Confidence Interval Estimation: Single

Population
Dr. A. Ramesh
Department of Management Studies
IIT ROORKEE

1
 Goals
After completing this lecture, you should be able to:
• Distinguish between a point estimate and a confidence interval estimate
• Construct and interpret a confidence interval estimate for a single
population mean using both the Z and t distributions
• Form and interpret a confidence interval estimate for a single population
proportion
• Create confidence interval estimates for the variance of a normal
population

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 Confidence Intervals
• Confidence Intervals for the Population Mean, μ
– when Population Variance σ2 is Known
– when Population Variance σ2 is Unknown
• Confidence Intervals for the Population Proportion, p̂ (large samples)
• Confidence interval estimates for the variance of a normal population

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 Definitions

• An estimator of a population parameter is

– a random variable that depends on sample information . . .
– whose value provides an approximation to this unknown parameter

• A specific value of that random variable is called an estimate

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 Point and Interval Estimates

• A point estimate is a single number,

• a confidence interval provides additional information about
variability

Lower Upper
Confidence Confidence
Point Estimate Limit
Limit
Width of
confidence interval
5
 Point Estimates

We can estimate a with a Sample

Population Parameter … Statistic
(a Point Estimate)

Mean μ x
Proportion P p̂

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 Unbiasedness

• A point estimator θ̂ is said to be an unbiased estimator of the

parameter  if the expected value, or mean, of the sampling
distribution of θ̂ is ,

E(θˆ )  θ
• Examples:
– The sample mean x is an unbiased estimator of μ
– The sample variance s2 is an unbiased estimator of σ2
– The sample proportion p̂ is an unbiased estimator of P

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 Unbiasedness
(continued)
• θ̂1 is an unbiased estimator, θ̂2 is biased:

θ̂1 θ̂2

θ θ̂
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 Bias

• Let θ̂ be an estimator of 

• The bias in θ̂ is defined as the difference between its mean and 

Bias(θˆ )  E(θˆ )  θ
• The bias of an unbiased estimator is 0

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 Most Efficient Estimator
• Suppose there are several unbiased estimators of 
• The most efficient estimator or the minimum variance unbiased estimator
of  is the unbiased estimator with the smallest variance
• Let θ̂1 and θ̂2 be two unbiased estimators of , based on the same number
of sample observations. Then,
– θ̂1 is said to be more efficient than θ̂2 if Var(θˆ 1 )  Var(θˆ 2 )

– The relative efficiency of θ̂1 with respect to θ̂2 is the ratio of

their variances:
Var( θˆ 2 )
Relative Efficiency 
Var( θˆ )
1

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 Confidence Intervals

• How much uncertainty is associated with a point estimate of a population

parameter?

• An interval estimate provides more information about a population

characteristic than does a point estimate

• Such interval estimates are called confidence intervals

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 Confidence Interval Estimate

• An interval gives a range of values:

– Takes into consideration variation in sample statistics from sample to
sample
– Based on observation from 1 sample
– Gives information about closeness to unknown population
parameters
– Stated in terms of level of confidence
• Can never be 100% confident

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 Confidence Interval and Confidence Level

• If P(a <  < b) = 1 -  then the interval from a to b is called a 100(1 -

)% confidence interval of .
• The quantity (1 - ) is called the confidence level of the interval (
between 0 and 1)

– In repeated samples of the population, the true value of the

parameter  would be contained in 100(1 - )% of intervals
calculated this way.
– The confidence interval calculated in this manner is written as a <  <
b with 100(1 - )% confidence

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 Estimation Process

Random Sample I am 95% confident

that μ is between 40
Population & 60.
Mean
(mean, μ, is X = 50
unknown)

Sample

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 Confidence Level, (1-)
(continued)
• Suppose confidence level = 95%
• Also written (1 - ) = 0.95
• A relative frequency interpretation:
– From repeated samples, 95% of all the confidence intervals that can
be constructed will contain the unknown true parameter
• A specific interval either will contain or will not contain the true
parameter

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 General Formula

• The general formula for all confidence intervals is:

Point Estimate  (Reliability Factor)(Standard Error)

• The value of the reliability factor depends on the desired level of confidence

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 Confidence Intervals

Confidence
Intervals

Population Population Population

Mean Proportion Variance

σ2 Known σ2 Unknown

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 Confidence Interval for μ (σ2 Known)
• Assumptions
– Population variance σ2 is known
– Population is normally distributed
– If population is not normal, use large sample
• Confidence interval estimate:
σ σ
x  z α/2  μ  x  z α/2
n n
(where z/2 is the normal distribution value for a probability of /2 in each tail)

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 Margin of Error
• The confidence interval,
σ σ
x  z α/2  μ  x  z α/2
n n

• Can also be written as x  ME

where ME is called the margin of error

σ
ME  z α/2
n

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 Reducing the Margin of Error

σ
ME  z α/2
n
The margin of error can be reduced if

• the population standard deviation can be reduced (σ↓)

• The sample size is increased (n↑)

• The confidence level is decreased, (1 – ) ↓

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 Finding the Reliability Factor, z/2
• Consider a 95% confidence interval:
1    .95

α α
 .025  .025
2 2

Z units: z = -1.96 0 z = 1.96

Lower Upper
X units: Confidence Point Estimate Confidence
Limit Limit

 Find z.025 = 1.96 from the standard normal distribution table

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 Common Levels of Confidence

• Commonly used confidence levels are 90%, 95%, and 99%

Confidence
Confidence
Coefficient, Z/2 value
Level
1 
80% .80 1.28
90% .90 1.645
95% .95 1.96
98% .98 2.33
99% .99 2.58
99.8% .998 3.08
99.9% .999 3.27
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 Intervals and Level of Confidence
Sampling Distribution of the Mean
/2 1  /2

Intervals
x
μx  μ
extend from 100(1-)%
x1
of intervals
σ
LCL  x  z x2 constructed
n contain μ;
to
σ 100()% do
UCL  x  z not.
n
Confidence Intervals
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 Example

• A sample of 11 circuits from a large normal population has a mean

resistance of 2.20 ohms. We know from past testing that the population
standard deviation is 0.35 ohms.

• Determine a 95% confidence interval for the true mean resistance of the
population.

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 Example
(continued)

• A sample of 11 circuits from a large normal population has a mean resistance

of 2.20 ohms. We know from past testing that the population standard
deviation is .35 ohms.
σ
x z
• Solution: n

 2.20  1.96 (.35/ 11)

 2.20  .2068

1.9932  μ  2.4068

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 Interpretation

• We are 95% confident that the true mean resistance is

between 1.9932 and 2.4068 ohms
• Although the true mean may or may not be in this interval,
95% of intervals formed in this manner will contain the true
mean

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 Confidence Intervals

Confidence
Intervals

Population Population Population

Mean Proportion Variance

σ2 Known σ2 Unknown

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