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 S TATISTICS Section 6-4Determining Sample Size Required to Estimate 

2. 2 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman zz  / 2 E =  n Sample Size for Estimating Mean 

3. 3 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman zz  / 2 E =  n (solve for n by algebra) Sample Size for Estimating Mean 

4. 4 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman zz  / 2 E =  n (solve for n by algebra) zz  / 2 E  n = 2 Formula 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 

5. 5 Chapter 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.

6. 6 Chapter 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)

7. 7 Chapter 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.)

8. 8 Chapter 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

9. 9 Chapter 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 22

10. 10 Chapter 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 22

11. 11 Chapter 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 22 If n is not a whole number, round it up to the next higher whole number.

12. 12 Chapter 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 22 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.

13. 13 Chapter 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 ?

14. 14 Chapter 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 ?

15. 15 Chapter 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 ?

16. 16 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman What happens when E is doubled ?

17. 17 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman What happens when E is doubled ?  / 2 zz 1  n = = 2 1  / 2 ( z )   2 E = 1 :

18. 18 Chapter 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z 1  n = = 2 1  / 2 ( z )   2  / 2 zz 2  n = = 2 4  / 2 ( z )   2 E = 1 : E = 2 :