1 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman zz / 2 E = n Sample Size for Estimating Mean
3 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman zz / 2 E = n (solve for n by algebra) Sample Size for Estimating Mean
4 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman zz / 2 E = n (solve for n by algebra) zz / 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 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 = = 217 (rounded up)
7 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 lb.)
8 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 lb.) = 0.01 z = E = 0.25 s = 1.065
9 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 lb.) = 0.01 z = E = 0.25 s = n = z = (2.575)(1.065) E
10 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 lb.) = 0.01 z = E = 0.25 s = n = z = (2.575)(1.065) E 0.25 = = 121 households 22
11 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 lb.) = 0.01 z = E = 0.25 s = n = z = (2.575)(1.065) E 0.25 = = 121 households 22 If n is not a whole number, round it up to the next higher whole number.
12 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 lb.) = 0.01 z = E = 0.25 s = n = z = (2.575)(1.065) E 0.25 = = 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 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman What happens when E is doubled ?
17 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman What happens when E is doubled ? / 2 zz 1 n = = 2 1 / 2 ( z ) 2 E = 1 :
18 Chapter 6. Section 6-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 zz 1 n = = 2 1 / 2 ( z ) 2 / 2 zz 2 n = = 2 4 / 2 ( z ) 2 E = 1 : E = 2 :