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Transcript
Section 7.4: Estimation of a
population mean m
(s is not known)
This section presents methods for
estimating a population mean when
the population standard deviation s
is not known.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
1
Sample Mean
_
The sample mean x is still
the best point estimate of
the population mean m.
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2
Construction of a confidence
intervals for m
(s is not known)
With σ unknown, we use the
Student t distribution instead
of normal distribution.
It involves a new feature:
number of degrees of freedom
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3
Definition
The number of degrees of freedom for a
collection of sample data is the number of
sample values that can vary after certain
restrictions have been imposed on all data
values.
The degree of freedom is often abbreviated df.
degrees of freedom = n – 1
in this section.
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4
Margin of Error E for Estimate of m
(With σ Not Known)
Formula 7-6
E = t /
s
2
n
where t/2 has n – 1 degrees of freedom.
t/2 = critical t value separating an area of /2
in the right tail of the t distribution
Table A-3 lists values for tα/2
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5
Confidence Interval for the
Estimate of μ (With σ Not Known)
x–E <µ<x +E
where
E = t/2 s
n
df = n – 1
t/2 found in Table A-3
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6
Important Properties of the
Student t Distribution
1. The Student t distribution is different for different sample sizes
(see the following slide, for the cases n = 3 and n = 12).
2. The Student t distribution has the same general symmetric bell
shape as the standard normal distribution but it reflects the
greater variability (with wider distributions) than that the
standard normal distribution does.
3. The Student t distribution has a mean of t = 0 (just as the
standard normal distribution has a mean of z = 0).
4. The standard deviation of the Student t distribution varies with
the sample size and is greater than 1 (unlike the standard
normal distribution, which has a s = 1).
5. As the sample size n gets larger, the Student t distribution gets
closer to the normal distribution.
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7
Student t Distributions for
n = 3 and n = 12
Figure 7-5
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8
Choosing the Appropriate Distribution
Use the normal (z)
distribution
Use t distribution
s known and normally
distributed population
or
s known and n > 30
s not known and
normally distributed
population
or
s not known and n > 30
Methods of Chapter 7
do not apply
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Population is not
normally distributed
and n ≤ 30
9
Confidence Intervals by TI-83/84
• Press STAT and select TESTS
• Scroll down to TInterval press ENTER
• choose Data or Stats. For Stats:
_
• Type in x: (sample mean)

Sx: (sample st. deviation)
•
n: (number of trials)
•
C-Level: (confidence level)
• Press on Calculate
• Read the confidence interval (…..,..…)
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10
Confidence Intervals by TI-83/84
• Press STAT and select TESTS
• Scroll down to TInterval press ENTER
• choose Data or Stats. For Data:
• Type in List: L1 (or L2 or L3)
• (specify the list containing your data)

Freq: 1 (leave it)
•
C-Level: (confidence level)
• Press on Calculate
• Read the confidence interval (…..,..…)
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11
Finding the Point Estimate
and E from a Confidence Interval
Point estimate of µ:
x = (upper confidence limit) + (lower confidence limit)
2
Margin of Error:
E = (upper confidence limit) – (lower confidence limit)
2
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12
Section 7-5
Estimating a Population
Variance
This section covers the estimation
2
of a population variance s and
standard deviation s.
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13
Estimator of s
2
The sample variance s2 is the best
point estimate of the population
variance s2.
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14
Estimator of s
The sample standard deviation s is a
commonly used point estimate of s .
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15
Construction of confidence
2
intervals for s
We use the chi-square distribution,
denoted by Greek character 2
(pronounced chi-square).
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16
Properties of the Chi-Square
Distribution
1. The chi-square distribution is not symmetric, unlike
the normal and Student t distributions.
degrees of freedom = n – 1
Chi-Square Distribution
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Chi-Square Distribution for
df = 10 and df = 20
17
Properties of the Chi-Square
Distribution
2. The values of chi-square can be zero or positive, but
they cannot be negative.
3. The chi-square distribution is different for each
number of degrees of freedom, which is df = n – 1.
In Table A-4, each critical value of 2 corresponds to
an area given in the top row of the table, and that
area represents the cumulative area located to the
right of the critical value.
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18
Example
A sample of ten voltage levels is obtained.
Construction of a confidence interval for the
population standard deviation s requires the
left and right critical values of 2
corresponding to a confidence level of 95%
and a sample size of n = 10.
Find the critical value of 2 separating an area
of 0.025 in the left tail, and find the critical
value of 2 separating an area of 0.025 in the
right tail.
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19
Example
Critical Values of the Chi-Square Distribution
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20
Confidence Interval for Estimating a
Population Variance
(n  1)s

2
R
2
s
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2
n  1)s
(


2
2
L
21
Confidence Interval for Estimating a
Population Standard Deviation
(n  1)s

2
R
2
s 
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(n  1)s

2
2
L
22
Requirement:
The population must have
normally distributed values
(even if the sample is large)
This requirement is very strict
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23
Round-Off Rule for Confidence
Intervals Used to Estimate s or s 2
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.
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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24
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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25
Determining Sample Sizes
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26
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.
How large should the sample be?
Assume that the population is normally
distributed.
From Table 7-2, 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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27