Chapter 6 Estimates and Sample Sizes

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ELEMENTARY
STATISTICS
Chapter 6
Estimates and Sample Sizes
EIGHTH
EDITION
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
MARIO F. TRIOLA
1
Chapter 6
Estimates and Sample Sizes
6-1
Overview
6-2
Estimating a Population Mean:
Large Samples
6-3
Estimating a Population Mean:
Small Samples
6-4
Sample Size Required to Estimate µ
6-5
Estimating a Population Proportion
6-6
Estimating a Population Variance
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
6-1
Overview
This chapter presents:
 methods for estimating
population means, proportions, and
variances
 methods for determining sample sizes
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
3
6-2
Estimating a Population Mean:
Large Samples
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
4
Assumptions
 n > 30
The sample must have more than 30 values.
 Simple Random Sample
All samples of the same size have an equal chance of
being selected.
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
Assumptions
 n > 30
The sample must have more than 30 values.
 Simple Random Sample
All samples of the same size have an equal chance of
being selected.
Data collected carelessly can be
absolutely worthless, even if the sample
is quite large.
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
 Estimator
Definitions
a formula or process for using sample data to estimate
a population parameter
 Estimate
a specific value or range of values used to
approximate some population parameter
 Point Estimate
a single value (or point) used to approximate a
population parameter
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
 Estimator
Definitions
a formula or process for using sample data to estimate
a population parameter
 Estimate
a specific value or range of values used to
approximate some population parameter
 Point Estimate
a single value (or point) used to approximate a
population parameter
The sample mean x is the best point estimate of
the population mean µ.
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Definition
Confidence Interval
(or Interval Estimate)
a range (or an interval) of values used to
estimate the true value of the population
parameter
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
9
Definition
Confidence Interval
(or Interval Estimate)
a range (or an interval) of values used to
estimate the true value of the population
parameter
Lower # < population parameter < Upper #
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
Definition
Confidence Interval
(or Interval Estimate)
a range (or an interval) of values used to
estimate the true value of the population
parameter
Lower # < population parameter < Upper #
As an example
Lower # <  < Upper #
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Definition
Degree of Confidence
(level of confidence or confidence coefficient)
the probability 1 -  (often expressed as the
equivalent percentage value) that is the relative
frequency of times the confidence interval
actually does contain the population parameter,
assuming that the estimation process is
repeated a large number of times
usually 90%, 95%, or 99%
( = 10%), ( = 5%), ( = 1%)
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
Interpreting a Confidence Interval
98.08 < µ < 98.32
o
o
Correct: We are 95% confident that the interval from
98.08 to 98.32 actually does contain the true value of
.
This means that if we were to select many different
samples of size 106 and construct the confidence
intervals, 95% of them would actually contain the
value of the population mean .
Wrong: There is a 95% chance that the true value of 
will fall between 98.08 and 98.32.
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
13
Confidence Intervals from 20 Different Samples
Figure 6-1
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
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 z/2 is a critical
value that is a z score with the property that it
separates an area /2 in the right tail of the
standard normal distribution.
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
15
The Critical Value
2
2
2
-z
Figure 6-2
z
2
z=0
z
2
Found from Table A-2
(corresponds to area of
0.5 - 2 )
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
16
Finding z2 for 95% Degree of Confidence
95%
 = 5%
2 = 2.5% = .025
.95
.025
-z2
.025
z2
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
17
Finding z2 for 95% Degree of Confidence
95%
 = 5%
2 = 2.5% = .025
.95
.025
.025
z2
-z2
Critical Values
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
18
Finding z2 for 95% Degree of Confidence
 = 0.05
 = 0.025
.4750
.025
Use Table A-2
to find a z score of 1.96
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
19
Finding z2 for 95% Degree of Confidence
 = 0.05
 = 0.025
.4750
.025
Use Table A-2
to find a z score of 1.96
z2 =  1.96
.025
- 1.96
.025
1.96
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
20
Definition
Margin of Error
is the maximum likely difference observed
between sample mean x and true population
mean µ.
denoted by E
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
21
Definition
Margin of Error
is the maximum likely difference observed
between sample mean x and true population
mean µ.
denoted by E
x -E
µ
x +E
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
22
Definition
Margin of Error
is the maximum likely difference observed
between sample mean x and true population
mean µ.
denoted by E
x -E
µ
x +E
x -E < µ < x +E
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
23
Definition
Margin of Error
is the maximum likely difference observed
between sample mean x and true population
mean µ.
denoted by E
x -E
µ
x +E
x -E < µ < x +E
lower limit
upper limit
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
24
Definition
Margin of Error
E = z/2 •
x -E

Formula 6-1
n
µ
x +E
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
25
Definition
Margin of Error
E = z/2 •
x -E

