1Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH
EDITIONEDITION
ELEMENTARY STATISTICS
Section 6-4 Determining Sample Size Required to Estimate
2Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
z/ 2 •E =
n
Sample Size for Estimating Mean
3Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
z/ 2 •E =
n
(solve for n by algebra)
Sample Size for Estimating Mean
4Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
z/ 2 •E =
n
(solve for n by algebra)
z/ 2 E
n =
2Formula 6-3
z/2 = critical z score based on the desired degree of confidence E = desired margin of error
= population standard deviation
Sample Size for Estimating Mean
5Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.
6Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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)
7Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.)
8Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
9Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
n = z = (2.575)(1.065) E 0.25
2 2
10Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
n = z = (2.575)(1.065) E 0.25
= 120.3 = 121 households
2 2
11Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
n = z = (2.575)(1.065) E 0.25
= 120.3 = 121 households
2 2
If n is not a whole number, round it up to the next higher whole number.
12Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
n = z = (2.575)(1.065) E 0.25
= 120.3 = 121 households
2 2
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.
13Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
1. Use the range rule of thumb to estimate the
standard deviation as follows: range 4
What if is Not Known ?
14Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.
What if is Not Known ?
15Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.
What if is Not Known ?
16Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
What happens when E is doubled ?
17Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
What happens when E is doubled ?
/ 2 z
1
n = =2
1/ 2
(z ) 2
E = 1 :
18Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
What happens when E is doubled ?
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.
/ 2 z
1
n = =2
1/ 2
(z ) 2
/ 2 z
2
n = =
2
4/ 2
(z ) 2
E = 1 :
E = 2 :