338 CHAPTER 7 Estimating Parameters and Determining Sample Sizes Finding the Sample Size Required to Estimate a Population Mean Objective Determine the sample size n required to estimate the value of a population mean m. Notation m = population mean s = population standard deviation x = sample mean E = desired margin of error za>2 = z score separating an area of a>2 in the right tail of the standard normal distribution Requirement The sample must be a simple random sample. Sample Size The required sample size is found by using Formula 7-4. Formula 7-4 n = c za>2s E d 2 Round-Off Rule If the computed sample size n is not a whole number, round the value of n up to the next larger whole number. KEY ELEMENTS Population Size Formula 7-4 does not depend on the size (N) of the population (except for cases in which a relatively large sample is selected without replacement from a finite population). Rounding The sample size must be a whole number because it is the number of sample values that must be found, but Formula 7-4 usually gives a result that is not a whole number. The round-off rule is based on the principle that when rounding is necessary, the required sample size should be rounded upward so that it is at least adequately large instead of being slightly too small. Dealing with Unknown S When Finding Sample Size Formula 7-4 requires that we substitute a known value for the population standard deviation s, but in reality, it is usually unknown. When determining a required sample size (not constructing a confidence interval), here are some ways that we can work around the problem of not knowing the value of s: 1. Range Rule of Thumb Use the range rule of thumb (see Section 3-2) to estimate the standard deviation as follows: s ≈ range>4, where the range is determined from sample data. (With a sample of 87 or more values randomly selected from a normally distributed population, range>4 will yield a value that is greater than or equal to s at least 95% of the time.) 2. Start and Improve Start the sample collection process without knowing s and, using the first several values, calculate the sample standard deviation s and use it in place of s. The estimated value of s can then be improved as more sample data are obtained, and the required sample size can be adjusted as you collect more sample data.
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