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.
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...
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Where is my mistake? Thanks
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Homework Statement
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Homework Statement
Hi,
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Homework Statement
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Hello Physicsforum
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Homework Statement
1.) When we collect more data for our confidence interval
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(B) Confidence Decreases
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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...
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.
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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...
Homework Statement
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Homework Statement
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Homework...
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
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.
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
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May I know what is the difference between
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any example to show me??
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