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M ARIO F . T RIOLA

S TATISTICS. E LEMENTARY. Section 6-4 Determining Sample Size Required to Estimate . M ARIO F . T RIOLA. E IGHTH. E DITION. . z . E =.  / 2 •. n. Sample Size for Estimating Mean . . z . E =.  / 2 •. n. Sample Size for Estimating Mean .

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M ARIO F . T RIOLA

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  1. STATISTICS ELEMENTARY Section 6-4 Determining Sample Size Required to Estimate  MARIO F. TRIOLA EIGHTH EDITION

  2. z E = / 2 • n Sample Size for Estimating Mean 

  3. z E = / 2 • n Sample Size for Estimating Mean  (solve for n by algebra)

  4. z E = / 2 • n Sample Size for Estimating Mean  (solve for n by algebra) 2 z Formula 6-3  / 2 n = E z/2 = critical z score based on the desired degree of confidence E = desired margin of error  = population standard deviation

  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.

  6. 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. 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.  = 0.01 z = 2.575 E = 0.25 s = 1.065 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.)

  9.  = 0.01 z = 2.575 E = 0.25 s = 1.065 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 n = z = (2.575)(1.065) E 0.25

  10.  = 0.01 z = 2.575 E = 0.25 s = 1.065 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 n = z = (2.575)(1.065) E 0.25 = 120.3 = 121 households

  11.  = 0.01 z = 2.575 E = 0.25 s = 1.065 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 n = z = (2.575)(1.065) E 0.25 = 120.3 = 121 households If n is not a whole number, round it upto the next higher whole number.

  12.  = 0.01 z = 2.575 E = 0.25 s = 1.065 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 n = z = (2.575)(1.065) E 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.

  13. 1. Use the range rule of thumb to estimate the standard deviation as follows: range 4 What if is Not Known ?

  14. 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. 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. What happens when E is doubled ?

  17. 2 (z ) 2  z  / 2 / 2 n = = 1 1 What happens when E is doubled ? E = 1 :

  18. Sample size nis decreased to 1/4 of its original value if E is doubled. Larger errors allow smaller samples. Smaller errors require larger samples. 2 (z ) 2  z  / 2 / 2 n = = 1 1 2 (z ) 2  z  / 2 / 2 n = = 4 2 What happens when E is doubled ? E = 1 : E = 2 :

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