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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 . (solve for n by algebra). 2. z . .  / 2 . n. = . E .

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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  (solve for n by algebra) 2 z  / 2 n = E z/2 = critical z score based on the desired degree of confidence E = desired margin of error  = population standard deviation

  3. Round-Off Rule for Sample Size n When finding the sample size n, if the formula does not result in a whole number, always increase the value of n to the next larger whole number. That’s because the formula gives the minimum sample size necessary for a given degree of confidence and margin of error. If the result were rounded down, it would end up less than the mininum. n = 216.09 →217 (rounded up)

  4. = 0.01 /2 = z = 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 = (2.576)(1.065) 0.005 E 2.576 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.

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

  6. Sample size nis decreased to 1/4 of its original value if E is doubled. 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 : • Larger errors allow smaller samples. • Smaller errors requirelarger samples.

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