Transcript Ch. 6.4 PowerPt

ELEMENTARY
STATISTICS
Section 6-4 Determining Sample Size Required to Estimate 
EIGHTH
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION
MARIO F. TRIOLA
1
Sample Size for Estimating Mean 

E = z/ 2 • n
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
Sample Size for Estimating Mean 

E = z/ 2 • n
(solve for n by algebra)
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
3
Sample Size for Estimating Mean 

E = z/ 2 • n
(solve for n by algebra)
n=
z/ 2 
2
Formula 6-3
E
z/2 = critical z score based on the desired degree of confidence
E = desired margin of error
 = population standard deviation
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
4
Round-Off Rule for Sample Size n
When finding the sample size n, if the use
of Formula 6-3 does not result in a whole
number, always increase the value of n to
the next larger whole number.
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
Round-Off Rule for Sample Size n
When finding the sample size n, if the use
of Formula 6-3 does not result in a whole
number, always increase the value of n to
the next larger whole number.
n = 216.09 = 217 (rounded up)
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
Example:
If we want to estimate the mean weight of
plastic discarded by households in one week, how many
households must be randomly selected to be 99%
confident that the sample mean is within 0.25 lb of the true
population mean? (A previous study indicates the
standard deviation is 1.065 lb.)
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
Example:
If we want to estimate the mean weight of
plastic discarded by households in one week, how many
households must be randomly selected to be 99%
confident that the sample mean is within 0.25 lb of the true
population mean? (A previous study indicates the
standard deviation is 1.065 lb.)
 = 0.01
z = 2.575
E = 0.25
s = 1.065
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Example:
If we want to estimate the mean weight of
plastic discarded by households in one week, how many
households must be randomly selected to be 99%
confident that the sample mean is within 0.25 lb of the true
population mean? (A previous study indicates the
standard deviation is 1.065 lb.)
2
2
 = 0.01
z = 2.575
E = 0.25
s = 1.065
n = z
E
= (2.575)(1.065)
0.25
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
9
Example:
If we want to estimate the mean weight of
plastic discarded by households in one week, how many
households must be randomly selected to be 99%
confident that the sample mean is within 0.25 lb of the true
population mean? (A previous study indicates the
standard deviation is 1.065 lb.)
2
2
 = 0.01
z = 2.575
E = 0.25
s = 1.065
n = z
E
= (2.575)(1.065)
0.25
= 120.3 = 121 households
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
Example:
If we want to estimate the mean weight of
plastic discarded by households in one week, how many
households must be randomly selected to be 99%
confident that the sample mean is within 0.25 lb of the true
population mean? (A previous study indicates the
standard deviation is 1.065 lb.)
2
2
 = 0.01
z = 2.575
E = 0.25
s = 1.065
n = z
E
= (2.575)(1.065)
0.25
= 120.3 = 121 households
If n is not a whole number, round it up
to the next higher whole number.
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Example:
If we want to estimate the mean weight of
plastic discarded by households in one week, how many
households must be randomly selected to be 99%
confident that the sample mean is within 0.25 lb of the true
population mean? (A previous study indicates the
standard deviation is 1.065 lb.)
2
2
 = 0.01
z = 2.575
E = 0.25
s = 1.065
n = z
E
= (2.575)(1.065)
0.25
= 120.3 = 121 households
We would need to randomly select 121 households and
obtain the average weight of plastic discarded in one
week. We would be 99% confident that this mean is within
1/4 lb of the population mean.
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
What if is Not Known ?
1. Use the range rule of thumb to estimate the
standard deviation as follows:   range
4
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
13
What if is Not Known ?
1. Use the range rule of thumb to estimate the
standard deviation as follows:   range
4
2. Conduct a pilot study by starting the sampling
process. Based on the first collection of at least
31 randomly selected sample values, calculate
the sample standard deviation s and use it in
place of . That value can be refined as more
sample data are obtained.
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
What if is Not Known ?
1. Use the range rule of thumb to estimate the
standard deviation as follows:   range
4
2. Conduct a pilot study by starting the sampling
process. Based on the first collection of at least
31 randomly selected sample values, calculate
the sample standard deviation s and use it in
place of . That value can be refined as more
sample data are obtained.
3. Estimate the value of  by using the results
of some other study that was done earlier.
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
15
What happens when E is doubled ?
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
16
What happens when E is doubled ?
2
2
(z/ 2 )
z/ 2 
E=1:
n=
1
=
1
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
17
What happens when E is doubled ?
2
2
(z/ 2 )
z/ 2 
E=1:
n=
E=2:
(z/ 2 )
z
/ 2 
n=
=
4
2
1
=
2
1
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.
Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
18