Formula 6-1
n
µ
x +E
also called the maximum error of the estimate
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
26
Calculating E When  Is Unknown
 If n > 30, we can replace  in Formula 61 by the sample standard deviation s.
 If n  30, the population must have
a normal distribution and we must know
 to use Formula 6-1.
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
27
Confidence Interval (or Interval Estimate)
for Population Mean µ
(Based on Large Samples: n >30)
x -E <µ< x +E
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
28
Confidence Interval (or Interval Estimate)
for Population Mean µ
(Based on Large Samples: n >30)
x -E <µ< x +E
µ=x +E
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
29
Confidence Interval (or Interval Estimate)
for Population Mean µ
(Based on Large Samples: n >30)
x -E <µ< x +E
µ=x +E
(x + E, x - E)
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
30
Procedure for Constructing a
Confidence Interval for µ
( Based on a Large Sample:
n
> 30 )
1. Find the critical value z2 that corresponds to the
desired degree of confidence.
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
31
Procedure for Constructing a
Confidence Interval for µ
( Based on a Large Sample:
n
> 30 )
1. Find the critical value z2 that corresponds to the
desired degree of confidence.
2. Evaluate the margin of error E = z2 •  / n .
If the population standard deviation  is
unknown, use the value of the sample standard
deviation s provided that n > 30.
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
32
Procedure for Constructing a
Confidence Interval for µ
( Based on a Large Sample:
n
> 30 )
1. Find the critical value z2 that corresponds to the
desired degree of confidence.
2. Evaluate the margin of error E = z2 •  / n .
If the population standard deviation  is
unknown, use the value of the sample standard
deviation s provided that n > 30.
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
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
33
Procedure for Constructing a
Confidence Interval for µ
( Based on a Large Sample:
n
> 30 )
1. Find the critical value z2 that corresponds to the
desired degree of confidence.
2. Evaluate the margin of error E = z2 •  / n .
If the population standard deviation  is
unknown, use the value of the sample standard
deviation s provided that n > 30.
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 roundoff rules.
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
34
Round-Off Rule for Confidence
Intervals Used to Estimate µ
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.
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
35
Round-Off Rule for Confidence
Intervals Used to Estimate µ
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 mean.
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
36
Example:
A study found the body temperatures of 106
healthy adults. The sample mean was 98.2 degrees and the
sample standard deviation was 0.62 degrees. Find the
margin of error E and the 95% confidence interval.
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
37
Example:
A study found the body temperatures of 106
healthy adults. The sample mean was 98.2 degrees and the
sample standard deviation was 0.62 degrees. Find the
margin of error E and the 95% confidence interval.
n = 106
x = 98.2o
s = 0.62o
 = 0.05
/2 = 0.025
z / 2 = 1.96
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
38
Example:
A study found the body temperatures of 106
healthy adults. The sample mean was 98.2 degrees and the
sample standard deviation was 0.62 degrees. Find the
margin of error E and the 95% confidence interval.
n = 106
x = 98.20o
s = 0.62o
E = z / 2 • 
n
= 1.96 • 0.62
106
= 0.12
 = 0.05
/2 = 0.025
z / 2 = 1.96
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
39
Example:
A study found the body temperatures of 106
healthy adults. The sample mean was 98.2 degrees and the
sample standard deviation was 0.62 degrees. Find the
margin of error E and the 95% confidence interval.
n = 106
x = 98.20o
s = 0.62o
 = 0.05
/2 = 0.025
z / 2 = 1.96
E = z / 2 • 
n
= 1.96 • 0.62
106
= 0.12
x -E << x +E
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
40
Example:
A study found the body temperatures of 106
healthy adults. The sample mean was 98.2 degrees and the
sample standard deviation was 0.62 degrees. Find the
margin of error E and the 95% confidence interval.
n = 106
x = 98.20o
s = 0.62o
 = 0.05
/2 = 0.025
z / 2 = 1.96
E = z / 2 • 
n
= 1.96 • 0.62
106
= 0.12
x -E << x +E
98.20o - 0.12
<<
98.20o + 0.12
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
41
Example:
A study found the body temperatures of 106
healthy adults. The sample mean was 98.2 degrees and the
sample standard deviation was 0.62 degrees. Find the
margin of error E and the 95% confidence interval.
n = 106
x = 98.20o
s = 0.62o
 = 0.05
/2 = 0.025
z / 2 = 1.96
E = z / 2 • 
n
= 1.96 • 0.62
106
= 0.12
x -E << x +E
98.20o - 0.12
98.08o
<<
<<
98.20o + 0.12
98.32o
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
42
Example:
A study found the body temperatures of 106
healthy adults. The sample mean was 98.2 degrees and the
sample standard deviation was 0.62 degrees. Find the
margin of error E and the 95% confidence interval.
n = 106
x = 98.20o
s = 0.62o
 = 0.05
/2 = 0.025
z / 2 = 1.96
E = z / 2 • 
n
= 1.96 • 0.62
106
= 0.12
x -E << x +E
98.08o
<  <
98.32o
Based on the sample provided, the confidence interval for the

population mean is 98.08o <
< 98.32o. If we were to select many
different samples of the same size, 95% of the confidence intervals
would actually contain the population mean .
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
43
Finding the Point Estimate and E
from a Confidence Interval
Point estimate of µ:
x = (upper confidence interval limit) + (lower confidence interval limit)
2
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
44
Finding the Point Estimate and E
from a Confidence Interval
Point estimate of µ:
x = (upper confidence interval limit) + (lower confidence interval limit)
2
Margin of Error:
E = (upper confidence interval limit) - (lower confidence interval limit)
2
Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
45

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