1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison...

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1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman MARIO F. TRIOLA EIGHTH EDITION ELEMENTARY STATISTICS Section 6-4 Determining Sample Size Required to Estimate

Transcript of 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison...

Page 1: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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

Page 2: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

2Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

z/ 2 •E =

n

Sample Size for Estimating Mean

Page 3: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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

Page 4: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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

Page 5: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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.

Page 6: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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)

Page 7: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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.)

Page 8: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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

Page 9: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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

Page 10: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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

Page 11: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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.

Page 12: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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.

Page 13: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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 ?

Page 14: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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 ?

Page 15: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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 ?

Page 16: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

16Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

What happens when E is doubled ?

Page 17: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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 :

Page 18: 1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.

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 :