1810DMTCYFS03229
HARSH RIMZA
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Confidence Interval is a range where we are certain that
true value exists. The selection of a confidence level for
an interval determines the probability that the
confidence interval will contain the true parameter
value. This range of values is generally used to deal
with population-based data, extracting specific,
valuable information with a certain amount of
confidence.
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Probability of including the true value of a parameter
within a confidence interval.
The confidence level describes the uncertainty
associated with a sampling method.
For Example : let’s suppose you were surveying an
average height of men in a particular city. To find that,
you set a 95% confidence level and find that the 95%
confidence interval is (168,182). That means if you
repeated this over and over, 95 percent of the time the
height of a man would fall somewhere between 168 cm
and 182 cm.
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Two extreme measurements within which an
observation lies. Or in other words end points of the
confidence interval.
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Step 1: Identify the sample problem. Choose the statistic (like
sample mean, etc) that you will use to estimate population
parameter.
Step 2: Select a confidence level. (Usually, it is 90%, 95% or
99%)
Step 3: Find the margin of error. (Usually given) If not given,
use the following formula:Margin of error = Critical value * Standard deviation
Step 4: Specify the confidence interval. The uncertainty is
denoted by the confidence level. And the range of the
confidence interval is defined by Eq-1.
Confidence Interval = Sample_Statistic + Margin_of_Error
where ,
Sample_Statistic : Can be sample mean.
Margin_of_Error : Is generally ± 2.5
-Eq-1
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Mean (μ) — Arithmetic mean is the average of
numbers. It is defined as the sum of n numbers divided
by the count of numbers till n. (Eq-2)
μ = (1+2+3+..+n)/n
- Eq 2
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Standard deviation (σ) — It is defined as the
summation of squared of the difference between each
number and the mean. (Eq-3)
Using t-distribution : We use t-distribution when the sample
size n<30.
Step 1: Subtract 1 from your sample size.[Eq-4] This gives the
degrees of freedom (df)
df=n-1
Eq-4
Step 2 - Subtract the confidence interval from 1, then divide by
two. [Eq-5] This gives the significance level (α)
α=(1- CL)/2
Eq-5
Step 3 - Use the values of α and df in the t-distribution table and
find the value of t.
Step 4 - Use the t-value obtained in step 3 in the formula given
for Confidence Interval with t-distribution. [Eq-6]
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μ ± t(σ/√n)
where,
μ = mean,
t = chosen t-value from the table above,
σ = the standard deviation,
n = number of observations.
Eq-6
Using z-distribution : We use z-distribution when the sample
size n>30. Z-test is more useful when the standard deviation is
known.
Step 1 : Find the mean.
Step 2 : Find the standard deviation.
Step 3 : Determine the z-value for the specified confidence
interval.
Step 4 - Use the z-value obtained in step 3 in the formula given
for Confidence Interval with z-distribution. [Eq-7]
μ ± z (σ/√n)
Eq-7
where, μ = mean,
z = chosen z-value from the table above
σ = the standard deviation
n = number of observations
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Confidence Interval is one of the foundational concepts
of statistics. It tells a statement about the data. Various
sampling methods such as mean, median etc. can be
used based on the data present. One can also determine
what distribution to use when in order to get the best
results.