0.01 Therefore 0.5 - 0.01/2 = 0.45 Reading 0.45 from standard distribution table i get a value of 1.65 therefore confidence interval should be...
43. ### Confidence Interval for Coefficient of Quadratic Fit

I have a bunch of noisy data points (x,y), and I want to model the data as y = ax2 + bx + c + noise where noise can probably be assumed to be Gaussian, or perhaps uniformly distributed. My data is firmly inside of an interval and I'm only interested in modeling correctly inside of this...
44. ### Confidence Interval for Population Mean (μ): Known vs Unknown σ

Hello May I know what is the difference between Confidence Interval for Population Mean (μ) when σ Known vs Confidence Interval for Population Mean (μ) when σ unknown.. any example to show me??
45. ### Need help with basic confidence interval question - TI 84 Plus

Hello, I need help using my TI 84 to figure out this 4 part question. My professor went over it way to quickly in class, when showing us how to do it on the TI 84, I believe its only a few quick steps for each part though. EDIT: I have figured out part d, but am still struggling with parts a...
46. ### How to find Confidence interval

A researcher surveys 25 people and finds that their average commute is 15 minutes with a standard deviation of 1.5 minutes. Calculate the 95 percent confidence interval on variance. Do most people have a commute of 20 minutes? Confidence interval for variance? should I just use the variance...
47. ### Confidence Interval for θ | Homework Solution

Homework Statement Let θ>0 and X1, X2,...,Xn be a random sample from the distribution with the pdf fx(x)=fx(x;θ)=(θ/(2√x))e-θ√x, x>0 Recall: Ʃ√xi, i=1, n has Gamma (α = n, "usual θ" = 1/θ) distribution. a) Suggest a confidence interval θ with (1-α)100% confidence level. b)...
48. ### Confidence interval interpretation mistake

Hi all. I've been thinking about this question a lot for the past few days and it seems to me that I'm committing a mistake somewhere along the way, but certainly can't figure out where. Here's one of the interpretations which I've encountered most frequently and think is the right one (here's...
49. ### Formula For Confidence Interval of Two Samples With Unequal Variances

Are there any online sources that would note (and define all variables, especially on how to calculate standard error) in the formula for calculating the confidence interval, a complement to the t-test, of two independent samples with unequal variances? (I want to see exactly how...
50. ### Finding a confidence interval.

Homework Statement A machine produces metal pieces that are cylindrical in shape. A sample of pieces is taken and the diameters are found to be 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, 1.03 centimeters. Find a 99% confidence interval for the mean diameter of pieces from this machine...