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ESTIMATION OF THE MEAN
INTRO :: ESTIMATION
Definition
The assignment of plausible value(s) to a
population parameter based on a value of
a sample statistic is called estimation.
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INTRO :: ESTIMATION
The sample statistic used to estimate a
population parameter is called estimator.
Examples …
sample mean estimator population mean
…
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INTRO :: ESTIMATION
The estimation procedure involves …
1.
2.
3.
4.
Select a sample.
Collect the required information from the
members of the sample.
Calculate the value of the sample statistic.
Assign plausible value(s) to the
corresponding population parameter.
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POINT ESTIMATES &
INTERVAL ESTIMATES


A Point Estimate
An Interval Estimate
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A Point Estimate
Definition
The value of a sample statistic that is used
to estimate a population parameter is
called a point estimate.
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INTERVAL ESTIMATES

Usually, whenever we use point estimation,
we calculate the margin of error associated with
that point estimation.
Point Estimate w/out M.E. not very useful!!!!
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Interval Estimates
Definition
In interval estimation, an interval is
constructed around the point estimate,
and it is stated that this interval is likely to
contain the corresponding population
parameter. Gives a range of plausible
values for the parameter of interest.
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Interval estimation.
x  
$1130
x  $1370
$1610
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Interval Estimates


Definition
Each interval is constructed with regard to a given
confidence level and is called a confidence
interval. The confidence level associated with a
confidence interval states how much confidence we
have that this interval contains the true population
parameter. The confidence level is denoted by
(1 – α)100%.
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INTERVAL ESTIMATION OF A
POPULATION MEAN:


The t Distribution
Confidence Interval for μ
Using the t Distribution
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The t Distribution
Conditions Under Which the t Distribution Is Used to Make a
Confidence Interval About μ
The t distribution is used to make a
confidence interval about μ if
The population from which the sample is drawn
is (approximately) normally distributed.
2. The population standard deviation, σ, unknown.
1.
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The t Distribution cont.
The t distribution is a specific type of bellshaped distribution with a lower height and a
wider spread than the standard normal
distribution. As the sample size becomes larger,
the t distribution approaches the standard normal
distribution. The t distribution has only one
parameter, called the degrees of freedom (df).
The mean of the t distribution is equal to 0 and
its standard deviation is df /( df  2) .
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The t distribution for df = 9 and the
standard normal distribution.
The standard deviation
of the t distribution is
The standard deviation of
the standard normal
distribution is 1.0
9 /(9  2)  1.134
μ=0
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Example
Find the value of t for 16 degrees of
freedom and .05 area in the right tail of a
t distribution curve.
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Determining t for 16 df and .05 Area in the Right Tail
Area in the right tail
Area in the Right Tail Under the t Distribution Curve
df
df
.10
.05
.025
…
.001
1
2
3
.
16
.
3.078
1.886
1.638
…
1.337
…
6.314
2.920
2.353
…
1.746
…
12.706
4.303
3.182
…
2.120
…
…
…
…
…
…
…
318.309
22.327
10.215
…
3.686
…
The required value of t for 16 df and
.05 area in the right tail
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The value of t for 16 df and .05 area in the right tail.
.05
df = 16
0
1.746
t
This is the required
value of t
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The value of t for 16 df and .05 area in the left tail.
df = 16
.05
-1.746
0
t
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Confidence Interval for μ
Using the t Distribution
The (1 – α)100% confidence interval
for μ is
x  tsx
s
where s x 
n
The value of t is obtained from the t
distribution table for n – 1 degrees of
freedom and the given confidence level.
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Examples …
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