Transcript Section 6-1, 6-2

Estimates and Sample Sizes
Chapter 6
M A R I O F. T R I O L A
Copyright ©Copyright
1998,
Triola, Elementary Statistics
© 1998, Triola, Elementary Statistics
Addison Wesley
Wesley Longman Longman
Addison
1
Chapter 6
Estimate and Sample Sizes
6-1 Overview
6-2 Estimating a Population Mean:
Large Samples
6-3 Estimating a Population Mean:
Small Samples
6-4 Determining Sample Size
6-5 Estimating a Population Proportion
6-6 Estimating a Population Variance
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6-1
Overview
This chapter presents:
methods for estimating population means,
proportions, and variances
methods for determining sample sizes
necessary to estimate the above
parameters.
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6-2
Estimating a Population Mean:
Large Samples
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Definitions
 Estimator
a sample statistic used to approximate a population
parameter
 Estimate
a specific value or range of values used to
approximate some population parameter
 Point Estimate
a single volume (or point) used to approximate a
population parameter
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Definitions
 Estimator
a sample statistic used to approximate a population
parameter
 Estimate
a specific value or range of values used to
approximate some population parameter
 Point Estimate
a single volue (or point) used to approximate a
popular parameter
The sample mean x
is the best point estimate of the
population mean µ.
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 Confidence Interval
(or Interval Estimate)
a range (or an interval) of values likely to
contain the true value of the population
parameter
Lower # < population parameter < Upper #
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 Confidence Interval
(or Interval Estimate)
a range (or an interval) of values likely to
contain the true value of the population
parameter
Lower # < population parameter < Upper #
As an example
Lower # < µ < Upper #
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
Definition
Degree of Confidence
(level of confidence or confidence coefficient)
the probability 1 – a (often expressed as the
equivalent percentage value) that the confidence
interval contains the true value of the population
parameter
 usually 95% or 99%
(a = 5%) (a = 1%)
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Confidence Intervals from
Different Samples
µ = 98.25 (but unknown to us)
98.00
x
•
98.08
•
Figure 6-3
•
•
98.50
•
98.32
•
•
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•
This confidence interval
does not contain µ
•
•
10
Definition

Critical Value
the number on the borderline separating sample
statistics that are likely to occur from those that
are unlikely to occur. The number
value that is a
z
za/2 is a critical
score with the property that it
separates an area a/2 in the right tail of the
standard normal distribution. There is an area of
1 – a between the vertical borderlines at
–za/2
and
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za/2.
11
If Degree of Confidence = 95%
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If Degree of Confidence = 95%
95%
a = 5%
a/2 = 2.5% = .025
.95
.025
–za/2
.025
za/2
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If Degree of Confidence = 95%
95%
a = 5%
a/2 = 2.5% = .025
.95
.025
.025
–za/2
za/2
Critical Values
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95% Degree of Confidence
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95% Degree of Confidence
a = 0.05
a/2 = 0.025
.4750
.025
Use Table A-2
to find a z score of 1.96
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95% Degree of Confidence
a = 0.05
a/2 = 0.025
.4750
.025
Use Table A-2
to find a z score of 1.96
za/2 = +- 1.96
.025
–1.96
.025
1.96
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Definition
Margin of Error
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Definition
Margin of Error
 is the maximum likely difference
 between the observed sample mean, x, and true
population mean µ.
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Definition
Margin of Error
 is the maximum likely difference
 between the observed sample mean, x, and true
population mean µ.
 denoted by E
x –E
µ
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x +E
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Definition
Margin of Error
 is the maximum likely difference
 between the observed sample mean, x, and true
population mean µ.
 denoted by E
x –E
µ
x +E
x–E<µ<x+E
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Definition
Margin of Error
 is the maximum likely difference
 between the observed sample mean, x, and true
population mean µ.
 denoted by E
µ
x –E
x +E
x–E<µ<x+E
lower limit
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Definition
Margin of Error
 is the maximum likely difference
 between the observed sample mean, x, and true
population mean µ.
 denoted by E
µ
x –E
x +E
x – E < µ < x +E
lower limit
upper limit
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Calculating the Margin of Error
When s Is Unknown
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Calculating the Margin of Error
When s Is known
E = za/2 •
s
n
Formula 6-1
If n > 30, we can replace s in Formula 6-1 by the
sample standard deviation s.
If n  30, the population must have a normal
distribution and we must know s to use Formula
6-1.
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Confidence Interval
Round off Rules
• 1. If using original data, round to one more
decimal place than used in data.
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Confidence Interval
Round off Rules
• 1. If using original data, round to one more
decimal place than used in data.
• 2. If given summary statistics (n, x, s), round to
same number of decimal places as in x.
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Procedure for Constructing a
Confidence Interval for µ
( based on a large sample:
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n
> 30 )
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Procedure for Constructing a
Confidence Interval for µ
( based on a large sample:
n
> 30 )
1. Find the critical value za/2 that corresponds to the
desired degree of confidence.
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Procedure for Constructing a
Confidence Interval for µ
( based on a large sample:
n
> 30 )
1. Find the critical value za/2 that corresponds to the
desired degree of confidence.
2. Evaluate the margin of error E= za/2 • s / n . If the
population standard deviation s is unknown and
n > 30, use the value of the sample standard
deviation s.
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Procedure for Constructing a
Confidence Interval for µ
( based on a large sample:
n
> 30 )
1. Find the critical value za/2 that corresponds to the
desired degree of confidence.
2. Evaluate the margin of error E = za/2 • s / n . If the
population standard deviation s is unknown and
n > 30, use the value of the sample standard
deviation s.
3. Find the values of x – E and x + E. Substitute those
values in the general format of the confidence
interval:
x –E <µ< x +E
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Procedure for Constructing a
Confidence Interval for µ
( based on a large sample:
n
> 30 )
1. Find the critical value za/2 that corresponds to the
desired degree of confidence.
2. Evaluate the margin of error E= za/2 • s / n . If the
population standard deviation s is unknown and
n > 30, use the value of the sample standard
deviation s.
3. Find the values of x – E and x + E. Substitute those
values in the general format of the confidence
interval:
x –E <µ< x +E
4. Round using the confidence intervals round off rules.
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Confidence Intervals from
Different Samples
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6-3
Determining Sample Size
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Determining Sample Size
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Determining Sample Size
s
E = za / 2 • n
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Determining Sample Size
s
E = za / 2 • n
(solve for n by algebra)
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Determining Sample Size
s
E = za / 2 • n
(solve for n by algebra)
n=
za / 2 s
E
2
Formula 6-2
If n is not a whole number, round it up to
the next higher whole number.
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What happens when E is doubled ?
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What happens when E is doubled ?
(za / 2s )
z
a / 2s
n= 1
=
1
2
E=1:
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What happens when E is doubled ?
E=1:
(za / 2s )
z
a / 2s
n= 1
=
1
E=2:
(za / 2s )
z
a / 2s
n=
=
4
2
2
2
2
2
Sample size n is decreased to 1/4 of its
original value if E is doubled.
Larger errors allow smaller samples.
Smaller errors require larger samples.
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What if s is unknown ?
1. Use the range rule of thumb to estimate the
standard deviation as follows: s  range / 4
or
2. Calculate the sample standard deviation s and
use it in place of s. That value can be refined as
more sample data are obtained.
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