Download Ch7-4

Section 7.4
Estimation of a Population Mean
(s is unknown)
This section presents methods for estimating
a population mean when the population
standard deviation s is not known.
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Best Point Estimate
_
The sample mean x is still
the best point estimate of
the population mean m.
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Student t Distribution
( t-dist )
When σ is unknown, we must use
the Student t distribution instead
of the normal distribution.
Requires new parameter
df = Degrees of Freedom
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Definition
The number of degrees of freedom (df) for
a collection of sample data is defined as:
“The number of sample values that can
vary after certain restrictions have been
imposed on all data values.”
In this section: df = n – 1
Basically, since σ is unknown, a data point has to
be “sacrificed” to make s. So all further
calculations use n – 1 data points instead of n.
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Using the Student t Distribution
The t-score is similar to the z-score but applies
for the t-dist instead of the z-dist. The same is
true for probabilities and critical values.
α (area)
0
-1 0
P(t < -1)
(Area under curve)
tα
(Critical value)
NOTE: The values depend on df
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Important Properties of the
Student t Distribution
1. Has a symmetric bell shape similar to the z-dist
2. Has a wider distribution than that the z-dist
3. Mean μ = 0
4. S.D.
σ > 1 (Note: σ varies with df)
5. As df gets larger, the t-dist approaches the z-dist
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Student t Distributions for
n = 3 and n = 12
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z-Distribution and t-Distribution
df = 2
Wider Spread
df = 100
Almost the same
As df increases,
the t-dist approaches the z-dist
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Progression of t-dist with df
df = 2
df = 3
df = 4
df = 6
df = 7
df = 8
df = 20
df = 5
df = 50
df = 100
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Choosing the Appropriate Distribution
s known and normally
Use the normal (Z)
distribution
distributed population
or
s known and n > 30
s not known and normally
Use t distribution
distributed population
or
s not known and n > 30
Methods of Ch. 7
do not apply
Population is not normally
distributed and n ≤ 30
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Calculating values from t-dist
Stat → Calculators → T
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Calculating values from t-dist
Enter Degrees of Freedom (DF) and t-score
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Calculating values from t-dist
P(t<-1) = 0.1646
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when df = 20
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Calculating values from t-dist
tα = 1.697
when α = 0.05 df = 20
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Margin of Error E for Estimate of m
(σ unknown)
Formula 7-6
where t/2 has n – 1 degrees of freedom.
t/2 = The t-value separating the right
tail so it has an area of /2
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C.I. for the Estimate of μ
(With σ Not Known)
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Finding the Point Estimate and E from a C.I.
Point estimate of µ:
Margin of Error:
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Example:
Find the 90%confidence interval for the population
mean using a sample of size 42, mean 38.4, and
standard deviation 10.0
s
Note: Same parameters as example used in Section 7-3
7-3: Etimating a population mean: σ known
Using σ = 10 ( instead of s = 10.0 )
we found the 90% confidence interval:
C.I. = (35.9, 40.9)
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Example:
Find the 90%confidence interval for the population
mean using a sample of size 42, mean 38.4, and
standard deviation 10.0
s
.0
Direct Computation:
T Calculator (df = 41)
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Example:
Find the 90%confidence interval for the population
mean using a sample of size 42, mean 38.4, and
standard deviation 10.0
s
.0
Using StatCrunch
Stat → T statistics → One Sample → with Summary
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Example:
Find the 90%confidence interval for the population
mean using a sample of size 42, mean 38.4, and
standard deviation 10.0
s
.0
Using StatCrunch
Enter Parameters, click Next
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Example:
Find the 90%confidence interval for the population
mean using a sample of size 42, mean 38.4, and
standard deviation 10.0
s
.0
Using StatCrunch
Select Confidence Interval and enter Confidence
Level, then click Calculate
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Example:
Find the 90%confidence interval for the population
mean using a sample of size 42, mean 38.4, and
standard deviation 10.0
s
.0
Using StatCrunch
Standard Error
Lower Limit
Upper Limit
From the output, we find the Confidence interval is
CI = (35.8, 41.0)
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Example:
Find the 90%confidence interval for the population
mean using a sample of size 42, mean 38.4, and
standard deviation 10.0
s
Results
If σ known
Used σ = 10 to obtain 90% CI:
(35.9, 40.9)
If σ unknown
Used s = 10.0 to obtain 90% CI:
(35.8, 41.0)
Notice: σ known yields a smaller CI (i.e. less uncertainty)
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