320 CHAPTER 7 Estimating Parameters and Determining Sample Sizes Determining Sample Size When planning to collect sample data in order to estimate some population proportion p, we must first determine the sample size, which is the number of sample units that must be collected. If we solve the formula for the margin of error E (Formula 7-1) for the sample size n, we get Formula 7-2 below. Formula 7-2 requires pn as an estimate of the population proportion p, but if no such estimate is known (as is often the case), we replace pn by 0.5 and replace qn by 0.5, with the result given in Formula 7-3. Replacing pn and qn with 0.5 results in the largest possible sample size, so we are sure that the sample size is adequate for estimating p. EXAMPLE 4 What Percentage of Adults Make Online Purchases? A 2016 Pew Research Center survey of 4787 randomly selected U.S. adults showed that 79% of the respondents shop online. If we want to conduct a new survey to determine whether that percentage has changed, how many adults must be surveyed in order to be 95% confident that the sample percentage is in error by no more than three percentage points? a. Assume that 79% of adults make online purchases (as in the earlier Pew survey). b. Assume that we have no prior information suggesting a possible value of the population proportion. Finding the Sample Size Required to Estimate a Population Proportion Objective Determine how large the sample size n should be in order to estimate the population proportion p. Notation p = population proportion pn = sample proportion n = number of sample values E = desired margin of error za>2 = z score separating an area of a>2 in the right tail of the standard normal distribution Requirements The sample must be a simple random sample of independent sample units. When an estimate pn is known: Formula 7-2 n = 3za>24 2 pnqn E2 When no estimate pn is known: Formula 7-3 n = 3za>24 20.25 E2 If a reasonable estimate of pn can be made by using previous samples, a pilot study, or someone’s expert knowledge, use Formula 7-2. If nothing is known about the value of pn, use Formula 7-3. Round-Off Rule for Determining Sample Size If the computed sample size n is not a whole number, round the value of n up to the next larger whole number, so the sample size is sufficient instead of being slightly insufficient. For example, round 708.135 to 709. KEY ELEMENTS
RkJQdWJsaXNoZXIy NjM5ODQ=