Download Chapter 7

Best Point Estimate
Section 7.4
Estimation of a Population Mean
(s is unknown)
_
The sample mean x is still
the best point estimate of
the population mean m.
This section presents methods for estimating
a population mean when the population
standard deviation s is not known.
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Student t Distribution
( t-dist )
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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.”
When σ is unknown, we must use
the Student t distribution instead
of the normal distribution.
In this section: df = n – 1
Requires new parameter
df = Degrees of Freedom
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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Important Properties of the
Student t Distribution
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.
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
α (area)
P(t < -1)
(Area under curve)
4. S.D.
0
-1 0
σ > 1 (Note: σ varies with df)
5. As df gets larger, the t-dist approaches the z-dist
tα
(Critical value)
NOTE: The values depend on df
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Student t Distributions for
n = 3 and n = 12
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
Choosing the Appropriate Distribution
df = 4
s known and normally
Use the normal (Z)
distribution
df = 6
df = 20
df = 5
df = 7
df = 50
df = 8
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
df = 100
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Calculating values from t-dist
Calculating values from t-dist
Stat → Calculators → T
Enter Degrees of Freedom (DF) and t-score
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Calculating values from t-dist
P(t<-1) = 0.1646
Calculating values from t-dist
when df = 20
tα = 1.697
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Margin of Error E for Estimate of m
(σ unknown)
Formula 7-6
𝑬 = 𝒕𝜶
2
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C.I. for the Estimate of μ
(With σ Not Known)
𝒔
𝒏
where t/2 has n – 1 degrees of freedom.
𝑬 = 𝒕𝜶
t/2 = The t-value separating the right
tail so it has an area of /2
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when α = 0.05 df = 20
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Finding the Point Estimate and E from a C.I.
Point estimate of µ:
2
𝒔
𝒏
𝒅𝒇 = 𝒏 − 𝟏
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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
Margin of Error:
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:
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
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:
Using StatCrunch
𝒅𝒇 = 𝒏 − 1 = 41
𝒕𝜶
2
= 𝒕0.05 = 1.6839
𝑬 = 𝒕𝜶
2
𝒔
10.0
= 1.6839
= 2.5983
𝒏
42
𝒙 − 𝑬 = 38.4 − 2.5983 = 35.8017
T Calculator (df = 41)
𝒙 + 𝑬 = 38.4 + 2.5983 = 40.9983
𝑪. 𝑰. = 𝒙 − 𝑬, 𝒙 + 𝑬 = (35.8, 41.0)
Stat → T statistics → One Sample → with Summary
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Example:
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
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
Using StatCrunch
Select Confidence Interval and enter Confidence
Level, then click Calculate
Enter Parameters, click Next
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Example:
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
Find the 90%confidence interval for the population
mean using a sample of size 42, mean 38.4, and
standard deviation 10.0
s
Using StatCrunch
Results
Standard Error
If σ known
Used σ = 10 to obtain 90% CI:
If σ unknown
Used s = 10.0 to obtain 90% CI:
Lower Limit
(35.9, 40.9)
Upper Limit
(35.8, 41.0)
From the output, we find the Confidence interval is
CI = (35.8, 41.0)
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Notice: σ known yields a smaller CI (i.e. less uncertainty)
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Best Point Estimate of s2
Section 7.5
Estimation of a Population
Variance
The sample variance s2 is
the best point estimate of
the population variance s2
This section presents methods for
estimating a population variance s2
and standard deviation s.
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Best Point Estimate of s
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The 2 Distribution
( 2-dist )
Pronounced “Chi-squared”
The sample standard deviation s
is the best point estimate of the
population standard deviation s
Also dependent on the number
degrees of freedom df.
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Calculating values from 2-dist
Properties of the 2 Distribution
1. The chi-square distribution is not symmetric,
unlike the z-dist and t-dist.
Stat → Calculators → Chi-Squared
2. The values can be zero or positive, they are nonnegative.
3. Dependent on the Degrees of Freedom: df = n – 1
Chi-Square Distribution
Chi-Square Distribution for
df = 10 and df = 20
Use StatCrunch to Calculate values (similar to z-dist and t-dist)
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Example:
Find the 90% left and right critical values
(2L and 2R) of the 2-dist when df = 20
Calculating values from 2-dist
Enter Degrees of Freedom DF and parameters
( same procedure as with t-dist )
Need to calculate values when the left/right areas are 0.05 ( i.e. α/2 )
2L = 10.851
P(2 < 10)= 0.5595 when df = 10
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Important Note!!
2R = 31.410
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Confidence Interval for Estimating a
Population Variance
The 2-distribution is used for
calculating the Confidence Interval of
the Variance σ2
Take the square-root of the values to
get the Confidence Interval of the
Standard Deviation σ
Note: Left and Right Critical values on opposite sides
( This is why we call it 2 instead of  )
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Confidence Interval for Estimating a
Population Standard Deviation
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Requirement for Application
The population MUST be
normally distributed to hold
(even when using large samples)
This requirement is very strict!
Note: Left and Right Critical values on opposite sides
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Example
Round-Off Rules for Confidence
Intervals Used to Estimate s or s 2
Suppose the scores a test follow a normal distribution. Given
a sample of size 40 with mean 72.8 and standard deviation
4.92, find the 95% C.I. of the population standard deviation.
1. When using the original set of data, round the
confidence interval limits to one more decimal
place than used in original set of data.
Direct Computation:
2. When the original set of data is unknown and
only the summary statistics (n, x, s) are
used, round the confidence interval limits to the
same number of decimal places used for the
sample standard deviation.
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Example
Chi-Squared Calculator (df = 39)
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Example
Suppose the scores a test follow a normal distribution. Given
a sample of size 40 with mean 72.8 and standard deviation
4.92, find the 95% C.I. of the population standard deviation.
Using StatCrunch
Suppose the scores a test follow a normal distribution. Given
a sample of size 40 with mean 72.8 and standard deviation
4.92, find the 95% C.I. of the population standard deviation.
Using StatCrunch
Sample Variance
Enter parameters, then click Next
Be sure to enter the sample variance s2 (not s)
Stat → Variance → One Sample → with Summary
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Example
40
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Example
Suppose the scores a test follow a normal distribution. Given
a sample of size 40 with mean 72.8 and standard deviation
4.92, find the 95% C.I. of the population standard deviation.
Using StatCrunch
Suppose the scores a test follow a normal distribution. Given
a sample of size 40 with mean 72.8 and standard deviation
4.92, find the 95% C.I. of the population standard deviation.
Using StatCrunch
Remember:
The result is the C.I for the Variance σ2
Take the square root for Standard Deviation σ
Variance Lower Limit: LLσ
2
Variance Upper Limit: ULσ
σ2 CI = ( LLσ , ULσ ) = (16.2, 39.9)
2
Select Confidence Interval, enter Confidence Level,
then click Calculate
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σ
41
2
CI = ( LLσ , ULσ ) = (4.03, 6.32)
2
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Table 7-2
Determining Sample Sizes
The procedure for finding the sample size
necessary to estimate s2 is based on Table 7-2
You just read the required sample size from an
appropriate line of the table.
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For s 95% confident and within 20%
Example
We want to estimate the standard deviation s.
We want to be 95% confident that our estimate is
within 20% of the true value of s.
Assume that the population is normally distributed.
How large should the sample be?
For s 95% confident and within 20%
From Table 7-2 (see next slide), we can see that
95% confidence and an error of 20% for s
correspond to a sample of size 48.
We should obtain a sample of 48 values.
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