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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